## Migration

My new website is under construction now. Be patient.

Expand

Expand

## Spam

### My Devil's Dictionary

#### Chinese

• 戾拔山兮气盖世: trolls.
• 力排众议: 参议院否决众议院通过的弹劾案.

#### Polish

• Cześć: CZ, escape!

### Interesting

#### Science and Math

• Almost Integer
• Most notably $$(28-\delta^{-1})(\alpha^{-1}-137)=1$$.

Expand

## Towards Modernization (I): Front-End Techniques

#### Initialize a Project

npm init -y
git init && git branch -m main


Create a file LICENSE and copy from a template, e.g. MIT or ISC.

#### Other Information

The following entries may be filled in package.json

"description": "A description",
"main": "./build/src/index.js",
"files": [
"build/**/*"
],
"repository": {
"type": "git",
"url": "git+https://github.com/user/repo.git"
},
"author": {
},
"engines": {
"node": ">=12.0"
},
"keywords": [
"something"
],
"bugs": {
"url": "https://github.com/user/repo/issues"
},


### Version Control

#### Husky

husky ensures that tests, lint, etc. are done before git commit. See installation procedure.

npx husky-init && npm install


#### Commitizen

commitizen generates the commit message upon each commit. Install via

npm install --save-dev commitizen


Then run

npx commitizen init cz-conventional-changelog --save-dev --save-exact


Install husky following the instruction here. Then integrate commitizen with husky by adding the following entry in package.json

"husky": {
"hooks": {
"prepare-commit-msg": "exec < /dev/tty && git cz --hook || true"
}
}


Run npx cz to commit.

Run npx cz --retry to retry a failed commit.

#### Semantic Release

Install via

npm install --save-dev semantic-release


In package.json we need the following entry

"repository": {
"type": "git",
}


In .releaserc.json put

{
"branches": ["main"],
"plugins": [
["@semantic-release/npm", {
"npmPublish": false
}]
]
}


Note that we need to git push for once to make sure that the branch exists in the remote repo.

Then we could run semantic release

npx semantic-release --no-ci # --dry-run


semantic-release works also with Nx. See nx-semantic-release-demo.

### Language

#### Typescript

Install with

npm install typescript --save-dev


Generate tsconfig.json

npx tsc --init


Add the following entry to package.json

"scripts": {
"typecheck": "tsc --noEmit"
}


I decided to postpone the configuration until we set up Nx.

#### TS-Node

Install with

npm install -D typescript


Then we could execute .ts files directly

npx ts-node foo.ts


#### @types/node

Install with

npm install -D @types/node


### Code Style

#### Prettier

Format the code, e.g. trailing semicolons.

See here for set-up.

Install and configure with

npm install --save-dev --save-exact prettier
echo {}> .prettierrc.json


Add the following to .prettierignore

# Ignore artifacts:
build
coverage


Since we will be using lint-staged later, we don't have to worry about .prettierignore yet.

Run npx prettier --write . to format files.

Run npx prettier --check . to check files.

Remember to enable prettier in the editor (see Editor Integration).

#### ESLint

It's the recommended practice to let Prettier handle formatting and ESLint for non-formatting issues (See What's the difference between prettier-eslint, eslint-plugin-prettier and eslint-config-prettier?).

Install using

npm install eslint --save-dev


Then interactively configure eslint using

npm init @eslint/config


Remember to enable ESLint in the editor (see Editor Integration).

We may also need the following for ESLint to work correctly

extends: [
"plugin:import/typescript", 'prettier'
]

rules: {
'import/extensions': ['error', 'never']
},


#### ESLint With Prettier

To handle conflicting rules of ESLint with Prettier, we install eslint-config-prettier using (see GitHub: eslint-config-prettier)

npm install --save-dev eslint-config-prettier


Then we edit the following in *.eslintrc.*

{
"extends": [
"some-other-config-you-use",
"prettier"
]
}


Make sure to put "prettier" last, so it gets the chance to override other configs.

It is debatable whether one should run Prettier as if it was a linter rule. Packages like eslint-plugin-prettier do that job. However, based on the discussion in Integrating with Linters, we don't use this plugin.

#### ESLint Plugins

##### ESLint-Node

Install with

npm install --save-dev eslint-plugin-node


Then in .eslintrc.js add (see eslint-plugin-node)

extends: [
"plugin:node/recommended",
]


However, I had not enabled this plugin yet because it has some problems when working with TypeScript (ECMAScript to be specific).

#### Lint-Staged

Make sure that Prettier and ESLint are both installed before installing lint-staged.

For Prettier and ESLint to automatically run upon git commit, we install lint-staged via (see lint-staged)

npx mrm@2 lint-staged


It may match only *.js files. Edit the lint-staged entry in package.json so that it covers *.ts as well.

EsLint automatically finds .eslintrc in the directory of the file to be linted, see Configuration Files.

### Documentation

#### Typedoc

Install with

npm install -D typedoc


Add to the root dir a file typedoc.json with

{
"out": "doc",
"theme": "default"
}


Generate doc with (e.g. in an Nx workspace)

typedoc --tsconfig libs/data/tsconfig.lib.json --out docs/data libs/data/src/index.ts


#### Immer

Remark: another option is using immutable. However, we don't bother using it since Redux Toolkit uses immer.

#### Ramda

A functional programming library. See ramda.

#### RxJS

See RxJS. Library for observables.

### Testing

#### Jest

Install with

npm install --save-dev jest


Add the following entry to package.json

{
"scripts": {
"test": "jest --coverage",
"test:watch": "jest --watch"
}
}


We may start a test using

npm t


For a tutorial, see Getting Started.

If husky is installed, tests will run automatically upon git commit.

To generate coverage report, run

npx jest --coverage


#### TS-Jest

Install with

npm i -D ts-jest @types/jest


Create config file with

npx ts-jest config:init


Edit jest.config.js

module.exports = {
preset: 'ts-jest',
testEnvironment: 'node',
testMatch: ['**/test/**/*.spec.ts'],
collectCoverageFrom: [
'<rootDir>/src/**/*.ts',
'!<rootDir>/src/types/**/*.ts',
],
globals: {
'ts-jest': {
diagnostics: false,
},
},
};


#### ESLint With Jest

See eslint-plugin-jest for installation and configuration. In .eslintrc.js we need

env: {
'jest/globals': true,
}

plugins: ['jest'],


#### Cypress

Cypress is for e2e test.

### Front-End

#### Redux

Redux Toolkits uses immer, making states immutable.

### Styling

#### Tailwindcss

Install the dev-dependencies first

npm install -D tailwindcss postcss autoprefixer vite


Then init at the component lib directory e.g. libs/ui with

npx tailwindcss init -p


### Transpiling and Bundling

For a brief introduction, see Setup react with webpack and babel.

### Back-End and Integration

#### Next.js

To let Next.js work with Nest one may try Next + Nest: Working together or NestJS + React (Next.js) in One MVC Repo for Rapid Prototyping. However, I don't deploy these techniques since I am gonna use Nx.

Next.js comes with webpack and babel, customization of which could be found here.

#### Nx

Create an Nx workspace

npx create-nx-workspace


Install Nx CLI

npm install -g nx


nx run-many --target=serve --projects=api,html --parallel

##### Storybook With Nx
npm install -D @nrwl/storybook
nx g @nrwl/react:storybook-configuration project-name
nx run project-name:storybook


#### Nest

Install with

npm i -g @nestjs/cli


Then we could start a new project by

nest new project-name


This will pop up an interactive setup program.

##### A Collection of Terminologies
• SOLID principles:
• The single-responsibility principle: "There should never be more than one reason for a class to change." In other words, every class should have only one responsibility.
• The open–closed principle: "Software entities ... should be open for extension, but closed for modification."
• The Liskov substitution principle: "Functions that use pointers or references to base classes must be able to use objects of derived classes without knowing it." See also design by contract.
• The interface segregation principle: "Many client-specific interfaces are better than one general-purpose interface."
• The dependency inversion principle: "Depend upon abstractions, [not] concretions."
• DTO: Data transfer object.
• CRUD: Create, Read, Update, Delete.
• IoC: inversion of control.
##### Controller

Create a controller using

nest g controller controller-name

##### Service

Create a service using

nest g service service-name

##### Module

Create a module using

nest g module module-name

##### CRUD Resource

Create a resource using

nest g resource users


This will pop up an interactive creator. Controllers, modules, services, and DTOs are automatically generated.

### Authentication

#### Passport

##### Using With NestJS

Install with

npm i @nestjs/passport passport passport-local
npm install -D @types/passport-local


### Deploy

#### Nginx

For a basic config see Next.js + API on same server.

#### Docker

To have docker alpine image node:alpine run with Next.js we may need the following

RUN apk add --no-cache libc6-compat


Moreover we need (for compiling)

RUN apk add --no-cache --virtual .gyp \
python3 \
make \
g++
RUN npm i
RUN npx nx run-many --target=build --projects=api,front --prod
RUN apk del .gyp


#### AWS Elastic Beanstalk

To deploy apps to AWS Elastic Bean, See @How to Deploy a NestJS app to AWS Elastic Beanstalk (with added HTTPS and CI/CD).

Expand

Expand

## SFINAE vs Concept

A brief introductory example may be found in Concepts versus SFINAE-based constraints.

#### A Toxic Example

##### Universal Reference is Too Greedy

Universal reference is the terminology in Modern Effect C++, which basically refer to T&& in template functions.

The following code should be short and clear enough.

#include <iostream>

class Person
{ };

class SpecialPerson: public Person
{ };

template<typename T>
void handle(T&& person)
{
std::cout << "Handle person." << std::endl;
}

void handle(int number)
{
std::cout << "Handle number." << std::endl;
}

int main()
{
Person p;
SpecialPerson sp;
int i = 0;
short s = 0;
handle(p);
handle(sp);
handle(i);
handle(s);
}


The behavior is, however, puzzling. It prints

Handle person.
Handle person.
Handle number.
Handle person.


Why did the last line print Handle person.? This is because handle(int) requires a conversion in this case while in handle(T&&) the compiler could infer T = short and provide a perfect match.

We may resort to the C++98 approach and declare handle like the following.

void handle(const Person &);
void handle(int);


However, what if we want to retain perfect forwarding? In such case universal reference is inevitable. The problem may be rephrased as follow: const Person& and Person&& are too restrictive as they matches only lvalue or rvalue reference. T&& is too generic as it matches reference to any type. We need something somewhere between.

##### Detour: How Do C# Handle Such Problem

A simlar problem exists in C#.

using System;

class Person
{ }

class SpecialPerson: Person
{ }

public class Program
{
private static void handle<T>(T person)
{
Console.WriteLine("Handle person.");
}

private static void handle(int number)
{
Console.WriteLine("Handle number.");
}

public static void Main()
{
Person p = new Person();
SpecialPerson sp = new SpecialPerson();
int i = 0;
short s = 0;
handle(p);
handle(sp);
handle(i);
handle(s);
}
}


The above code prints the following.

Handle person.
Handle person.
Handle number.
Handle person.


What surprises us is the last line, where

short s = 0;
handle(s);


prints

Handle person.


Handle number.


Imposing constraints on handle solves the problem.

This code is tested with C#6.0. Compilation may fail with older versions.

using System;

class Person
{ }

class SpecialPerson: Person
{ }

public class Program
{
private static void handle<T>(T person) where T: Person
{
Console.WriteLine("Handle person.");
}

private static void handle(int number)
{
Console.WriteLine("Handle number.");
}

public static void Main()
{
Person p = new Person();
SpecialPerson sp = new SpecialPerson();
int i = 0;
short s = 0;
handle(p);
handle(sp);
handle(i);
handle(s);
}
}


Now the program correctly prints

Handle person.
Handle person.
Handle number.
Handle number.


Question: What is the counterpart of where in C++?

#### Solution: SFINAE vs Concept

##### The SFINAE Solution

SFINAE stands for substitution failure is not an error. In this case, we make use of std::enable_if and implement handle in the following way.

#include <iostream>

class Person
{ };

class SpecialPerson: public Person
{ };

template<typename T,
typename = std::enable_if_t<
std::is_base_of_v<Person, std::decay_t<T>>
>
>
void handle(T&& person)
{
std::cout << "Handle person." << std::endl;
}

void handle(int number)
{
std::cout << "Handle number." << std::endl;
}

int main()
{
Person p;
SpecialPerson sp;
int i = 0;
short s = 0;
handle(p);
handle(sp);
handle(i);
handle(s);
}


It works correctly now and prints

Handle person.
Handle person.
Handle number.
Handle number.

##### The Concept Solution

Concept is introduced in C++20 and enables the following more straightforward implementation closer to where in C#.

#include <iostream>
#include <concepts>

class Person
{ };

class SpecialPerson: public Person
{ };

template<typename T> requires std::derived_from<std::decay_t<T>, Person>
void handle(T&& person)
{
std::cout << "Handle person." << std::endl;
}

void handle(int number)
{
std::cout << "Handle number." << std::endl;
}

int main()
{
Person p;
SpecialPerson sp;
int i = 0;
short s = 0;
handle(p);
handle(sp);
handle(i);
handle(s);
}


It works also correctly now and prints

Handle person.
Handle person.
Handle number.
Handle number.


Expand

## Green's Functions (I)

Definitions and Diagrams

This post is not aiming at giving a systematic introduction to Green's functions in condensed matter physics. Rather, it is written in a spaghetti pattern that intertwines many related or unrelated materials. For a systematic approach, see my note on quantum field theory.

#### Motivation: Why Do We Need Green's Function

##### The Zero-Temperature Case

In condensed matter physics, it's a common task for us to evaluate the expectation value of a certain operator (for the ground state) \begin{align*} \langle \mathscr{J}(\vb{x}) \rangle &= \frac{\bra{\Omega} \mathscr{J}(\vb{x}) \ket{\Omega}}{\bra{\Omega}\ket{\Omega}} \\ &= \pm i \lim_{t\rightarrow t^+} \lim_{\vb{x}' \rightarrow \vb{x}} \sum_{\alpha\beta} J_{\beta\alpha}(\vb{x}) G_{\alpha\beta}(\vb{x}t, \vb{x}'t') \\ &= \pm i \lim_{t\rightarrow t^+} \lim_{\vb{x}' \rightarrow \vb{x}} \tr[J(\vb{x}) G(\vb{x}t,\vb{x}'t')]. \end{align*} for $\mathscr{J}(\vb{x}) = \sum_{\alpha\beta} \psi_\beta^\dagger(\vb{x})J_{\beta\alpha}(x) \psi_\alpha(\vb{x})$ where we have defined the Green's function $iG_{\alpha\beta}(\vb{x}t;\vb{x'}t') = \frac{\bra{\Omega} T[\psi_\alpha(\vb{x}t) \psi^\dagger_\beta(\vb{x}'t')]\ket{\Omega}}{\bra{\Omega}\ket{\Omega}}.$

A few remarks are in order here:

• The field operators are complete in principle: all the observables could be constructed from field operators.
• If the Hamiltonian is time-independent, the Green's function depends only on $$(t-t')$$. After taking the limit $$t\rightarrow t'$$ it doesn't matter at which $$t$$ we evaluate the Green's function.
• The limit $$\vb{x}'\rightarrow \vb{x}$$ should act before $$J_{\alpha\beta}(\vb{x})$$ since $$J$$ may contain gradient.
• The number density is given by $n_{\alpha\beta}(\vb{x}) = \pm i \tr G(\vb{x}t; \vb{x}t^+).$
• The spin density is given by $\langle \vb*{\sigma}(\vb{x}) \rangle = \pm i \tr[\vb*{\sigma} G(\vb{xt;\vb{x}t^+})].$

Surprisingly, we don't need the four-point function in order to evaluated the interaction term. After some calculation we find for $V(\vb{x},\vb{x}')_{\alpha\alpha',\beta\beta'} = V(\vb{x}',\vb{x})_{\beta\beta',\alpha\alpha'}.$ that $\langle H \rangle = \pm \frac{1}{2}i \int \dd{^3 x} \lim_{t'\rightarrow t^+}\lim_{\vb{x}' \rightarrow \vb{x}} \qty[i\hbar \pdv{}{t} - \frac{\hbar^2 \grad^2}{2m}] \tr G(\vb{x}t; \vb{x'}t').$

Useful, right? Now the question becomes how do we evaluate Green's functions. Before we open the Pandora's box of Feynman diagrams, let's see how do we handle the case of finite-temperature.

##### The Finite-Temperature Case

At non-zero temperature, we have the similar result \begin{align*} \langle J \rangle &= \Tr(\rho_G J) \\ &= \mp \sum_{\alpha\beta} \int \dd{^3 x} \lim_{\vb{x}'\rightarrow\vb{x}} \lim_{\tau'\rightarrow\tau^+} J_{\beta\alpha}(x) \mathscr{G}_{\alpha\beta}(\vb{x}\tau; \vb{x}'\tau') \\ &= \mp \int \dd{^3 x} \lim_{\vb{x}' \rightarrow \vb{x}} \lim_{\tau'\rightarrow \tau^+} \tr[J(x)\mathscr{G}(\vb{x}\tau; \vb{x}'\tau')], \end{align*} where we have defined the Matsubara Green's function $\mathscr{G}_{\alpha\beta}(\vb{x}\tau;\vb{x}'\tau') = -\Tr[\rho_G T_\tau[\psi_{\alpha}(\vb{x}\tau) \psi^\dagger_{\beta}(\vb{x}'\tau')]],$ the modified Heisenberg picture (i.e. $$\tau = it$$) \begin{align*} \psi_{\alpha}(\vb{x}\tau) &= e^{K\tau/\hbar} \psi_\alpha(\vb{x}) e^{-K\tau/\hbar}, \\ \psi_{\alpha}^\dagger(\vb{x}\tau) &= e^{K\tau/\hbar} \psi_\alpha^\dagger(\vb{x}) e^{-K\tau/\hbar}, \end{align*} and the statistical properties \begin{align*} K &= H - \mu N, \\ Z_G &= e^{-\beta\Omega} = \Tr e^{-\beta K}, \\ \rho_G &= Z^{-1}_G e^{-\beta K} = e^{\beta(\Omega - K)}. \end{align*}

#### Example: Degenerate Electron Gas

With average density $$\langle n \rangle = N/V$$ fixed, the expectation value of interation energy is given by $\langle V \rangle = \frac{1}{2} \int \dd{^3 x} \int \dd{^3 x'} V(\vb{x} - \vb{x}')[\langle \tilde{n}(\vb{x}) \tilde{n}(\vb{x}') \rangle + \langle n(\vb{x})\rangle \langle n(\vb{x}) \rangle - \delta(\vb{x} - \vb{x}')\langle n(\vb{x}) \rangle].$ where we have defined $$\tilde{n}(\vb{x}) = n(\vb{x}) - \langle n(\vb{x})\rangle$$. The last two terms are essentially constants and may therefore be discarded. Now our goal becomes evaluating the polarization propagator (or the four-point function) $iD(x, x') = \frac{\bra{\Omega} T[\tilde{n}_H(x') \tilde{n}_H(x)] \ket{\Omega}}{\bra{\Omega} \ket{\Omega}}.$ Its non-interacting counterpart is defined by $iD^0(x',x) = \bra{0} T[\tilde{n}_I(x')\tilde{n}_I(x)] \ket{0}.$ Now the expectation value $$\langle V \rangle$$ could be divided into the non-interacting contribution and the interating contribution \begin{align*} \langle V \rangle &= \frac{1}{2} \int \dd{^3 x} \dd{^3 x'} V(\vb{x} - \vb{x}')[iD^0(\vb{x}'t, \vb{x}t) + n^2 - \delta(\vb{x} - \vb{x}')n] + \frac{1}{2}\int \dd{^3 x} \dd{^3 x'} V(\vb{x} - \vb{x}')[iD(\vb{x}'t, \vb{x}t) - iD^0(\vb{x}'t, \vb{x}t)] \\ &= \bra{\Phi_0} V \ket{\Phi_0} + \frac{1}{2}\int \dd{^3 x} \dd{^3 x'} V(\vb{x} - \vb{x}')[iD(\vb{x}'t, \vb{x}t) - iD^0(\vb{x}'t, \vb{x}t)]. \end{align*}

Expand

## Category Theory

#### Definition and Examples

• A category $$C$$ consists of the following.
• A class $$\operatorname{ob} C$$ of objects.
• For each ordered pair of objects $$(A,B)$$, a set of $$\mathrm{hom}_{C}(A,B)$$ whose elements are called morphisms with domain $$A$$ and codomain $$B$$.
• For each ordered triple of objects $$(A,B,C)$$, a map $$(f,g)\mapsto gf$$ of the product set $$\mathrm{hom}(A,B) \times \mathrm{hom}(B,C)$$ into $$\mathrm{hom}(A,C)$$.
• The objects and morphisms satisfy the following conditions:
• If $$(A,B) \neq (C,D)$$, then $$\mathrm{hom}(A,B)$$ and $$\mathrm{hom}(C,D)$$ are disjoint.
• If $$f\in\mathrm{hom}(A,B)$$, $$g\in\mathrm{hom}(B,C)$$ and $$h\in\mathrm{hom}(B,C)$$, then $$(hg)f = h(gf)$$.
• For every object $$A$$ we have an element $$\mathbb{1}_A\in \mathrm{hom}(A,A)$$ such that $$f\mathbb{1}_A = f$$ for every $$f\in \mathrm{hom}(A,B)$$ and $$\mathbb{1}_A g = g$$ for every $$g\in \mathrm{hom}(B,A)$$.
• A category $$D$$ is called a subcategory of $$C$$ if
• $$\operatorname{ob} C$$ is a subclass of $$\operatorname{ob}D$$ and
• for any $$A,B\in \operatorname{ob} D$$, $$\mathrm{hom}_D(A,B) \subset \mathrm{hom}_C(A,B)$$.
• $$\mathbb{1}_A$$ is defined in the same way as in $$D$$.
• A subcategory $$D$$ is called full if for every $$A,B\in D$$ $\mathrm{hom}_D(A,B) = \mathrm{hom}_C(A,B).$

Monoids are identified with categories whose object classes are single-element sets.

• A category $$C$$ is called small if $$\operatorname{ob} C$$ is a set.
• An element $$f\in \mathrm{hom}(A,B)$$ is called an isomorphism if there exists a $$g\in \mathrm{hom}(B,A)$$ such that $$fg = \mathbb{1}_B$$ and $$gf = \mathbb{1}_A$$.
• $$g$$ is uniquely defined by $$f$$ and is denoted $$f^{-1}$$.
• $$(f^{-1})^{-1} = f$$.
• If $$f$$ and $$h$$ are isomorphisms, and $$fh$$ is defined, then $$fh$$ is also an isomorphism.

Groups are identifies with categories whose object classes are single-element sets and all morphisms are isomorphisms. A groupoid is a small category in which morphisms are isomorphisms.

• The dual category $$C^{\mathrm{op}}$$ of $$C$$ is defined by
• $$\operatorname{ob} C^{\mathrm{op}} = \operatorname{ob} C$$.
• $$\mathrm{hom}_{C^{\mathrm{op}}}(A,B) = \mathrm{hom}_{C}(B,A)$$.
• Composition $$gf$$ in $$C^{\mathrm{op}}$$ is defined by $$fg$$ in $$C$$.
• $$\mathbb{1}_A$$ as in $$C$$.
• The product category $$C\times D$$ is defined by
• $$\operatorname{ob} C\times D = \operatorname{ob}C \times \operatorname{ob} D$$.
• $$\mathrm{hom}_{C\times D}((A,A'),(B,B')) = \mathrm{hom}_{C}(A,B)\times \mathrm{hom}_{C}(A',B')$$.
• Composition defined component-wise.
• $$\mathbb{1}_{(A,A')}$$ defined component-wise.

Expand

## LnOPnCh₂ Series

#### Structure

• LnO layer + PnCh₂ layer.
• Usually P4 symmetry.

#### Previous Work

##### LaOBiS₂
• Thermoelectric $$\ce{LaO_{1-x}F_{x}BiS2}{}$$ (Omachi2014).
• Thermoelectric $$\ce{LaOBiS_{2-x}Se_x}{}$$ (Mizuguchi2014)(Nishida2015).
• $$\kappa \sim 1 \mathrm{W}\cdot \mathrm{m}^{-1} \cdot \mathrm{K}^{-1}$$.
• $$zT\sim 0.36$$ at 370°C.
##### LnOSbSe₂
• Thermoelectric $$\ce{LaO_{1-x}F_xSbSe2}{}$$ and $$\ce{CeO_{1-x}F_xSbSe2}{}$$ (Goto2018).
• $$\ce{NdOSbS2}$$ is likely an insulator.

#### Calculation

• $$\ce{LnOAsSe2}{}$$, with predicted $$zT\sim 2$$ (Ochi2017), synthesis failed due to the stable phase $$\ce{Ln4O4Se3}{}$$ (Goto2018).
• $$\ce{NdOSbS2}$$, already synthesized (Tanryverdiev1996), large PF (Ochi2017).
• Designing principles: (Ochi2017)
• Small SOC;
• Small Bi-Bi and large Bi-S hopping amplitudes;
• Small on-site energy difference between the Bi-$$p_{x,y}{}$$ orbitals and S-$$p_{x,y}{}$$ orbitals.
• $$\Rightarrow$$ replacing Bi with lighter elements and S with heavier elements (putting them closer in the periodic table);
• Band structures of $$\ce{LnO_{1-x}F_x BiS2}{}$$ calculated (Morice2016).
• The change of rare earth does not affect the Fermi surface.
• Conduction electrons are confined to the BiS₂ planes.

#### Proposal

##### (Ce,Pr)OBiS₂
• Also replacing S with Se.
• Thermoelectric properties should be similar to that of LaOBiS₂.
##### NdOSbS₂
• Try to make it a semiconductor.
##### LnOAsSe₂
• Try to synthesize.

Expand

## Quantum Transport

### Linear Response

#### Formulations

##### Kubo Formula
• System undergoes a perturbation $\hat{H}' = \hat{V}(t) \Theta(t-t_0).$
• Up to the first order, $\langle \hat{A}(t) \rangle = \langle \hat{A} \rangle_0 - i \int_{t_0}^t \dd{t'} \langle [ \hat{A}(t), \hat{V}(t') ] \rangle_0.$
##### Green-Kubo Relations
• Transport coefficient is given by $\gamma = \int_0^\infty \langle \dot{A}(0) \dot{A}(t) \rangle \dd{t}.$

### Mesoscopic Transport

#### Landauer Formula

• Ideal conductors (two terminals) $I = 2e \int_{\mu_2}^{\mu_1} v_k \dv{k}{\epsilon_k} \frac{\dd{\epsilon_k}}{2\pi} = \frac{2e}{h}(\mu_1 - \mu_2)$ and therefore $G = \frac{I}{V} = \frac{2e^2}{h}.$
• Inserting a mesoscopic conducting device with transmission coefficient $$T$$, we find the Büttiker formula $G = \frac{2e^2}{h}T.$ The conductivity of the device itself is given by the Landauer formula $G = \frac{2e^2}{h}\frac{T}{1-T}.$
• Büttiker formula $I_p = \sum_q G_{pq}(V_q - V_p).$

Expand

## Useful Packages for Measurements

#### PyVISA

PyVISA controls your instruments using raw commands.

#### Keithley2600

Keithley2600 controls your Keithley 2600 instruments with abstraction.

#### PyMeasure

PyMeasure controls a wide range of instruments.

Expand

## DMRG (I): Theory

トニカクタダシイ

### Simple DMRG

#### Basic Theory

Quantum renormalization in real space applies only to single-particle systems.

On each step, the Hamiltonian undergoes a truncation $H^S(M) = V^\dagger H(M\cdot M_S) V,$ where $$V$$ contains the most significant $$M$$ eigenstates of $\rho^S (M\cdot M_S),$ given by the ground state of $H(M\cdot M_S).$

In the algorithm, we should keep track of the creation operators on the boundary sites.

##### Structure of Infinite-Size DMRG
1. Set up an initial chain of length $$L$$.
2. Calculate the ground state of this chain.
3. Get the reduced density matrix $$\rho^S$$.
4. Diagonalize $$\rho^S{}$$ and keep the most significant $$M$$ eigenstates as $$V$$.
5. Reduce $$H$$ to a matrix of size $$M\times M$$, as well as the creation operators at the boundary.
7. Find the ground state of the new system.
##### Structure of Finite-Size DMRG
1. Carry out infinite-size DMRG until the size reaches $$L$$. During this step, store all the Hamiltonians and other necessary steps in the memory.
2. Reduce the matrices (Hamiltonian and creations operators) of $$S$$ to size $$M$$. Before doing the projection, we should first trace out the environment part.
3. Add two sites to $$H_S$$, and reduce the environment to length $$L/2 - 2$$.
4. The procedure of the last two steps is repeated until $$S$$ has size $$L-3$$. The matrices for $$S$$ should still be stored for each size.
5. Reduce the size of $$S$$. For the environment, the stored matrices are used, whereas for the system, the matrices are updated at eac hstep. Carry on until the system $$S$$ has reached size $$1$$.
6. Repeat the up and down sweeps of the previous steps until convergence has been achieved.

### Matrix Product State

Prerequisites: for SVD and QR Decomposition, see Mathematics Miscellany.

###### Conventions
• $$M$$ for general matrices;
• $$\Lambda$$ and $$S$$ for diagonal matrices;
• $$A$$ for left-canonical ones;
• $$B$$ for right-canonical ones.
• Label $$l$$ in $$\ket{a_l}$$ indicates the bond.
• Superscript $$[l]$$ in $$O^{[l]}$$ indicates that the operator lives on site $$l$$.
• OBS unless otherwise stated. Site index from $$1$$ to $$L$$ while bond index from $$1$$ to $$L-1$$.
###### Graphical Notations
 $$\Lambda_{a_{l-1}, a_l}$$ Diagonal Matrix $$A^{\sigma_1}_{1, a_1}{}$$ Row Matrix (MPS) $$A^{\sigma_L}_{a_{L-1}, 1}{}$$ Column Matrix (MPS) $$A^{\sigma_l}_{a_{l-1}, a_l}{}$$ Left or Right Normalized Matrix (MPS) $$A^{\sigma_l *}_{a_{l-1}, a_l}{} = A^{\sigma_l \dagger}_{a_{l}, a_{l-1}}{}$$ Conjugation $$\sum_{\sigma} A^{\sigma \dagger} A^\sigma = \mathbb{1}{}$$ Left Normalization $$\sum_{\sigma} B^{\sigma} B^{\sigma\dagger} = \mathbb{1}{}$$ Right Normalization $$\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} M^{\sigma_1} \cdots M^{\sigma_L} \ket{\sigma_1\cdots \sigma_L}{}$$ MPS String $$W^{\sigma_1 \sigma'_1}_{1,b_1}{}$$ Row Matrix (MPO) $$W^{\sigma_L \sigma'_L}_{b_{L-1},1}{}$$ Column Matrix (MPO) $$W^{\sigma_l \sigma'_l}_{b_{l-1}, b_l}{}$$ Matrix (MPO) $$\hat{O} = \sum_{\vb*{\sigma},\vb*{\sigma}'} W^{\sigma_1 \sigma'_1} \cdots W^{\sigma_L \sigma'_L} \ket{\vb*{\sigma}}\bra{\vb*{\sigma}'}$$ MPO String $$\hat O \ket{\psi} = \sum_{\vb*{\sigma}} N^{\sigma_1} \cdots N^{\sigma_L} \ket{\vb*{\sigma}}$$ MPO Acting on MPS

#### Representations and Conversions

##### Classical Representation

$\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} c_{\sigma_1,\cdots,\sigma_L} \ket{\sigma_1, \cdots, \sigma_L}.$

##### Left-Canonical

$\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} A^{\sigma_2} \cdots A^{\sigma_{L-1}} A^{\sigma_L} \ket{\sigma_1,\cdots,\sigma_L}$ where

• for every $$l$$, $\sum_{\sigma_l} A^{\sigma_l \dagger} A^{\sigma_l} = \mathbb{1};$
• $$A^{\sigma_1}$$'s are row matrices while $$A^{\sigma_L}$$'s are column matrices.
###### From Classical Representation

\begin{align*} c_{\sigma_1,\cdots,\sigma_L} &\xlongequal[\mathrm{reshape}]{} \Psi_{\sigma_1,(\sigma_2,\cdots,\sigma_L)} \\ &\xlongequal{\mathrm{SVD}} \sum_{a_1}^{r_1} U_{\sigma_1,a_1} S_{a_1,a_1} (V^\dagger)_{a_1,(\sigma_2,\cdots,\sigma_L)} \\ &\xlongequal[\mathrm{reshape}]{A^{\sigma_1}_{a_1} = U_{\sigma_1,a_1}} \sum_{a_1}^{r_1} A^{\sigma_1}_{a_1} \Psi_{(a_1\sigma_2),(\sigma_3,\cdots,\sigma_L)} \\ &\xlongequal{\mathrm{SVD}} \sum_{a_1}^{r_1} \sum_{a_2}^{r_2} A^{\sigma_1}_{a_1} U_{(a_1\sigma_2),a_2} S_{a_2,a_2} (V^\dagger)_{a_2,(\sigma_3,\cdots,\sigma_L)} \\ &\xlongequal[\mathrm{reshape}]{A^{\sigma_2}_{a_1,a_2} = U_{(a_1\sigma_2),a_2}} \sum_{a_1}^{r_1} \sum_{a_2}^{r_2} A^{\sigma_1}_{a_1} A^{\sigma_2}_{a_1,a_2} \Psi_{(\sigma_2 a_3),(\sigma_4,\cdots,\sigma_L)} \\ &= \cdots \\ &= \sum_{a_1,\cdots,a_{L-1}} A^{\sigma_1}_{a_1} A^{\sigma_2}_{a_1,a_2} \cdots A^{\sigma_{L-1}}_{a_{L-2},a_{L-1}} A^{\sigma_L}_{a_{L-1}} \end{align*}

###### From General MPS

\begin{align*} \sum_{\sigma_1,\cdots,\sigma_L} \sum_{a_1,\cdots,a_L} M_{(\sigma_1,1),a_1} M^{\sigma_2}_{a_1,a_2} \cdots \ket{\sigma_1\cdots \sigma_L} &= \sum_{\sigma_1,\cdots,\sigma_L} \sum_{a_1,\cdots,a_L} \sum_{s_1} A_{(\sigma_1,1),s_1} S_{s_1,s_1} V^\dagger_{s_1,a_1} M^{\sigma_2}_{a_1,a_2} \cdots \ket{\sigma_1,\cdots,\sigma_L} \\ &= \sum_{\sigma_1,\cdots,\sigma_L} \sum_{a_2,\cdots,a_L} \sum_{s_1} A^{\sigma_1}_{1,s_1} \tilde{M}^{\sigma_2}_{s_1,a_2} M^{\sigma_3}_{a_2,a_3} \cdots \ket{\sigma_1,\cdots,\sigma_L} \\ &= \cdots. \end{align*}

###### From Vidal Notation

$\qty(\Lambda^{[0]} \Gamma^{\sigma_1})\qty(\Lambda^{[1]} \Gamma^{\sigma_2})\qty(\Lambda^{[2]} \Gamma^{\sigma_3})\cdots \rightarrow A^{\sigma_1} A^{\sigma_2} A^{\sigma_3} \cdots.$

##### Right-Canonical

$\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} B^{\sigma_1} B^{\sigma_2} \cdots B^{\sigma_{L-1}} B^{\sigma_L} \ket{\sigma_1,\cdots,\sigma_L}$ where

• for every $$l$$, $\sum_{\sigma_l} B^{\sigma_l} B^{\sigma_l \dagger} = \mathbb{1};$
• $$B^{\sigma_1}$$'s are row matrices while $$B^{\sigma_L}$$'s are column matrices.
###### From Classical Representation

Similar to that of left-canonical ones.

###### From General MPS

Similar to that of left-canonical ones.

###### From Vidal Notation

$\qty(\Gamma^{\sigma_1}\Lambda^{[1]}) \qty(\Gamma^{\sigma_2}\Lambda^{[2]}) \qty(\Gamma^{\sigma_3}\Lambda^{[3]})\cdots \rightarrow B^{\sigma_1} B^{\sigma_2} B^{\sigma_3} \cdots.$

##### Mixed-Canonical

$\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_l} S B^{\sigma_{l+1}} \cdots B^{\sigma_L}.$

###### From Classical Representation

\begin{align*} c_{\sigma_1,\cdots,\sigma_L} &= \sum_{a_l} (A^{\sigma_1} \cdots A^{\sigma_l})_{a_l} S_{a_l,a_l} (V^\dagger)_{a_l,(\sigma_{l+1},\cdots,\sigma_L)} \\ &= \sum_{a_l} (A^{\sigma_1} \cdots A^{\sigma_l})_{a_l} S_{a_l,a_l} (B^{\sigma_{l+1}} \cdots B^{\sigma_L})_{a_l}. \end{align*}

##### Vidal Notation

$\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} \Gamma^{\sigma_1} \Lambda^{[1]} \Gamma^{\sigma_2} \Lambda^{[2]} \cdots \Gamma^{\sigma_{L-1}} \Lambda^{[L-1]} \Gamma^{\sigma_L} \ket{\sigma_1,\cdots,\sigma_L},$ where

• the states on block $$A$$ and $$B$$ given by \begin{align*} \ket{a_l}_A &= \sum_{\sigma_1,\cdots,\sigma_l} \qty(\Gamma^{\sigma_1} \Lambda^{[1]} \Gamma^{\sigma_2} \cdots \Lambda^{[l-1]}\Gamma^{\sigma_l})_{a_l} \ket{\sigma_1,\cdots,\sigma_l}, \\ \ket{a_l}_B &= \sum_{\sigma_{l+1},\cdots,\sigma_L} \qty(\Gamma^{\sigma_{l+1}} \Lambda^{[l+1]} \Gamma^{\sigma_{l+2}} \cdots \Lambda^{[L-1]} \Gamma^{\sigma_L})_{a_l} \ket{\sigma_{l+1} \cdots \sigma_L} \end{align*} are orthonormal respectively.
• The reduced density matrices are given by $\rho^{[l]}_A = \rho^{[l]}_B = \qty(\Lambda^{[l]})^2.$
###### From Classical Representation

\begin{align*} c_{\sigma_1,\cdots,\sigma_L} &\xlongequal[\mathrm{reshape}]{} \Psi_{\sigma_1,(\sigma_2,\cdots,\sigma_L)} \\ &\xlongequal{\mathrm{SVD}} \sum_{a_1} A^{\sigma_1}_{a_1} \Lambda^{[1]}_{a_1,a_1}(V^\dagger)_{a_1,(\sigma_2,\cdots,\sigma_L)} \\ &\xlongequal[\mathrm{reshape}]{A^{\sigma_1}_{a_1} = \Gamma^{\sigma_1}_{a_1}} \sum_{a_1} \Gamma^{\sigma_1}_{a_1} \Psi_{(a_1\sigma_2),(\sigma_3,\cdots,\sigma_L)} \\ &\xlongequal{\mathrm{SVD}} \sum_{a_1,a_2} \Gamma^{\sigma_1}_{a_1} A^{\sigma_2}_{a_1,a_2} \Lambda^{[2]}_{a_2,a_2} (V^\dagger)_{a_2,(\sigma_3,\cdots,\sigma_L)} \\ &\xlongequal[\color{orange} \mathrm{problematic\ division}]{A^{\sigma_2}_{a_1,a_2} = \Lambda^{[1]}_{a_1,a_1}\Gamma^{\sigma_2}_{a_1,a_2}} \sum_{a_1,a_2} \Gamma^{\sigma_1}_{a_1} \Lambda^{[1]}_{a_1,a_1} \Gamma^{\sigma_2}_{a_1,a_2} \Psi_{(a_2,\sigma_3),(\sigma_4,\cdots,\sigma_L)} \\ &= \cdots \\ &= \sum_{a_1,\cdots,a_L} \Gamma^{\sigma_1}_{a_1} \Lambda^{[1]}_{a_1,a_1} \Gamma^{\sigma_2}_{a_1,a_2} \Lambda^{[2]}_{a_2,a_2} \cdots \Lambda^{[L-1]}_{a_{l-1},a_{l-1}} \Gamma^{\sigma_L}_{a_{l-1},a_l}. \end{align*}

• Substitution starting from the left: $A^{\sigma_l}_{a_{l-1},a_l} = \Lambda^{[l-1]}_{a_{l-1},a_{l-1}} \Gamma^{\sigma_l}_{a_{l-1},a_l}.$
• Substitution starting from the right: $B^{\sigma_l}_{a_{l-1},a_l} = \Gamma^{\sigma_l}_{a_{l-1},a_l} \Lambda^{[l]}_{a_l,a_l}.$
• Dummy $\Lambda^{[0]} = \Lambda^{[L]} = 1.$
###### From Left-Canonical

Similar to that of right-canonical ones.

###### From Right-Canonical

\begin{align*} B^{\sigma_1} B^{\sigma_2} B^{\sigma_3} \cdots &\xlongequal{\mathrm{SVD}} (A^{\sigma_1} \Lambda^{[1]} V^\dagger) B^{\sigma_2} B^{\sigma_3} \cdots \\ &\xlongequal{A^{\sigma_1} = \Lambda^{[0]}\Gamma^{\sigma_1}} \Gamma^{\sigma_1} M^{\sigma_2} B^{\sigma_3} \cdots \\ &\xlongequal{\mathrm{SVD}} \Gamma^{\sigma_1} (A^{\sigma_2} \Lambda^{[2]}V^\dagger) B^{\sigma_3} \cdots \\ &\xlongequal[\color{orange}\mathrm{problematic\ division}]{A^{\sigma_2} = \Lambda^{[1]}\Gamma^{\sigma_2}} \Gamma^{\sigma_1} \Lambda^{[1]}\Gamma^{\sigma_2} M^{\sigma_3} B^{\sigma_3} \cdots. \end{align*}

#### Calculation With MPS

##### Block States
• Left block: defined recursively by \begin{align*} \ket{a_l}_A &= \sum_{a_{l-1}} \sum_{\sigma_l} A^{\sigma_l}_{a_{l-1}, a_l} \ket{a_{l-1}}_A \ket{\sigma_l} \\ &= \cdots \\ &= \sum_{\sigma_1,\cdots,\sigma_l} (A^{\sigma_1} \cdots A^{\sigma_l})_{1,a_l} \ket{\sigma_1} \cdots \ket{\sigma_l}. \end{align*}
• Right block: defined recursively by \begin{align*} \ket{a_{l}}_B &= \sum_{a_{l+1}} \sum_{\sigma_{l+1}} B^{\sigma_{l+1}}_{a_l,a_{l+1}} \ket{\sigma_{l+1}} \ket{a_{l+1}}_B \\ &= \cdots \\ &= \sum_{\sigma_{l+1}, \cdots, \sigma_L} (B^{\sigma_{l+1}} \cdots B^{\sigma_L})_{a_{l+1},1} \ket{\sigma_{l+1}} \cdots \ket{\sigma_L}. \end{align*}
• The $$\qty{\ket{a_l}_A}$$ (as well as $$\qty{\ket{a_l}_B}$$) constructed in this way are orthonormal.
• A general state may be written as $\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_l} \Psi B^{\sigma_{l+1}} \cdots B^{\sigma_L} \ket{\sigma_1} \cdots \ket{\sigma_L},$ where $$\Psi$$ is a matrix.
##### Vidal Notation and DMRG
• Vidal notation to blocks: $\begin{gather*} \qty(\Lambda^{[0]}\Gamma)\cdots \qty(\Lambda^{[l-2]}\Gamma)\qty(\Lambda^{[l-1]}\Gamma) \Lambda^{[l]} \qty(\Gamma \Lambda^{[l+1]}) \qty(\Gamma \Lambda^{[l+2]}) \cdots \qty(\Gamma \Lambda^{L}) \\ \rightarrow \ket{\psi} = \sum_{a_l} \ket{a_l}_A s_{a_l} \ket{a_l}_B. \end{gather*}$
• Vidal notation to one-site DMRG: $\begin{gather*} \qty(\Lambda^{[0]}\Gamma)\cdots \qty(\Lambda^{[l-2]}\Gamma)\qty(\Lambda^{[l-1]}\Gamma \Lambda^{[l]}) \qty(\Gamma \Lambda^{[l+1]}) \qty(\Gamma \Lambda^{[l+2]}) \cdots \qty(\Gamma \Lambda^{L}) \\ \rightarrow \ket{\psi} = \sum_{a_{l-1},a_l,\sigma_l} \ket{a_{l-1}}_A \Psi^{\sigma_l}_{a_{l-1},a_l} \ket{a_l}_B. \end{gather*}$
• Vidal notation to two-site DMRG: $\begin{gather*} \qty(\Lambda^{[0]}\Gamma)\cdots \qty(\Lambda^{[l-2]}\Gamma)\qty(\Lambda^{[l-1]}\Gamma \Lambda^{[l]} \Gamma \Lambda^{[l+1]}) \qty(\Gamma \Lambda^{[l+2]}) \cdots \qty(\Gamma \Lambda^{L}) \\ \rightarrow \ket{\psi} = \sum_{a_{l-1},a_{l+1},\sigma_l,\sigma_{l+1}} \ket{a_{l-1}}_A \Psi^{\sigma_l\sigma_{l+1}}_{a_{l-1},a_{l+1}} \ket{a_{l+1}}_B. \end{gather*}$
##### Matrix Elements
###### Notations
• $$\ket{\psi}$$ as MPS: $\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} M^{\sigma_1} \cdots M^{\sigma_L} \ket{\sigma_1} \cdots \ket{\sigma_L}.$
• $$\ket{\phi}$$ as MPS: $\ket{\phi} = \sum_{\sigma_1,\cdots,\sigma_L} N^{\sigma_1} \cdots N^{\sigma_L} \ket{\sigma_1} \cdots \ket{\sigma_L}.$
• Operator living on site $$l$$: $\hat{O}^{[l]} = \sum_{\sigma_l,\sigma'_l} O^{\sigma_l,\sigma'_l} \ket{\sigma_l} \bra{\sigma'_l}.$
###### Overlap
• $$C^{[0]} = 1$$.
• $$C^{[l]}$$ defined recursively by $C^{[l]} = \sum_{\sigma_l} N^{\sigma_l\dagger} C^{[l-1]} M^{\sigma_l}.$
• Transfer operator: specifically for $$N=M$$, $\hat{E}^{[l]} = \sum_{a_{l-1},a'_{l-1}} \sum_{a_l,a'_l} \qty(\sum_{\sigma_l} N^{[l] \sigma_{l*}} \otimes M^{[l]\sigma_l})_{a_{l-1} a'_{l-1}, a_l a'_l} (\ket{a'_{l-1}} \bra{a_{l-1}})(\ket{a_l}\bra{a'_l}).$
• Reformulated with transfer opeartor: $C^{[l]} = E^{[l]}\qty[C^{[l-1]}].$
• Finally we have $C^{[L]} = \bra{\phi}\ket{\psi}.$
###### Matrix Element
• $$C^{[0]} = 1$$.
• For $$l\neq i, j$$, $$C^{[l]}$$ is defined recursively by $C^{[l]} = \sum_{\sigma_l} N^{\sigma_l\dagger} C^{[l-1]} M^{\sigma_l}.$
• For $$l=i$$ or $$l=j$$, $$C^{[l]}$$ is defined by $C^{[l]} = \sum_{\sigma_l,\sigma'_l} O^{\sigma_l,\sigma'_l} N^{\sigma_l\dagger} C^{[l-1]} M^{\sigma'_l}.$
• Transfer operator: specifically for $$N=M$$, $\hat{E}^{[l]}_O = \sum_{a_{l-1},a'_{l-1}} \sum_{a_l,a'_l} \qty(\sum_{\sigma_l,\sigma'_l} O^{\sigma_l,\sigma'_l} N^{[l] \sigma_{l*}} \otimes M^{[l]\sigma_l})_{a_{l-1} a'_{l-1}, a_l a'_l} (\ket{a'_{l-1}} \bra{a_{l-1}})(\ket{a_l}\bra{a'_l}).$
• Reformulated with transfer opeartor: $C^{[l]} = E^{[l]}\qty[C^{[l-1]}].$
• Finally we have $C^{[L]} = \bra{\phi} O^{[i]} O^{[j]} \ket{\psi}.$
###### Properties of Transfer Matrices
• If $$N=M$$ is a left-normalized or right-normalized matrix, then all eigenvalues of $$E$$ satisfies $\abs{\lambda} \le 1.$

#### Compression of MPS

##### SVD Compression
• Starting from a left-canonical MPS.
• $$\tilde{O}$$ denoting $$O$$ truncated, \begin{align*} \ket{\psi} &= \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L} \\ &= \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_{L-1}} USB^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L} \\ &\rightarrow \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots (A^{\sigma_{L-1}} \tilde{U}\tilde{S})B^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L} \\ &= \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_{L-2}} US B^{\sigma_{L-1}} B^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L} \\ &\rightarrow \cdots \\ &\rightarrow \sum_{\sigma_1,\cdots,\sigma_L} (A^{\sigma_1} \cdots A^{\sigma_l} U) S (B^{\sigma_{l+1}}\cdots B^{\sigma_L}) \ket{\sigma_1\cdots \sigma_l} \ket{\sigma_{l+1}\cdots \sigma_L}. \end{align*}
##### Variational Compression

Section not done, cf. §4.5.2, Schollwöck (2011).

#### MPO

$\hat O \ket{\psi} = \sum_{\vb*{\sigma}} N^{\sigma_1} \cdots N^{\sigma_L} \ket{\vb*{\sigma}}.$

• Left/right normalized by taking $$(\sigma_l,\sigma'_l)$$ as a large index.

$\hat{O} \ket{\psi} = \sum_{\vb*{\sigma}} N^{\sigma_1} \cdots N^{\sigma_L} \ket{\vb*{\sigma}}$ where $N^{\sigma_i} = \sum_{\sigma'_i} W^{\sigma_i \sigma'_i} \otimes M^{\sigma'_i}.$

#### Ground State Calculation

##### MPO Representation of Hamiltonians
1. Hamiltonian defined by $\hat{H} = \sum_{i=1}^{L-1} \qty(\frac{J}{2} \hat{S}_i^{+} \hat{S}_{i+1}^{-} + \frac{J}{2} \hat{S}_{i}^{-} \hat{S}_{i+1}^+ + J^z \hat{S}^z_{i} \hat{S}^z_{i+1}) - h\sum_i \hat{S}^z_i.$
2. Hamiltonian represented as $\hat{O} = \hat{W}^{[1]} \cdots \hat{W}^{[L]}$ where $$W^{[i]}$$'s are Operator-valued matrices $\hat{W}_{bb'} = \sum_{\sigma \sigma'} W^{\sigma \sigma'}_{bb'} \ket{\sigma}\bra{\sigma}{}.$
3. For $$2 \le l \le L-1$$, the linear space that $$W^{[l]}$$ acts on may be generated by $$\qty{\vb*{e}^{[l]}_{b'}}$$ defined by \begin{align*} O^{[l]}_1 &= \mathbb{1} \otimes \cdots \otimes \mathbb{1}, \\ O^{[l]}_2 &= \hat{S}^+ \otimes \mathbb{1} \otimes \cdots \otimes \mathbb{1}, \\ O^{[l]}_3 &= \hat{S}^- \otimes \mathbb{1} \otimes \cdots \otimes \mathbb{1}, \\ O^{[l]}_4 &= \hat{S}^z \otimes \mathbb{1} \otimes \cdots \otimes \mathbb{1}, \\ O^{[l]}_5 &= \mathrm{sum\ of\ completed\ terms\ to\ the\ right}. \\ \end{align*}
4. We have the recurrence relation \begin{align*} O^{[l-1]}_1 &= \mathbb{1} \otimes O^{[l]}_1, \\ O^{[l-1]}_2 &= \hat{S}^+ \otimes O^{[l]}_1, \\ O^{[l-1]}_3 &= \hat{S}^- \otimes O^{[l]}_1, \\ O^{[l-1]}_4 &= \hat{S}^{z} \otimes O^{[l]}_1, \\ O^{[l-1]}_1 &= -h \hat{S}^z \otimes O^{[l]}_1 + \frac{J}{2} \hat{S}^- \otimes O^{[l]}_2 + \frac{J}{2} \hat{S}^+ \otimes O^{[l]}_3 + J^z \hat{S}^z O^{[l]}_4 + \mathbb{1}\otimes O^{[l]}_5. \\ \end{align*}
5. We could therefore define $$W^{[l]}$$ by $W^{[l]} = \begin{pmatrix} \hat{1} & & & & \\ \hat{S}^+ & & & & \\ \hat{S}^- & & & & \\ \hat{S}^z & & & & \\ -h\hat{S}^z & (J/2) \hat{S}^- & (J/2) \hat{S}^+ & J^z \hat{S}^z & \mathbb{1} \end{pmatrix}{}$ such that $\begin{pmatrix} O^{[l-1]}_1 \\ O^{[l-1]}_2 \\ O^{[l-1]}_3 \\ O^{[l-1]}_4 \\ O^{[l-1]}_5 \end{pmatrix} = \hat{W}^{[l]} \begin{pmatrix} O^{[l]}_1 \\ O^{[l]}_2 \\ O^{[l]}_3 \\ O^{[l]}_4 \\ O^{[l]}_5 \end{pmatrix},$ where the multiplication in the dot product is given by kron.
6. At the border we have $\hat{W}^{[1]} = \begin{pmatrix} -h\hat{S}^z & (J/2) \hat{S}^- & (J/2) \hat{S}^+ & J^z \hat{S}^z & \mathbb{1} \end{pmatrix}$ and $\hat{W}^{[L]} = \begin{pmatrix} \mathbb{1} \\ \hat{S}^+ \\ \hat{S}^- \\ \hat{S}^z \\ -h\hat{S}^z \end{pmatrix}.$

The lesson:

• It's not that hard to contruct $$W^{[l]}$$. Instead of thinking $$W^{[l]}$$ as matrices labeled by $$\sigma$$ and $$\sigma'$$, we take it as operator-valued matrices labeld by $$b$$ and $$b'$$.
• $$W^{[l]}$$ is sparse.
• The MPO representation of Hamiltonian stores the exact Hamiltonian in a small space without compression.
##### Applying a Hamiltonian MPO to a Mixed Canonical State

$\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_{l-1}} \Psi^{\sigma_l} B^{\sigma_{l+1}} \cdots B^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L}.$

$\bra{a_{l-1} \sigma_l a_l} \hat{H} \ket{a'_{l-1}\sigma'_l a'_l} = \sum_{b_{l-1},b_l} L^{a_{l-1},a'_{l-1}}_{b_{l-1}} W^{\sigma_l \sigma'_l}_{b_{l-1},b_l} R^{a_l,a'_l}_{b_l}.$

1. Target: with a Lagrangian multiplier $$\lambda$$, we minimize $\bra{\psi} H \ket{\psi} - \lambda \bra{\psi} \ket{\psi}.$

2. Iterative search: only $$M^{[l]}$$ is allowed to vary.

3. Boost: we'd like
1. $$M^{[l]}$$'s be already a good guess, and
2. The matrices left and right are left-normalized and right-normalized, respectively, so that the above equation is a eigenvalue problem instead of a generalized one.
###### Algorithm
1. Start from some initial guess for $$\ket{\psi}$$, right-normalized.
2. Calculate the $$R$$-expressions iteratively for all sites from $$L-1$$ to $$1$$.
3. Right sweep: sweeping from site $$l=1$$ to $$L-1$$.
1. At each site, solve the eigenvalue problem with starting vector given by the current state, getting $$M^{\sigma_l}$$.
2. Obtain $$A^{\sigma_l}$$ by SVD, with the remaining $$SV^\dagger$$ multiplied to the $$M^{\sigma_{l+1}}$$ to the right as the starting point of the eigensolver for the next site.
3. Build iteratively the $$L$$-expression.
4. Left sweep: sweeping from site $$l=L$$ to $$2$$.
1. Again, solve the eigenvalue problem at each site.
2. Obtain $$B^{\sigma_l}$$ by SVD, with the remaining $$US$$ multiplied to the $$M^{\sigma_l}$$ to the left.
3. Build iteratively the $$R$$-expression.
5. Repeat left and right sweeps, until convergence is achieved. Convergence condition given by $\bra{\psi} \hat{H}^2 \ket{\psi} - \qty(\bra{\psi} \hat{H} \ket{\psi})^2 \rightarrow 0.$

### Computational Tricks

For tricks that speed up the computation, see Computational Tricks, DMRG (II).

#### Avoid Trapping

Add a correction to $$\hat{\rho}^{A\bullet}$$ before doing SVD: $\hat{\rho}^{A\bullet} = \tr_B \ket{\psi} \bra{\psi} + \alpha \sum_{b_l} \tr_B \hat{H}^{A\bullet}_{b_l}\ket{\psi} \bra{\psi} \hat{H}^{A\bullet}_{b_l},$ where $$\alpha$$ takes small value $$O(10^{-4})$$ and will be taken slowly to zero as sweeping goes on.

The correction term $$\hat{H}_{b_l}^{A\bullet}\ket{\psi}$$ is evaluated as follows.

### Miscellaneous

#### Entanglement

• Maximally entangled state: $$\rho \in H_A \otimes H_B$$ has the form for each of the subsystems $\begin{pmatrix} 1/n & & & \\ & 1/n & & \\ & & \ddots & \\ & & & 1/n \end{pmatrix}.$

Expand

## Mathematics Miscellany

#### Matrix Decomposition

##### SVD

$M = USV^\dagger$ where

• columns of $$U$$ are orthonormal;
• $$S$$ is diagonal with non-negative entries;
• rows of $$V^\dagger$$ are orthonormal.
##### QR Decomposition
###### Full QR

$M = QR$ where

• $$Q$$ is unitary;
• $$R$$ is upper-triangular.
###### Thin QR

$M = Q_1 R_1$ where

• columns of $$Q_1$$ are orthonormal;
• $$R_1$$ is upper-triangular.

#### Kronecker Product

##### Properties
• Associativity: $(A\otimes B) \otimes C = A \otimes (B\otimes C).$
• Mixed-product: $(A\otimes B)\cdot (C\otimes D) = (A\cdot C)\otimes (B\cdot D).$ In particular, $A \otimes B = (A \otimes \mathbb{1}) \cdot (\mathbb{1}\otimes B).$
• Inverse, Transpose, etc: $\begin{gather*} (A\otimes B)^{-1} = A^{-1} \otimes B^{-1}, \\ (A\otimes B)^\dagger = A^\dagger \otimes B^\dagger. \end{gather*}{}$
• Trace, Determinant, etc: $\begin{gather*} \det (A_{n\times n}\otimes B_{m\times m}) = (\det A)^m (\det B)^n, \\ \tr (A \otimes B) = (\tr A)(\tr B). \end{gather*}{}$
• The kron of two matrices of orthonormal columns/rows is again a matrix of orthogonal columns/rows, respectively.
##### Implementations
numpy.kron(a, b)
scipy.sparse.kron(a, b)
scipy.linalg.kron(a, b)


Expand

## DMRG (II)

Implementations

### Simple DMRG

#### Simple DMRG Program

The Hamiltonian is given by the XXZ model $H = -\frac{1}{2} \sum_{j=1}^N (J \sigma_j^x \sigma_{j+1}^x + J \sigma_j^y \sigma_{j+1}^y + J_z \sigma_{j}^z \sigma_{j+1}^z).$

##### Finite System

$$d$$ is defined to be the number of states of a single site.

In line 40 and 41 two matrices are defined, $S^z_{1} = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad S^+_{1} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ as well as $H_1 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}.$ $$H_1$$ will be used as the single-site Hamiltonian at initialization.

The two-site Hamiltonian is defined in line 45 to be \begin{align*} H_2(S_1, S_2) &= \frac{J}{2} S^+_1 \otimes S^{-}_2 + \frac{J}{2} S^{-}_1 \otimes S^+_2 + J_z S^z_1 \otimes S^z_2 \\ &= J(S_1^x \otimes S_2^x + S_1^y \otimes S_2^y) + J_z S_1^z \otimes S_2^z. \end{align*}

###### Enlarging Block

The enlarging process renews the operaors as \begin{align*} H &\rightarrow H \otimes \mathbb{1}_{d\times d} + H_2(S_{\mathrm{edge}}, S_1), \\ S_{\mathrm{edge}}^z &\rightarrow \mathbb{1}_{M\times M} \otimes S^z_1, \\ S_{\mathrm{edge}}^+ &\rightarrow \mathbb{1}_{M\times M} \otimes S^+_1. \end{align*}

###### Single DMRG Step

At each DMRG step, the following steps are carried out:

1. The system block and the environment block is enlarged once.
2. The superblock Hamiltonian is constructed as $H = H_{\mathrm{sys}} \otimes \mathbb{1}_{M_{\mathrm{env}}\times M_{\mathrm{env}}} + \mathbb{1}_{M_{\mathrm{sys}} \times M_{\mathrm{sys}}} \otimes H_{\mathrm{env}} + H_2(S_{\mathrm{sys,edge}}, S_{\mathrm{env,edge}}).$
3. Calculate the lowest eigenvalue $$E$$ and $$\psi$$ of $$H$$.
4. Calculate the projected density matrix: $\rho = \tr_E (\ket \psi \bra \psi).$
5. Calculate the eigenvectors of the lowest $$m$$ eigenvalues. If $$m$$ is greater then the dimension of the matrix, we calculate all the eigenvectors.
6. Construct the transformation matrix $$T$$, where the columns are given by the eigenvectors in the previous step. $$T$$ is not necessarily a square matrix.
7. All the operators of the system, $$S_{\mathrm{sys}}$$ and $$H_{\mathrm{sys}}$$, should undergo a transformation $H \rightarrow T^\dagger H T.$

We may take system and environment to be the same. This may reduce half of the calculations.

How is the projected density matrix calculated? Although $$\psi$$ is a vector in the direct product of two Hilbert spaces, it was represented by a column vector returned from eigsh. The following snippet traces out the environment part:

psi = psi.reshape([m_sys, -1])
rho = np.dot(psi0, psi0.conjugate.transpose())


#### Hubbard Model on a Specific Sector

This section is deprecated. It's a total failure.

##### Notations
1. By $$[N_\uparrow \otimes N_\downarrow]_L$$ we mean the Hilbert space of $$N_\uparrow$$ spin-up and $$N_\downarrow$$ spin-down electrons on a chain of length $$L$$.
• If either of $$N_\uparrow$$ or $$N_\downarrow$$ is negative or greater than $$L$$ then the space is taken as an zero-dimensional one, i.e. $${0}$$.
• If $$L<0$$ then the space is taken as an zero-dimensional one, i.e. $${0}$$.
2. By $$R[N_\uparrow \otimes N_\downarrow]_L$$ we mean the projected Hilbert space of $$[N_\uparrow \otimes N_\downarrow]_L$$ onto a subspace determined by the DMRG process.

With the above notation the dimension of $$R[N_\uparrow\otimes N_\downarrow]$$ is at most $$M$$.

##### Block Solution

By a block solution of $$[0\otimes 0]_L$$ where $$L$$ may be $$0$$ we mean

1. A Hilbert space (not stored) $\mathcal{H} = \langle \ket{0} \rangle.$
2. A Hamiltonian $H = \begin{pmatrix} 0 \end{pmatrix}.$
3. Creation operators on the boundary, $$c_{L-1,\uparrow}^\dagger$$ and $$c_{L-1,\downarrow}^\dagger$$ are both $$1\times 0$$ zero matrices in this case.
4. A projection matrix $V = \begin{pmatrix} 1 \end{pmatrix}.$

By a block solution of $$[N_\uparrow \otimes 0]_L$$ where $$N_\uparrow > 0$$ we mean

1. A Hilbert space (before projection) $R[N_\uparrow \otimes 0]_{L-1} \otimes \ket{00} \oplus R[(N_\uparrow - 1) \otimes 0]_{L-1} \otimes \ket{\uparrow 0}.$
2. A Hamiltonian $H: R[N_\uparrow \otimes 0]_L \rightarrow R[N_\uparrow \otimes 0]_L.$
3. Creation operators on the boundary,
• $c_{L-1,\uparrow}^\dagger: R[(N_\uparrow - 1) \otimes 0]_L \rightarrow R[N_\uparrow \otimes 0]_L.$
• $$c_{L-1,\downarrow}^\dagger$$ an $$1\times 0$$ zero matrix.
4. A projection matrix $V: R[N_\uparrow \otimes 0]_{L-1} \otimes \ket{00} \oplus R[(N_\uparrow - 1) \otimes 0]_{L-1} \otimes \ket{\uparrow 0} \rightarrow R[N_\uparrow \otimes 0]_L.$

For a block solution of $$[0 \otimes N_\downarrow]_L$$ where $$N_\downarrow > 0$$ just substitute $$\downarrow$$ for $$\uparrow$$ in the above definition.

By a block solution of $$[N_\uparrow \otimes N_\downarrow]_L$$ where $$N_\uparrow > 0$$ and $$N_\downarrow > 0$$ we mean

1. A Hilbert space (before projection) \begin{align*} \qty[(N_\uparrow) \otimes (N_\downarrow)]_{L} &= R\qty[(N_\uparrow) \otimes (N_\downarrow)]_{L-1} \otimes \ket{00} \\ &\oplus R\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L-1} \otimes \ket{\uparrow 0} \\ &\oplus R\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L-1} \otimes \ket{0\downarrow} \\ &\oplus R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L-1} \otimes \ket{\uparrow \downarrow}. \end{align*}
2. A Hamiltonian $H: R[N_\uparrow \otimes N_\downarrow]_{L} \rightarrow R[N_\uparrow \otimes N_\downarrow]_{L}.$
3. Creation operators on the boundary,
• $c_{L-1,\uparrow}^\dagger: R[(N_\uparrow - 1) \otimes N_\downarrow]_L \rightarrow R[N_\uparrow \otimes N_\downarrow]_L.$
• $c_{L-1,\downarrow}^\dagger: R[N_\uparrow \otimes (N_\downarrow - 1)]_L\rightarrow R[N_\uparrow \otimes N_\downarrow]_L.$
4. A projection matrix \begin{align*} & V: R\qty[(N_\uparrow) \otimes (N_\downarrow)]_{L} \otimes \ket{00} \\ &\oplus R\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \otimes \ket{\uparrow 0} \\ &\oplus R\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{0\downarrow} \\ &\oplus R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow \downarrow} \\ &\rightarrow R[N_\uparrow\otimes N_\downarrow]_L. \end{align*}
##### Obtaining Block Solutions

Now we are gonna obtain the block solution of $$(L+1,N_\uparrow,N_\downarrow)$$ recursively. We assume that $$N_\uparrow > 0$$ and $$N_\downarrow > 0$$.

We assume that block solutions of length $$L$$ are already known, as well as the block solutions of $$(L+1, N^*_\uparrow, N^*_\downarrow)$$ where $$N^*_\uparrow \le N_\uparrow$$, $$N^*_\downarrow \le N_\downarrow$$, and at least one of the inequalities strictly holds.

The creation operators on the boundary are given by

• $c_{L,\uparrow}^\dagger = \begin{array}{ccccc} & R\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \otimes \ket{00} & R\qty[(N_\uparrow - 2) \otimes (N_\downarrow)]_{L} \otimes \ket{\uparrow 0} & R\qty[(N_\uparrow-1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{0\downarrow} & R\qty[(N_\uparrow - 2) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow \downarrow} \\ R\qty[(N_\uparrow) \otimes (N_\downarrow)]_{L} \otimes \ket{00} \\ R\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \otimes \ket{\uparrow 0} & \mathbb{1} \\ R\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{0\downarrow} \\ R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow \downarrow} & & & \mathbb{1} \end{array}.$
• $c_{L,\downarrow}^\dagger = \begin{array}{ccccc} & R\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{00} & R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow 0} & R\qty[(N_\uparrow) \otimes (N_\downarrow - 2)]_{L} \otimes \ket{0\downarrow} & R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 2)]_{L} \otimes \ket{\uparrow \downarrow} \\ R\qty[(N_\uparrow) \otimes (N_\downarrow)]_{L} \otimes \ket{00} \\ R\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \otimes \ket{\uparrow 0} \\ R\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{0\downarrow} & \mathbb{1} \\ R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow \downarrow} & & \mathbb{1} \end{array}.$

Now we have to find the creation operators next to the boundary.

• $c_{L-1,\uparrow}^\dagger = \begin{array}{ccccc} & R\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \otimes \ket{00} & R\qty[(N_\uparrow - 2) \otimes (N_\downarrow)]_{L} \otimes \ket{\uparrow 0} & R\qty[(N_\uparrow-1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{0\downarrow} & R\qty[(N_\uparrow - 2) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow \downarrow} \\ R\qty[(N_\uparrow) \otimes (N_\downarrow)]_{L} \otimes \ket{00} & c_{L-1,\uparrow}^\dagger\qty[(N_\uparrow) \otimes (N_\downarrow)]_{L} \\ R\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \otimes \ket{\uparrow 0} & & c_{L-1,\uparrow}^\dagger\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \\ R\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{0\downarrow} & & & c_{L-1,\uparrow}^\dagger\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \\ R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow \downarrow} & & & & c_{L-1,\uparrow}^\dagger\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \end{array}.$
• $c_{L-1,\downarrow}^\dagger = \begin{array}{ccccc} & R\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{00} & R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow 0} & R\qty[(N_\uparrow) \otimes (N_\downarrow - 2)]_{L} \otimes \ket{0\downarrow} & R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 2)]_{L} \otimes \ket{\uparrow \downarrow} \\ R\qty[(N_\uparrow) \otimes (N_\downarrow)]_{L} \otimes \ket{00} & c_{L-1,\downarrow}^\dagger\qty[(N_\uparrow) \otimes (N_\downarrow)]_{L} \\ R\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \otimes \ket{\uparrow 0} & & c_{L-1,\downarrow}^\dagger\qty[(N_\uparrow - 1) \otimes (N_\downarrow)]_{L} \\ R\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{0\downarrow} & & & c_{L-1,\downarrow}^\dagger\qty[(N_\uparrow) \otimes (N_\downarrow - 1)]_{L} \\ R\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \otimes \ket{\uparrow \downarrow} & & & & c_{L-1,\downarrow}^\dagger\qty[(N_\uparrow - 1) \otimes (N_\downarrow - 1)]_{L} \end{array}.$

We are now able to construct $$H$$ in principle. Then, following the standard DMRG procedure, we obtain a projection matrix $$V$$ consists of column vectors.

1. Hamiltonian: $H = V^\dagger H V.$
2. Creations operators:
• $c^\dagger_{L,\uparrow} = V^\dagger[(N_\uparrow) \otimes (N_\downarrow)]_L c^\dagger_{L,\uparrow} V[(N_\uparrow - 1) \otimes (N_\downarrow)]_L.$
• $c^\dagger_{L,\downarrow} = V^\dagger[(N_\uparrow) \otimes (N_\downarrow)]_L c^\dagger_{L,\downarrow} V[(N_\uparrow) \otimes (N_\downarrow - 1)]_L.$

### Computational Tricks

#### Speed-Up

Section not done.

Expand

Expand

Expand

## Sino-Japanese Words and Phrases

#### 名詞

##### 系統

###### In Japanese

The first of the definitions of 系統 given in 日本国語大辞典 reads

The first quotation in Japanese dates back to 1827, the end of 江戸時代, given in 日本外史.

###### In Chinese

The second definition of 系統 given in 辞海 reads

The first quotation in Chinese dates back to the Jiajing period, given in 青霞集 by 沈鍊 (1497–1557).

###### Conclusion: No Conclusion

The word system is derived from Late Latin (from the 3rd to the 6th centuries CE).

It's really hard for me to draw a conclusion here. Note that there are multiple questions here:

• Who first invented the word 系統?
• Who first used 系統 as the translation of system?
• Is 系統 a homophonic translation of system?

#### 動詞

##### 例を挙げる/舉例

It looks really strange that in both languages the phrase to give an example reads literally to raise an example. Is it accidental, or this definition is borrowed from one language to another?

###### In Japanese

Definition 4 and 4.5 of げる given in 日本国語大辞典 reads respectively

⇒実例、証拠などをはっきり表面に表す。

Quotations date back to 1785, in

「わざと悪事に一味して、まっかう手めを上げようため」〈浄・先代萩〉

as well as to 1787–1788, in

「歴史上の事実を挙げ」〈花間鶯〉

This definition gives rise to the usage 例を挙げる.

There are also compounds like きょれい or きょぐう.

###### In Chinese

Definition 5 of 舉 in 王力古漢語字典 reads

Quotations found in 論語·述而 (the 5th century BC)

in 韓非子·外儲說左上 (mid 3rd century BC)

as well as in 史記·屈原賈生列傳 (c. 90 BC)

The usage in the first quotation gives rise to 舉隅.

###### Conclusion: the Coincidence in Accidental

The definition to put forward seems to arise naturally from its original definition to physically raise something. However, the compounds きょれい or きょぐう may be imported from Chinese.

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## 广告与悬赏广告之法律性质

#### 广告与要约

##### 台湾法下的广告

73年台上字2540號 ... 惟查廣告文字僅為要約之引誘.

##### 中国大陆法下的广告

（一）内容具体确定；

（二）表明经受要约人承诺，要约人即受该意思表示约束。

#### 悬赏广告

###### 悬赏广告之定性问题

• 契约行为说 悬赏广告系对不特定人之要约, 经行为人完成一定行为予以承诺而成立契约.
• 单独行为说 广告人因其单方面意思而负担债务.
##### 德国法下的悬赏广告

BGB将「悬赏广告」置于§8.11, 与契约, 无因管理, 不当得利并列为债之发生事由.

BGB第657条 通过公开通告, 对完成某行为, 特别是对产生结果悬赏之人, 负有向完成此行为之人支付报酬的义务, 即使行为人完成该行为时未考虑到此悬赏广告.

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## Japanese Miscellaneous

### Grammar

#### Therefore

See だから／従って／よって／故に.

##### だから
 Formality ☆☆☆☆☆ Objectiveness ☆☆☆☆☆
• 夕べはよく寝た。だから今日はとても調子がいい。
• ここは環境がいい。だから住みたいという人がとても多い。
##### 従って
 Formality ★★★★☆ Objectiveness ★★★☆☆
• この学校は進学率が高い。したがって志望者も多い。
• このホテルはサービスがいい。したがって料金も高い。
##### よって
 Formality ★★★★☆ Objectiveness ★★★★★
• aはbに等しく、bはcに等しい。よってaとcは等しい。
• 貴殿はわが社の発展に多大な貢献をなされました。よってこれを賞します。
##### 故に
 Formality ★★★★★ Objectiveness ★★★★★
• 日本は経済大国である。ゆえに外国から働きに来る人も多い。
• 我思う、ゆえに我あり。

#### Because

##### から
 Formality ★☆☆☆☆ Objectiveness ★★☆☆☆
• 操作ミスから事故が生じた。
• 危ないから、窓から手を出さないでください。
• あんまり安いから、つい買う気になった。
##### ので
 Formality ★★★☆☆ Objectiveness ★★★☆☆
• 辛い物を食べたので、のどが渇いた。
• 朝が早かったので、ついうとうとする。
• 盆地なので、夏は暑い。
##### ため
 Formality ★★★★★ Objectiveness ★★★★☆
• 雨のため、試合は中止になりました。
• 風が強かったために、昨日は船が出ませんでした。

#### However

##### でも
• 今朝は大雨だった。でも、学校は休みにならなかった。
##### しかし
• 君の言い分はよく解るよ。しかし、それを周囲に理解してもらおうというのは難しいと思うな。
##### だが
• 今回は失敗をしてしまった。だが、次回は必ず成功すると確信している。
##### ただし
• これはすぐれた説だ。ただし疑えば疑える点もある。

#### Although

##### が
 Mood Disjunctive
• みんな行った、彼は行かなかった。
• 営業も大変だ、事務はもっと大変です。
##### のに
 Mood Unexpected
• トムさんは２年日本へ留学したのに、全然日本語が話せません。
• 昨日、早くたくさん寝たのに、まだ眠いです。
##### けど
 Mood Anyway
• もう夜中だけど、もう少し勉強しよう。
• 食べたいけど、太るのが怖い。
##### くせに
 Mood Fuck him
• 知っているくせに、知らないふりをしている。
• あいつは馬鹿のくせに、偉そうにしてる。

#### If

##### と
 Certainty ★★★★★ Objectiveness ★★★★★
• 右へ曲がる、パンやがあります。
• 春になる、お花見ができます。
##### ば
 Certainty ★★☆☆☆ Objectiveness ★★☆☆☆
• 明日もし雨が降れ、どうしますか。
##### たら
 Certainty ★★★★☆ Objectiveness ★★★☆☆

たら is the 仮定形 of た.

• お家に帰ったら、連絡してください。
• パリに行ったら、凱旋門にも行ってみたい。
• 帰宅したら、必ずお風呂に入りなさい。
• 飲んだら乗るな。乗るなら飲むな。
##### なら
 Certainty ★☆☆☆☆ Objectiveness ★★☆☆☆
• スーパーに行くのなら、しょうゆを買ってきて。
• 旅行に行くのなら、カメラを持っていくといいですよ。
• 飲んだら乗るな。乗るなら飲むな。

#### In Other Words

##### 詰まり
 Formality ★★☆☆☆ Coherence ★★★★★
• 地図の上方、詰まり北方は山岳地帯である。
##### 即ち
 Formality ★★★★★ Coherence ★★★★★
• 日本の首都、すなわち東京

#### Regardless of

See に限らず／によらず／を問わず.

##### を問わず
 Objectiveness ★★☆☆☆
• 休日、平日を問わず、一年中人出が絶えない。
• 経験の有無を問わず、人材を広く世に求める。
##### に関わらず
 Objectiveness ★★★★☆
• 晴雨に関わらず実施する。
##### によらず
 Objectiveness ★★★★★

「によらず」は、「何」「だれ」などの不定称の代名詞を受けて、どの…に対しても区別をつけないで一様に、どんな…でもみな、という意味を表わす。

• 妹は何によらず臆病なところが心配です。
• だれによらず人の不始末の尻ぬぐいなどしたくはない。

#### Limited to, Only

##### だけ
• 千円だけ貸してあげる。
• 好きなだけ食べた。
• 勉強すればしただけ成績は上がる。
##### のみ
• あとは結果を待つのみである。
• 日本のみならず全世界の問題だ。
##### ばかり
• 彼は勉強ばかりしている。
• 彼は子供とばかり遊んでいる。
• ばかりの贈り物。

#### Not Limited to

##### に限らず
• 彼女は肉に限らず動物性たんぱくはいっさいとらない。
• 目上の人と話す場合に限らず、人との付き合いには敬語は欠かせない。

#### As One Can

##### だけ
• できるだけ多くの本を読みなさい。
##### 限り
• 私はできる限りあなたの援助をします。

To be filled.

### Common Verbs

#### To Close

##### 閉める

To physically close something.

• ドアを閉める
• 店を閉める
##### 閉まる

For something to physically close.

• ドアが閉まる
• 店が閉まる
##### 閉じる

For something to close or to come to an end.

• ドアが閉じる
• 店が閉じる
• 会が閉じる
• 目を閉じる
• 傘を閉じる
• 歴史を閉じる
##### 閉ざす

To (figuratively) close.

• 道を閉ざす
• 門を閉ざす
• 国を閉ざす

#### To Walk

See 歩く vs 歩む and 歩く／歩む.

##### 歩く

「歩く」は、足を使って前に進む意。具体的な動作を表わす。

##### 歩む

「歩む」は、物事が進行する意。具体的な動作を表わすよりは、抽象的な意味で用いられるのが普通。

### Miscellaneous

#### Vocabulary

• さら名詞
• review
• rehearsal
• あま副詞
• not much
• まり 副詞
• to wit, that is to say
• さら副詞
• moreover
• すなわ副詞
• namely, i.e.
• 副詞
• if, supposing
• したがって 接続詞
• therefore
• のみ 助詞
• only, limited to
• 坩堝るつぼ
• Crucible.

#### Grammar

• ておる or とる
• to be (doing): indicates a progressive or continuous sense
• ている or てる
• indicating the present continuous tense
• indicating the resulting state of an action
• としても
• even if
• Usage:
• 動詞普通形＋としても
• イ形容詞普通形＋としても
• ナ形語幹＋（だ／である）としても
• 名詞＋（だ／である）としても
• にわたって
• to reach a certain point
• AをBにする
• with A as B, with A at B, or using A as B
• なので/ので
• as a result
• には , also
• emphatic
• in order to
• が欠かせない/に欠かせない
• to be indispensable
• ておく
• to do something in advance
• to leave something undone
• AをBとする/AをBとして , also , , or
• with A as B, with A at B, or using A as B
• to proceed to perform a new action (by quitting what one is currently doing)
• (volitional form) + として: trying to do
• assuming
• になる/となる
• to become, to be determined as
• によらず , かかわらず
• regardless of
• わけ/わけだ , also , or
• no wonder
• to emphasize
• くらい
• かどうか
• whether or not
• について
• または/しくは
• or
• 意志形としている
• To try to.
• Something is gonna ...
• のだ/のです
• 大雑把
• Rough, approximate.
• 大変役に立つ
• Very useful.
• というのも
• (At the beginning of a sentence) because.
• 役割を果たす
• To play a role.
• 必ずしもXわけではない
• Not necessarily X.
• Xざるを得ない
• To have to do X.
• って "女の子の名前で雪花って変ですか？" "de" and "tte" usage
• An informal form of は (topic marker).
• たりして ～たりして
• Something that seems unlikely but might be possible.
• おかげで
• Thanks to.

#### Phrases

• あたえられる
• be given by
• 次のように
• as follows
• 代表無くして課税なし
• No Taxation Without Representation

#### 外来語

• ランドセル
• A type of backpack worn by Japanese elementary schoolchildren.

#### Onomatopoeia

• キュンキュン
• Heartwrenchingly.

#### Remarks on Etymology

• て and is the 連用形 of .
• itself is a 終止形, with たら as its 仮定形. た is from たる, a contraction of てあり.
• is either from である or なり, with なら as its 仮定形 of the latter etymology.

#### Final-Over-Final Constraint

See 语言学中都有哪些定律？. Japanese is the epitome of a head final language.

#### Remarks on Translation

##### Phono-Semantic Matching
• for club.
• かっ for capa (Portuguese).
##### Semantic Loan

See 意味借用.

• あそぶ: 「離れた土地に行って学ぶ」 from 漢文.
• おそう: 「地位を受け継ぐ」 from 漢文.
##### Sino-Japanese Words

See 《语词的漂移：近代以来中日之间的知识互动与共有》——陈力卫, as well as 如何看待《驳所谓〈离开了日本外来词，中国人无法说话〉的言论》？

Expand

## Models for Topological Insulators

「二次元のメインヒロインも好きだけど、でも、三次元ほんものの加藤恵が一番好きなんだよ」

#### Category

Model Dimension Description
Thouless Charge Pump / Rice-Mele Model 1D Charge pump
Fu-Kane Spin Pump 1D Spin pump
Su-Schrieffer-Heeger Model 1D TI model for polyacetylene
Haldane Model 2D IQHE in graphene
Kane-Mele Model 2D QSHE in graphene
Bernevig-Hughes-Zhang Model 2D QSHE in HgTe/CdTe quantum well
$$\ce{Pb_{x}Sn_{1-x}Te}{}$$ 3D Weak TI $$\ce{Pb_{x}Sn_{1-x}Te}{}$$
$$\ce{Bi_{1-x}Sb_{x}}{}$$ 3D Strong TI $$\ce{Bi_{1-x}Sb_{x}}{}$$
$$\ce{Bi_{2}Q_{3}}{}$$ 3D Strong TI $$\ce{Bi_{2}Q_{3}}{}$$ with a single Dirac cone

#### Foundations

##### Dirac Matrices
• One-dimensional: $\alpha_x = \sigma_x, \quad \beta = \sigma_z.$
• Two-dimensional: $\alpha_x = \sigma_x,\quad \alpha_y = \sigma_y,\quad \beta = \sigma_z.$
• Three-dimensional: $\alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix},\quad \beta = \begin{pmatrix} \sigma_0 & 0 \\ 0 & -\sigma_0. \end{pmatrix}$
##### Quantum Mechanics
• Time reversal: $\Theta = i\sigma_y K.$
• Parity: in the Dirac equation, we let $P = \pi \beta$ such that $P\alpha_i P = -\alpha_i,\quad P\beta P = \beta,$ i.e. the Dirac equation is invariant under $$P$$.
##### Second Quantization
• Fourier transformation: \begin{align*} c_{i,\sigma} &= \frac{1}{\sqrt{Na}} \sum_{k_n} e^{ik_n R_i} c_{k_n,\sigma}, \\ c_{i,\sigma}^\dagger &= \frac{1}{\sqrt{Na}} \sum_{k_n} e^{-ik_n R_i} c_{k_n,\sigma}^\dagger. \end{align*}
• Hamiltonian diagonalized: $H = \sum_{k_n} \epsilon(k_n) c^\dagger_{k_n,\sigma} c_{k_n,\sigma}.$
##### From Continuous Models via Replacement: It Works but I Don't Know Why
• Replacements: \begin{align*} k_i &\rightarrow \frac{1}{a} \sin k_i a, \\ k_i^2 &\rightarrow \frac{4}{a^2} \sin^2 \frac{k_i a}{2}. \end{align*}
##### Periodic Hamiltonian
• Periodicity condition: $H(k, t+T) = H(k, t).$
• Time-reversal invariants: at $t = 0 + nT;\quad t = T/2 + nT$ we have $\Theta H(t) \Theta^{-1} = H(t).$
##### Evolution of Electric Polarization
• Evolution of Polarization under Periodic Hamiltonian: $\Delta P_\alpha = e\sum_n \int_0^T \dd{t} \int_{\mathrm{BZ}} \frac{\dd{\vb*{q}}}{(2\pi)^d} \Omega^n_{q_\alpha,t}.$
• Curvature: $\Omega^n_{q_\alpha,t} = \partial_{q_\alpha} A^n_t - \partial_t \vb*{A}^n_{q_\alpha}.$
• I'm not sure what $$\vb*{A}^n_t$$ mean. But if I were to guess, it should be $\vb*{A}^n_{t} = \bra{u_n} \frac{\partial}{\partial t} \ket{u_n}.$
• Polarization with Wannier functions: $\vb*{P} = -e \sum_n \bra{\vb*{R} = 0, n} \vb*{r} \ket{\vb*{R} = 0, n} = -\frac{e}{2\pi} \oint \dd{\vb*{k}} \cdot \vb*{A}(\vb*{k}).$

#### Minimal Lattice Models

• $$c_i^\dagger$$ denotes the row vector $\begin{pmatrix} c_{i,1}^\dagger & \cdots & c_{i,D}^\dagger \end{pmatrix}$ where $$D$$ is the dimension of the Dirac matrices.
• $$\qty(\beta,\vb*{\alpha})$$ denote Dirac matrices.
• Model Hamiltonian:
• $$k$$-space: $H = \frac{\hbar v}{a} \sum_{i} \sin(k_i a) \alpha_i + \qty(mv^2 - B\frac{4\hbar^2}{a^2}\sum_i \sin^2 \frac{k_i a}{2})\beta.$
• Sites: $H = \sum_i \Delta c_i^\dagger \beta c_i - t\sum_{\langle i,j\rangle} c_j^\dagger\beta c_i + i t' \sum_{i,a} \qty[c^\dagger_{i+a} \alpha_a c_i - c_i^\dagger \alpha_a c_{i+a}]$
• Subscript $$i+a$$ denotes the site $$\vb*{R}_i + \vb*{R}_a$$.
• Model parameters: with $$a=1$$ and $$\hbar=1$$, \begin{align*} t' &= \frac{\hbar v}{2a} = \frac{v}{2}, \\ \Delta &= mv^2 + 2dt, \\ t &= -\frac{B\hbar^2}{a^2} = -B. \end{align*}{}
##### One-Dimensional Lattice Model
• Hamiltonian:
• Second quantized: $H = \Delta \sum_{j=1}^N c^\dagger_j \sigma_z c_j - t\sum_{j=1}^{N-1} \qty(c^\dagger_{j+1} \sigma c_j + c^\dagger_j \sigma_z c_{j+1}) + it' \sum_{j=1}^{N-1} \qty{c^\dagger_{j+1} \sigma_x c_j - c_j^\dagger \sigma_x c_{j+1}}.$
• Matrix form: $H = \begin{pmatrix} \Delta \sigma_z & T & 0 & 0 & \cdots & 0 \\ T^\dagger & \Delta \sigma_z & T & 0 & \cdots & 0 \\ 0 & T^\dagger & \Delta \sigma_z & T & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & T^\dagger & \Delta \sigma_z & T \\ 0 & 0 & 0 & 0 & T^\dagger & \Delta \sigma_z \end{pmatrix},$ where $T = -t\sigma_z - it' \sigma_x.$
• Spcecial case: $$N=\infty$$, i.e. semi-infinite chain with one end at $$j=1$$.
• Component form: $\Delta \sigma_z \Psi_j + T\Psi_{j+1} + T^\dagger\Psi_{j-1} = E\Psi_j.$
• Ansatz: $\Psi_{j+1} = \lambda \Psi_j = \lambda^{j+1} \Psi,$ where $$\lambda$$ is a scalar.
• Solution: $$s=\pm1$$, $\Psi = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -is \end{pmatrix},\quad \lambda_\pm(s) = \frac{\Delta}{2(t+st')}\qty[1\pm \sqrt{1-\frac{4(t^2 - t'^2)}{\Delta^2}}].$
• End state: $$\abs{\lambda_\pm}\le 1$$ as $$\Psi_j\rightarrow 0$$, $s = \operatorname{sign}(t'/t).$
• $$\lambda\pm$$ complex: $4(t^2 - t'^2) > \Delta^2.$
• $$\lambda_\pm$$ real: for end state to exist, $4(t^2 - t'^2) < \Delta^2 < 4t^2.$
• With boundary condition $$\Psi_0 = 0$$, $\Psi_j = (\lambda^j_+ - \lambda^j_-) \Psi.$
• Special case: $$N$$ finite, $$\Delta = 0$$ and $$t=t'$$,
• Two solutions: $\Psi_L = \begin{pmatrix} \varphi_1 \\ 0 \\ \vdots \\ 0 \end{pmatrix},\quad \Psi_R = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ \varphi_N \end{pmatrix},$ where $\varphi_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix},\quad \varphi_N = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}.$
###### Structure of Solution (One-Dimensional)
• Hamiltonian: invariant under parity, $H = A \sin(k_x a) \alpha_x + \qty(\Delta - 4B \sin^2 \frac{k_x a}{2})\beta.$
• Dispersion relations: doubly degenerate, $E_\pm = \pm \sqrt{A^2 \sin^2 (k_x a) + \qty(\Delta - 4B \sin^2 \frac{k_x a}{2})^2}.$
• The two occupied states of $$E<0$$ are time-reversal partners.
• One of the occupied solutions: $\psi_1 = \begin{pmatrix} { -\frac{A \sin k_x a}{\sqrt{2E_+ (E_+ + \Delta - 4B \sin^2 (k_x a/2))}} } \\ 0 \\ 0 \\ { \frac{\Delta - 4B \sin^2(k_x a/2) + E_+}{\sqrt{2E_+ (E_+ + \Delta - 4B \sin^2 (k_x a/2))}} } \end{pmatrix}.$
• Time-reversal invariant momenta: $\Gamma_1 = k = 0$ and $\Gamma_2 = k = \frac{\pi}{a}.$
• Eigenvalues under parity: $\bra{\psi_1} P \ket{\psi_1} = \operatorname{sign}\qty(-\Delta + 4B \sin^2 \frac{\Gamma_i a}{2}).$
• Eigenvalues at time-reserval invariant momenta: $\delta\vert_{1} = \operatorname{sign}(-\Delta),\quad \delta\vert_{2} = \operatorname{sign}(-\Delta + 4B).$
• Parity changes sign at $$\Delta = 0$$ and $$\Delta = 4B$$, where the gap closes.
• $$Z_2$$ index: $$\nu$$ defined by $(-1)^\nu = \operatorname{sign}(\Delta) \operatorname{sign}{(\Delta - 4B)}.$
• Topologically nontrivial: $$\nu = 1$$ for $0 < \Delta^2 < 4\Delta B.$
##### Two-Dimensinal Lattice Model
###### Integer Quantum Hall Effect
• Effective Hamiltonian: $H = \vb*{d}(\vb*{k}) \cdot \sigma,$ where \begin{align*} d_x &= A \sin k_x a, \\ d_y &= A \sin k_y a, \\ d_z &= \Delta - 4B \sin^2 \frac{k_x a}{2} - 4B \sin^2 \frac{k_y a}{2}. \end{align*}
• Dispersion relations: $E_{k,\pm} = \pm\abs{\vb*{d}(\vb*{k})}.$
• For zeros to exist: i.e. topological phase transition,
• $$\Delta = 0$$ with $k_x a = k_y a = 0.$
• $$\Delta = 4B$$ with $k_x a = 0,\quad k_y a = \pi,\quad \text{or}\quad k_x a = \pi = \pi,\quad k_y a = 0.$
• $$\Delta = 8B$$ with $k_x a = k_y a = \pi.$
• Edge states: infinite ribbon along $$x$$ while open boundary along $$y$$,
• Hamiltonian: $$k$$-space on $$x$$ and site on $$y$$, \begin{align*} H(k_x) &= \sum_{j=1}^N c^\dagger_{k_x,j} h_{j,j}(k_x) c_{k_x,j} \\ &\phantom{{}={}} + \sum_{j=1}^{N-1} \qty[c^\dagger_{k_x,j} h_{j,j+1}(k_x) c_{k_x,j+1} + c^\dagger_{k_x,j+1}h_{j+1,j}(k_x)c_{k_x,j}], \end{align*}{} where \begin{align*} h_{j,j}(k_x) &= A\sin k_x \sigma_x + \qty(\Delta - 2B - 4B \sin^2\frac{k_x a}{2}) \sigma_z, \\ h_{j,j+1}(k_x) &= B\sigma_z - \frac{i}{2} A\sigma_y, \\ h_{j+1,j}(k_x) &= B\sigma_z + \frac{i}{2}A\sigma{y}. \end{align*}{}
• Problem reduced: to one-dimensional case for a specific $$k_x$$.
###### Quantum Spin Hall Effect
• Time reversal pair glued together: \begin{align*} H_{\mathrm{QSHE}} &= \begin{pmatrix} \vb*{d}(\vb*{k})\cdot \sigma & 0 \\ 0 & -\vb*{d}(\vb*{k})\cdot \sigma \end{pmatrix} \\ &= A \sin (k_x a) \sigma_x \otimes \sigma_0 + A \sin (k_y a) \sigma_y \otimes \sigma_0 \\ &{\phantom{{}={}}} + \qty(\Delta - 4B \sin^2\frac{k_x a}{2} - 4B \sin^2 \frac{k_y a}{2}) \sigma_z \otimes \sigma_0. \end{align*}
• Edge states persist: checked numerically, even when spin-orbit coupling introduced.
###### Structure of Solution (Two-Dimensional)
• Hamiltonian: $H = A \sum_{i=x,y} \sin(k_i a)\alpha_i + \qty(\Delta - 4B \sum_{i=x,y} \sin^2 \frac{k_i a}{2})\beta.$
• Dispersion relations: doubly degenerate, $E_\pm = \pm \sqrt{A^2 \sum_{i=x,y} \sin^2 k_i a + \qty(\Delta - 4B \sum_{i=x,y} \sin^2 \frac{k_i a}{2})^2}.$
• The two occupied states of $$E<0$$ are time-reversal partners.
• One of the occupied solutions: $\psi_1 = \begin{pmatrix} \frac{-A(\sin (k_x a) - i\sin (k_y a))}{\sqrt{2E_+(E_+ + \Delta - 4B(\sin^2(k_x a/2) + \sin^2(k_y a/2))) }} \\ 0 \\ 0 \\ \frac{\Delta - 4B(\sin^2(k_x a/2) + \sin^2(k_y a/2)) + E_+}{\sqrt{2E_+(E_+ + \Delta - 4B(\sin^2(k_x a/2) + \sin^2(k_y a/2))) }} \end{pmatrix}.$
• Eigenvalues under parity: $\bra{\psi_1} P \ket{\psi_1} = \operatorname{sign}\qty(-\Delta + 4B \sum_{i=x,y} \sin^2 \frac{\Gamma_i a}{2}).$
• Eigenvalues at time-reversal invariant momenta: \begin{align*} \delta\vert_{(0,0)} &= -\operatorname{sign}(\Delta), \\ \delta\vert_{(0,\pi)} &= \operatorname{sign}(-\Delta + 4B), \\ \delta\vert_{(\pi,0)} &= \operatorname{sign}(-\Delta + 4B), \\ \delta\vert_{(\pi,\pi)} &= \operatorname{sign}(-\Delta + 8B). \end{align*}
• $$Z_2$$ index: $$\nu$$ defined by $(-1)^\nu = \operatorname{sign}(\Delta) \operatorname{sign}(-\Delta + 4B)^2 \operatorname{sign} (-\Delta + 8B).$
• Topologically nontrivial: $$\nu=1$$ for $0 < \Delta^2 < 8\Delta B.$
##### Three-Dimensional Lattice Model
• $$k$$-Space Hamiltonian: $H = A \sum_{i} \alpha_i \sin(k_i a) + \beta \qty(\Delta - 4B \sum_{i} \sin^2 \frac{k_i a}{2}).$
• Dispersion: $E_{k,\pm} = \pm \sqrt{A^2 \sum_{i} \sin^2(k_i a) + \qty(\Delta - 4B \sum_{i} \sin^2 \frac{k_i a}{2})^2}.$
• For zeros to exist: i.e. topological phase transition:
• $$\Delta = 0$$,
• $$\Delta = 4B$$,
• $$\Delta = 8B$$.
• Surface states: semi-infinite $$x$$-$$y$$ plane,
• Fourier transformation on $$x$$ and $$y$$: \begin{align*} c_{k_x,k_y,j_z} &= \frac{1}{\sqrt{N_x N_y}} \sum_{j_x, j_y} c_{j_x,j_y,j_z} \exp\qty[i(k_x j_x + k_y j_y)], \\ c_{j_x,j_y,j_z} &= \frac{1}{\sqrt{N_x N_y}} \sum_{k_x, k_y} c_{k_x,k_y,k_z} \exp\qty[-i(k_x j_x + k_y j_y)]. \end{align*}{}
• $$c_{k_x,k_y,j_z}$$ is a four-component spinor.
• One-dimensional effective Hamiltonian: \begin{align*} H(k_x,k_y) &= \sum_i c^\dagger_{k_x,k_y,j_z} \epsilon(k_x,k_y) c_{k_x,k_y,j_z} \\ &{\phantom{{}={}}} + \sum_i \qty[c^\dagger_{k_x,k_y,j_z+1} \qty(i\frac{A}{2}\alpha_z - 2B\beta) c_{k_x,k_y,j_z} + \mathrm{h.c.}], \end{align*} where $\epsilon(k_x,k_y) = (A \sin k_x \alpha_x + A \sin k_y \alpha_y) + \qty(\Delta - 2B - 4B\sum_{i=x,y} \sin^2 \frac{k_i a}{2}) \beta.$
• Surface state found by exact diagonalization.
###### Structure of Solution (Three-Dimensional)
• Hamiltonian: $H = A \sum_{\alpha=x,y,z} \sin(k_\alpha a) \alpha_\alpha + \qty(\Delta - 4B \sum_{\alpha = x,y,z} \sin^2 \frac{k_a a}{2})\beta.$
• Dispersion relations: doubly degenerate, $E_\pm = \pm \sqrt{A^2 \sum_{i=x,y,z} \sin^2 k_i a + \qty(\Delta - 4B \sum_{i=x,y,z} \sin^2 \frac{k_i a}{2})^2}.$
• The two occupied states of $$E<0$$ are time-reversal partners.
• One of the occupied solutions: $\psi_1 = \begin{pmatrix} \frac{-A(\sin k_x a - i \sin k_y a)}{\sqrt{2E_+(E_+ + \Delta - 4B( \sum_{\alpha=x,y,z}\sin^2(k_\alpha a/2) )) }} \\ \frac{A \sin k_z}{\sqrt{2E_+(E_+ + \Delta - 4B( \sum_{\alpha=x,y,z}\sin^2(k_\alpha a/2) )) }} \\ 0 \\ \frac{\Delta - 4B \sum_{\alpha=x,y,z} \sin^2(k_\alpha a/2) + E_+}{\sqrt{2E_+(E_+ + \Delta - 4B( \sum_{\alpha=x,y,z}\sin^2(k_\alpha a/2) )) }} \end{pmatrix}.$
• Eigenvalued under parity: $\bra{\psi_1} P \ket{\psi_1} = \operatorname{sign}\qty(-\Delta + 4B \sum_{\alpha=x,y,z} \sin^2 \frac{\Gamma_i a}{2}).$
• Eigenvalues at time-reversal invariant momenta: \begin{align*} \delta\vert_{(0,0,0)} &= -\operatorname{sign}(\Delta), \\ \delta\vert_{(0,0,\pi)} &= \delta\vert_{(0,\pi,0)} = \delta\vert_{(\pi,0,0)} = \operatorname{sign}(-\Delta + 4B), \\ \delta\vert_{(\pi,\pi,0)} &= \delta\vert_{(\pi,0,\pi)} = \delta\vert_{(0,\pi,\pi)} = \operatorname{sign}(-\Delta + 8B), \\ \delta\vert_{(\pi,\pi,\pi)} &= \operatorname{sign}(-\Delta + 12B). \end{align*}
• $$Z_2$$ index: a tuple of $$\mathbb{Z}_2$$ elements, \begin{align*} (-1)^{\nu_1} &= \operatorname{sign}(\Delta) \operatorname{sign}(\Delta - 8B), \\ (-1)^{\nu_2} &= \operatorname{sign}(\Delta) \operatorname{sign}(\Delta - 8B), \\ (-1)^{\nu_3} &= \operatorname{sign}(\Delta) \operatorname{sign} (\Delta - 8B), \\ (-1)^{\nu_1'} &= \operatorname{sign}(\Delta - 4B) \operatorname{sign}(\Delta - 12B), \\ (-1)^{\nu_0} &= (-1)^{\nu_1 + \nu'_1}. \end{align*}
• Topological indices: \begin{align*} (\nu_0;\nu_1,\nu_2,\nu_3) &= (0;0,0,0),\quad \text{for}\quad \Delta < 0, \\ (\nu_0;\nu_1,\nu_2,\nu_3) &= (1;1,1,1),\quad \text{for}\quad 0 < \Delta < 4B, \\ (\nu_0;\nu_1,\nu_2,\nu_3) &= (0;1,1,1),\quad \text{for}\quad 4B < \Delta < 8B, \\ (\nu_0;\nu_1,\nu_2,\nu_3) &= (1;0,0,0),\quad \text{for}\quad 8B < \Delta < 12B, \\ (\nu_0;\nu_1,\nu_2,\nu_3) &= (0;0,0,0),\quad \text{for}\quad \Delta > 12B. \end{align*}

#### One-Dimensional Cases

##### Thouless Charge Pump: Rice-Mele Model
• Hamiltonian:
• Site: $H = h_{\mathrm{st}}(t) \sum_i (-1)^i c^\dagger_i c_i + \qty(\frac{1}{2} \sum_{i=1}^N \qty[t_0 + \delta(t) (-1)^i] c^\dagger_i c_{i+1} + \mathrm{h.c.}).$ where \begin{align*} \delta(t) &= \delta_0 \cos \frac{2\pi t}{T}, \\ h_{\mathrm{st}}(t) &= h_0 \sin \frac{2\pi t}{T}. \end{align*}
• $$k$$-space: $H = \sum_k \begin{pmatrix} a^\dagger_k \\ b^\dagger_k \end{pmatrix} \vb*{d}(k,t) \cdot \vb*{\sigma} \begin{pmatrix} a_k \\ b_k \end{pmatrix},$ where \begin{align*} d_x(k,t) &= \frac{1}{2}(t_0 + \delta(t)) + \frac{1}{2}(t_0 - \delta(t)) \cos k, \\ d_y(k,t) &= -\frac{1}{2}(t_0 - \delta(t)) \sin k, \\ d_z(k,t) &= h_{\mathrm{s.t.}}(t). \end{align*}
• $$k$$-space annhilation: \begin{align*} a_k &= \frac{1}{\sqrt{N}} \sum_{j\in 2n} c_j e^{-ikj}, \\ b_k &= \frac{1}{\sqrt{N}} \sum_{j\in 2n+1} c_j e^{-ikj}. \end{align*}
• Dispersion relations: $\epsilon_\pm(k,t) = \pm\abs{\mathcal{d}(k,t)}.$
• Evolution of polarization: $\Delta P = n_c e a$ where \begin{align*} n_c &= \int_0^T \dd{t} \int_{\mathrm{BZ}} \frac{\dd{k}}{2\pi} \Omega^n_{k,t} \\ &= -\frac{1}{4\pi} \int \dd{k} \int_0^T \frac{\vb*{d}(k,t) \cdot \qty[\partial_k \vb*{d}(k,t) \times \partial_t \vb*{d}(k,t)]}{\abs{\vb*{d}(k,t)}^3} \\ &= -\operatorname{sign}(t_0 h_0 \delta_0). \end{align*}
##### Fu-Kane Spin Pump
• Hamiltonian:
• Site: $H = h_{\mathrm{st}}(t) \sum_{i,\sigma,\sigma'} (-1)^i c^\dagger_{i,\sigma} \sigma^z_{\sigma \sigma'} c_{i,\sigma'} + \qty(\frac{1}{2} \sum_{i,\sigma} \qty[t_0 + \delta(t) (-1)^i] c^\dagger_{i,\sigma} c_{i+1,\sigma'} + \mathrm{h.c.}).$
• $$k$$-space: $H = \sum_k \begin{pmatrix} \phi^\dagger_{k,\uparrow} & \phi^\dagger_{k,\downarrow} \end{pmatrix} \begin{pmatrix} \vb*{d}_+ \cdot \vb*{\sigma} ^ 0 \\ 0 & \vb*{d}_- \cdot \vb*{\sigma} \end{pmatrix} \begin{pmatrix} \phi_{k,\uparrow} \\ \phi_{k,\downarrow} \end{pmatrix},$ where \begin{align*} d_x(k,t)_\pm &= \frac{1}{2}(t_0 + \delta(t)) + \frac{1}{2}(t_0 - \delta(t)) \cos k, \\ d_y(k,t)_\pm &= -\frac{1}{2}(t_0 - \delta(t)) \sin k, \\ d_z(k,t)_\pm &= \pm h_{\mathrm{s.t.}}(t). \end{align*}
• $$k$$-space creation: $\phi^\dagger_{k,\sigma} = \begin{pmatrix} a^\dagger_{k,\sigma} & b^\dagger_{k,\sigma} \end{pmatrix}.$
• Evolution of polarization: polarizations contributed by opposite spins cancel out, $\Delta P_{\uparrow} = +ea;\quad \Delta P_{\downarrow} = -ea.$

Expand

## CuSeI Compounds

#### Chalcogenido-Halogenido-Bismuthates

• Cu/Bi/Q/I (where Q = S, Se, Te): a rich family.
• Small band gap, tunable by substituting anions with more or less electronegative ones.
• Cu⁺ disordered and mobile + heavy elements + a comparatively high electrical conductivity ⇒ good thermoelectric properties.
##### Cu₄BiSe₄I
• Band gap: (Literature) 0.1meV at low T, 0.3eV at high T (Topological Materials Database) 0.02eV Semiconductor TI(NLC)
• Projection of the unit cell:
• Bi-Se strands: comparison with Te-I strands,
• Conductivity:
• Copper cation conductivity suppressed.
##### CuBi₂Se₃I, Cu₃Bi₆S₁₀I
• Band gap (CuBi₂Se₃I): (Literature) ∼1.2eV Semiconductor
• Band gap (Cu₃Bi₆S₁₀I): (Literature) ∼1.2eV Semiconductor
• Band structure: CuBi₂Se₃I (left) and Cu₃Bi₆S₁₀I (right)
• Thermoelectric of CuBi₂Se₃I: Seeback coefficient 460-575μV/K, electrical conductivity 0.02S/cm, thermal conductivity 0.22W/(m·K).
• Structure of CuBi₂Se₃I: two distinct layers alternating along the b-axis,
• Structure of Cu₃Bi₆S₁₀I: three individual columns,
##### Cu⁎Bi⁎Se⁎Br⁎
• Band gap: Unavailable
• Structure: built up of two alternating types of layered modules, A and B.
• A: consists of paired rods [Bi⁎Se⁎X⁎].
• Copper ion transport along [010] in the S-Br case.
• $$\ce{Cu_{1.5}Bi_{2.64}S_{3.42}Br_{2.58}}{}$$, $$\ce{Cu_{1.57}Bi_{2.37}Se_{2.68}Br_{3.32}}{}$$: Seeback coefficient 280μV/K, electrical resistivity 23Ω/m, thermal conductivity 0.77W/(m·K), $ZT = 1.3\times 10^{-6}.$
• Structure of $$\ce{Cu_{1.5}Bi_{2.64}S_{3.42}Br_{2.58}}{}$$:
• $$\ce{Cu_{1.57}Bi_{4.69}Se_{7.64}I_{0.36}}{}$$, $$\ce{Cu_{2.31}Bi_{5}Se_{8.31}I_{0.69}}{}$$: thermoelectric not given.
• Structure of $$\ce{Cu_{1.57}Bi_{4.69}Se_{7.64}I_{0.36}}{}$$:

#### P⁎Se⁎ Cages

• Cages:
• P₄Se₃ cages (left) and P₄Se₄ cages (right),
• P₈Se₃,
• Possibly high Cu⁺ ion conductivity.
##### (CuCl)P₄Se₃, (CuI)P₄Se₃, (CuBr)₃(P₄Se₃)₂, (CuI)₃(P₄Se₃)₂
• Band gap of (CuI)P₄Se₃: (AFLOW) 1.37eV Semiconductor (Topological Materials Database) 0.82eV Semiconductor
• Cage molecules P₄Se₃ introduced into CuX networks.
• Section of 2D structure of (CuCl)P₄Se₃: [CuCl]ₙ chains along the c-axis,
• Section of 3D structure of (CuBr)₃(P₄Se₃)₂:
• Section of 3D structure of (CuI)₃(P₄Se₃)₂:
• Section of 3D structure of (CuI)P₄Se₃: structured as one-dimensional polymers,
##### (CuI)P₄Se₄
• Band gap: (AFLOW) 1.75eV Semiconductor (Topological Materials Database) 1.61eV Semiconductor
• Cage molecules P₄Se₄ introduced into CuI networks.
• P₄Se₄ strands connected by CuI.
• Layered structure, connected by vdW interactions.
##### (CuI)₂P₈Se₃
• Band gap: (Topological Materials Database) 1.2eV Semiconductor
• Structure:
##### (CuI)₃P₄Se₄
• Band gap: Unavailable
• Cage molecules P₄Se₄ introduced into CuI networks.
• Projection of the crystal structure: stacked along the c-axis,

#### Others

##### CuSeTeCl, CuSeTeBr, CuSeTeI
• Band gap: Unavailable
• Layered structure, with one-dimensional infinite screws of Se and Te.
##### Cu⁎SiS⁎I⁎, Cu⁎GeSe⁎I⁎
• Band gap: Unavailable
• Copper argyrodites, ion conductors.

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## Statistical Physics (I)

#### Boltzmann Distribution

##### Combinatorics
• Counting microscopic states: $\Omega_{\mathrm{M.R.}} = \frac{N!}{\prod_l a_l!} \prod_l \omega_l^{a_l}.$
• Boltzmann distribution: $a_l = \omega_l e^{-\alpha - \beta \epsilon_l}.$
##### The Partition Function
• Partition function: of a single particle, $Z_l = \sum_l w_l e^{-\beta \epsilon_l}.$
• Single particle free energy: $f_1 = -\ln Z_1.$
• Generalized force: $Y = \frac{N}{\beta}\pdv{}{y} \ln Z_1.$
• $\delta \epsilon_l = f_1 \cdot \delta y.$
• $Y = \sum_l a_l f_l.$
• Work done: $\dd{W} = \sum_l a_l \dd{\epsilon_l}.$
• Internal energy: \begin{align*} U &= \frac{N}{Z_1} \sum_l \epsilon_l w_l e^{-\beta\epsilon_l} \\ &= -N \pdv{}{\beta} \ln Z_1 \\ &= N \pdv{f_1}{\beta}. \end{align*}
• Helmholtz free energy: $F = NkTf_1.$
• Entropy: for Boltzmann distribution, $S = Nk\qty(\beta \pdv{f_1}{\beta} - f_1).$
• In terms of microscopic states: $S = k \ln \Omega_{\mathrm{M.B.}}.$
• For Fermi-Dirac and Bose-Einstein distributions, under classical limit $S_{\mathrm{B.E.}} = S_{\mathrm{F.D.}} = k \ln \frac{\Omega_{\mathrm{M.B.}}}{N!}.$
• Condition for classical limit: $e^\alpha = \frac{z_1}{N} \gg 1.$

For the $$(p,V)$$ pair we find $Y = -p = -\frac{N}{\beta} \pdv{}{V}\ln Z_1 = \frac{N}{\beta}\pdv{}{V} f_1.$

##### Application to Ideal Gas
• Nearly independent system: phase space described by the single particle $(x,y,z,p_x,p_y,p_z).$
• Energy levels: $\epsilon = \frac{p^2}{2m}.$
• Single particle partition function: for monoatomic gas, $f_1 = -\ln V + \frac{3}{2}\ln \beta - \frac{3}{2} \ln \qty(\frac{2\pi m}{h^2}).$
• EoS: $p = \frac{NkT}{V}.$
• Internal energy: $U = \frac{3}{2} NkT.$
###### Velocity Distribution
• Maxwell's velocity distribution: for vector, $f(\vb*{v}) = \qty(\frac{m}{2\pi kT})^{3/2} e^{-m\vb*{v}^2/(2kT)}.$
• Maxwell's velocity distribution: for magnitude, $f(v) = \sqrt{\frac{2}{\pi}} \qty(\frac{m}{kT})^{3/2} v^2 e^{-mv^2/(2kT)}.$
• Statistical: \begin{align*} v_{\mathrm{m}} &= \sqrt{\frac{2kT}{m}}, \\ \overline{v} &= \sqrt{\frac{8kT}{m}}, \\ v_{\mathrm{rms}} &= \sqrt{\frac{3kT}{m}}. \end{align*}{}
###### Equipartition Theorem
• For every square mode in the kinetic energy, $\epsilon_{\vb*{p}} = \frac{1}{2} a_i \vb*{p}_i^2,$ where $$a_i$$ may or may not depends on $$\vb*{q}$$, and for every square mode in the potential energy $\frac{1}{2} b_i q_i^2,$ where $$b_i$$ may or may not depends on other $$\vb*{q}_j$$'s, the average value is given by $\overline{\frac{1}{2} c x^2} = \frac{1}{2} kT.$

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This is spam. I am not gonna remove it because 1) this is a problem that many bloggers may encounter; 2) this funny spammer seems to forget to attach his link. See this quora question for why such comments occur.

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## Group Theory in Solid State Physics

#### Space Group

• Symmorphic space group: a semidirect product of its point group with its translation subgroup.
• Nonsymmorphic space group: not symmorphic.
• May contain glide planes, where the translation is not a lattice vector.
##### Representations of Space Groups
• Small representation: small representation for the group of wavevector $$\vb*{k}$$: $D^{\Gamma_i}_{\vb*{k}}(\qty{g_{\vb*{k}} | \vb*{t}}) = e^{i\vb*{k}\cdot \vb*{t}} D^{\Gamma_i}(g_{\vb*{k}}).$

#### Example: Cubic Lattices (Symmorphic Space Group)

##### Brillouin Zones of Cubic Lattices
SC FCC BCC
\begin{align*}\Delta&: \Gamma \rightarrow X \\ \Sigma&: \Gamma \rightarrow M\\ \Lambda&: \Gamma \rightarrow R\end{align*}{} \begin{align*}\Delta&: \Gamma \rightarrow X \\ \Sigma&: \Gamma \rightarrow K\\ \Lambda&: \Gamma \rightarrow L\end{align*}{} \begin{align*}\Delta&: \Gamma \rightarrow H \\ \Sigma&: \Gamma \rightarrow N\\ \Lambda&: \Gamma \rightarrow P\end{align*}{}
$$S: X \rightarrow R$$
#221 #225 #229
##### Symmetry at Γ
• The whole point group $$\mathrm{O}_h$$.
• Large representation.
• Representation: superscript $$\pm$$ denotes parity,
• $$C_4$$ and $$C_4^2$$ around $$\qty{100}{}$$.
• $$C_2'$$ around $$\qty{110}{}$$.
• $$C_3$$ around $$\qty{111}{}$$.
$$\mathrm{O}_h$$ $$E$$ $$3C_4^2$$ $$6C_4$$ $$6C'_2$$ $$8C_3$$ $$i$$ $$3iC_4^2$$ $$6iC_4$$ $$6iC'_2$$ $$8iC_3$$
$$\Gamma_{25}^+$$ $$3$$ $$-1$$ $$-1$$ $$1$$ $$0$$ $$3$$ $$-1$$ $$-1$$ $$1$$ $$0$$
The rest $$9$$ IRs not listed.
##### Symmetry at X
• Symmetry $$D_{4h}{}$$.
• Small representation.
• Representation:
• $$C_{4\parallel}^2$$ around $$\Gamma X$$.
• $$C_{4\perp}^2$$ perpendicular to $$\Gamma X$$.
$$\mathrm{O}_h$$ $$E$$ $$2C_{4\perp}^2$$ $$C_{4\parallel}^2$$ $$2C_{4\parallel}{}$$ $$2C_2$$ $$i$$ $$2iC_{4\perp}^2$$ $$iC_{4\parallel}^2$$ $$2iC_{4\parallel}{}$$ $$2iC_2$$
$$X_1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$X_3$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$X_5$$ $$2$$ $$0$$ $$-2$$ $$0$$ $$0$$ $$2$$ $$0$$ $$-2$$ $$0$$ $$0$$
The rest $$7$$ IRs not listed.
##### Symmetry at M
• Symmetry $$D_{4h}{}$$.
• Small representation.
• Representation: almost the same as that at $$X$$, except that the conjugacy classes are relabeled.
##### Symmetry at R
• Symmetry $$\mathrm{O}_{h}$$.
• Large representation.
• Representation: almost the same as that at $$\Gamma$$, except that the IRs are relabeled.
##### Symmetry Along Δ
• Symmetry $$C_{4v}{}$$.
• Small representation.
• Representation:
$$\mathrm{C}_{4v}{}$$ $$E$$ $$C_4^2$$ $$2C_4$$ $$2iC_4^2$$ $$2iC'_2$$
$$\Delta_1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\Delta_2$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
$$\Delta'_2$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
$$\Delta'_1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$\Delta_5$$ $$2$$ $$-2$$ $$0$$ $$0$$ $$0$$
• Compatibility relation: $\Gamma_{25}^+ \rightarrow (\Delta'_2) \oplus (\Delta_5) \rightarrow (X_3) \oplus (X_5).$
• It's possible that: the band admits a three-fold degeneracy at $$\Gamma$$, where the wavefunctions span a space of dimension $$3$$, transforming under $$\mathrm{O}_h$$ according to the representation of $$\Gamma_{25}^+$$,
• as we move off $$\Gamma$$ along $$\Delta$$, the band splits into two, one of which transforms under $$C_{4v}{}$$ according to $$\Delta'_2$$, another (2-fold degenerate) according to $$\Delta_5$$, and
• as we reach $$X$$, the bands remain split, where the one-fold that originally transforms according to $$\Delta'_2$$ now transforms according to $$X_3$$, while the two-fold that originally transforms according to $$\Delta_5$$ now transforms according to $$X_5$$.
##### Symmetry Along Σ
• Symmetry $$C_{2v}{}$$.
• Small representation.
• Representation:
$$\mathrm{C}_{2v}{}$$ $$E$$ $$C_2$$ $$iC_4^2$$ $$iC_2$$
$$\Sigma_1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\Sigma_2$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$\Sigma_3$$ $$1$$ $$-1$$ $$-1$$ $$1$$
$$\Sigma_4$$ $$1$$ $$-1$$ $$1$$ $$-1$$
• Compatibility relation: $\Gamma^+_{25} \rightarrow (\Sigma_1) \oplus (\Sigma_2 \oplus \Sigma_3) \rightarrow (M_1) \oplus (M_5).$
• It's possible that: the band admits a three-fold degeneracy at $$\Gamma$$, where the wavefunctions span a space of dimension $$3$$, transforming under $$\mathrm{O}_h$$ according to the representation of $$\Gamma_{25}^+$$,
• as we move off $$\Gamma$$ along $$\Sigma$$, the band splits into three, one of which transforms under $$C_{2v}{}$$ according to $$\Sigma_1$$, one $$\Sigma_2$$, and another $$\Sigma_3$$, and
• as we reach $$M$$, the bands suddenly become partly degenerate, where the one-fold that originally transforms according to $$\Sigma_1$$ now transforms according to $$M_1$$, while the two that originally transforms separately according to $$\Sigma_2$$ and $$\Sigma_3$$ now become degenerate and combine to give rise to a space of dimension $$2$$ that transforms according to $$M_5$$.
##### Symmetry Along Λ
• Symmetry $$C_{3v}{}$$.
• Small representation.
• Representation:
$$\mathrm{C}_{3v}{}$$ $$E$$ $$2C_3$$ $$3iC_2$$
$$\Lambda_1$$ $$1$$ $$1$$ $$1$$
$$\Lambda_2$$ $$1$$ $$1$$ $$-1$$
$$\Lambda_3$$ $$2$$ $$-1$$ $$0$$
• Compatibility relation: $\Gamma_{25}^+ \rightarrow (\Lambda_1 \oplus \Lambda_3) \rightarrow (R_{15}).$
• It's possible that: the band admits a three-fold degeneracy at $$\Gamma$$, where the wavefunctions span a space of dimension $$3$$, transforming under $$\mathrm{O}_h$$ according to the representation of $$\Gamma_{25}^+$$,
• as we move off $$\Gamma$$ along $$\Delta$$, the band splits into two, one of which transforms under $$C_{3v}{}$$ according to $$\Lambda_1$$, another (2-fold degenerate) according to $$\Lambda_3$$, and
• as we reach $$R$$, the bands suddenly become degenerate, returning to a three-fold degeneracy (i.e. a space of dimension $$3$$) that transforms according to $$R_{15}$$.
##### Symmetry Along S
• Symmetry $$C_{2v}{}$$.
• Small representation.
• Representation: the same as that long $$\Sigma$$.

#### Compatibility Relations

It seems better to take $$\vb*{k}$$ as an external parameter like $$\vb*{E}$$ in the Stark effect. Therefore, the band structure is just the energy levels as we enumerate through all $$\vb*{k}$$'s.

##### Tight-binding Basis
• With the $$s$$-orbital: transforming like identity.
• In SC: $\Gamma_1 \rightarrow \Lambda_1 \rightarrow R_1 \rightarrow X_1 \rightarrow \Delta_1 \rightarrow \Gamma_1.$

• Lower band: $R_1 \rightarrow \Lambda_1 \rightarrow \Gamma_1^+ \rightarrow \Delta_1 \rightarrow X_1.$
• Upper band: $(R_{12}^+) \oplus (R_1^+) \rightarrow (\Lambda_3) \oplus (\Lambda_1) \rightarrow \Gamma_{25}^+ \rightarrow (\Delta'_2) \oplus (\Delta_5) \rightarrow (X'_2) \oplus (X_5).$

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## Physicists Suck at Maths

#### Abuse of Terminology

##### Representation Theory
• Physicists always say that a vector space $$V$$ is a representation of a group $$G$$. This completely makes no sense to someone who really knows representation theory. A representation is always a homomorphism from $$G$$ to linear transformations on $$V$$. Specifying $$V$$ is not equivalent to specifying the representation. However, physicists usually work with the natural representation $g\psi(x) = \psi(g^{-1} x)$ where $V = \set{\ket{\psi}}.$ In this sense, specifying $$V$$ does determine the representation.
##### Algebra
• Physicists tend to write every set in pair form $$(x,y)$$ as a direct product, though in many cases they are not.

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## Representation of Lie Groups

SU(3)は君の嘘

#### Lie Groups

##### Haar Measure
• Left Haar measure: left-invariant volume form $$\sigma$$.
• Wedge product of the duals of Lie algebras.
• Guaranteed that $\int_{gU} \sigma = \int_U \sigma.$
• Right Haar measure: right-invariant volume form $$\sigma$$.
• Haar measure: both left- and right-invariant.
• $A(1) = \set{\begin{pmatrix} x & y \\ 0 & 1 \end{pmatrix}}.$
• Lie algebras (left-invariant vector fields): $X = x\pdv{}{x},\quad Y = x\pdv{}{y}.$
• Dual: $\sigma = \frac{\dd{x}\wedge \dd{y}}{x^2}.$
##### Lie Algebra
• Casimir operator: an operator which commutes with all elements of a Lie group.
• Casimir operators are represented by scalar matrices.
• See Schur's lemma.

#### T(1)

##### Lie Algebra of T(1)
• Generators: $$P$$.
• Infinitesimal transform: $T(\dd{x}) = \mathbb{1} - iP\dd{x}.$
• Exponential map: $T(x) = e^{-iPx}.$
• IR of the Lie algebra: $P = p.$
##### Irreducible Representations of T(1)
• IRs: $U^p(x)\ket{p} = e^{-ipx}\ket{p}.$
• Haar measure: $$\dd{x}$$.
• Orthonormality relation: $\int_{-\infty}^{\infty} \dd{x} U^\dagger_p(x) U^{p'}(x) = N \delta(p-p').$
• Completeness relation: $\int_{-\infty}^{\infty} \dd{p} U^p(x) U^\dagger_p(x') = N \delta(x-x').$
##### Conjugate Basis Vectors to T(1)
• $$\ket{O}$$ expanded as $\bra{p}\ket{O} = 1.$
• $\ket{x} = U(x)\ket{O}.$
• Basis transformation $\ket{x} = \frac{1}{2\pi} \int_{-\infty}^{\infty} \ket{p} e^{-ipx} \dd{p} \Leftrightarrow \ket{p} = \int_{-\infty}^{\infty} \ket{x} e^{ipx} \dd{\phi}.$
• Wavefunction transformation $\psi(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \psi(p) e^{ipx} \dd{p} \Leftrightarrow \psi(p) = \int_{-\infty}^{\infty} \psi(x) e^{-ipx} \dd{x}.$
• Action of $$P$$: $P\ket{x} = i\dv{}{x} \ket{x}.$

#### SO(2)

##### Lie Algebra of SO(2)
• Generators: $$J$$.
• Infinitesimal transform: $R(\dd{\phi}) = \mathbb{1} - iJ\dd{\phi}.$
• Exponential map: $R(\phi) = e^{-i\phi J}.$
• IR of the Lie algebra: $$m\in \mathbb{Z}$$, $J = m.$
##### Irreducible Representations of SO(2)
• IRs: $$m\in \mathbb{Z}$$, $U^m(\phi)\ket{m} = e^{-im\phi}\ket{m}.$
• Haar measure: $$\dd{\phi}$$.
• Orthonormality relation: $\frac{1}{2\pi} \int_0^{2\pi} \dd{\phi} U^\dagger_n(\phi) U^m(\phi) = \delta^m_n.$
• Completeness relation: $\sum_n U^n(\phi) U^\dagger_n(\phi') = \delta(\phi - \phi').$
• Multivalued representations: $U_{n/m}(\phi) = e^{-in\phi/m}.$
##### Conjugate Basis Vectors to SO(2)
• $$\ket{O}$$ expanded as $\bra{m}\ket{O} = 1.$
• $\ket{\phi} = U(\phi)\ket{O}.$
• Basis transformation $\ket{\phi} = \sum_m \ket{m} e^{-im\phi} \Leftrightarrow \ket{m} = \frac{1}{2\pi} \int_0^{2\pi} \ket{\phi} e^{im\phi} \dd{\phi}.$
• Wavefunction transformation $\psi(\phi) = \sum_m e^{im\phi} \psi_m \Leftrightarrow \psi_m = \frac{1}{2\pi}\int_0^{2\pi} \psi(\phi) e^{-im\phi} \dd{\phi}.$
• Action of $$J$$: $J\ket{\phi} = i\dv{}{\phi} \ket{\phi}.$

#### SO(3)

##### Parametrization of SO(3)
• Angle-and-axis: $R(\psi,\hat{\vb*{n}}) = R(\psi,\theta,\phi).$
• Euler angles: body frame and laboratory frame, $R(\alpha,\beta,\gamma) = R_{z''}(\gamma) R_{y'}(\beta) R_z(\alpha) = R_z(\alpha) R_y(\beta) R_z(\gamma).$
• Conversion: \begin{align*} \phi &= \frac{\pi + \alpha - \gamma}{2}, \\ \tan\theta &= \frac{\tan (\beta/2)}{\tan (\gamma+\alpha)/2}, \\ \cos\psi &= 2\cos^2 \frac{\beta}{2}\cos^2\qty(\frac{\alpha+\gamma}{2}) - 1. \end{align*}{}
##### Lie Algebra of SO(3)
• Generators: $$J_1,J_2,J_3$$, $[J_k, J_l] = i\epsilon_{klm} J^m.$
• Infinitesimal transform: $R(\dd{\vb*{\psi}}) = \mathbb{1} - i \vb*{J}\cdot \dd{\vb*{\psi}}.$
• Exponential map: $R(\psi\vb*{n}) = e^{-i\psi \vb*{J}\cdot \vb*{n}} = e^{-i\alpha J_3} e^{-i\beta J_2} e^{-i\gamma J_3}.$
• Casimir operator: $$J^2$$.
• IR of the Lie algebra: \begin{align*} J^2\ket{jm} &= j(j+1)\ket{jm}, \\ J_z \ket{jm} &= m\ket{jm}, \\ J_\pm \ket{jm} &= \sqrt{j(j+1) - m(m\pm 1)} \ket{j,m\pm 1}. \end{align*}
##### Irreducible Representations of SO(3)
• IRs: $U^j(\alpha,beta,\gamma)\ket{jm} = D^j(\alpha,\beta,\gamma){^{m'}}{_m} \ket{jm'}.$
• $D^j(\alpha,\beta,\gamma){^{m'}}{_m} = e^{-i\alpha m'} d^{j}(\beta){^{m'}}{_m} e^{-i\gamma m}.$
• $d^j(\beta){^{m'}}{_m} = \bra{jm'} e^{-i\beta J_y} \ket{jm}.$
• Single-valued for $$j$$ a positive integer. Double-valued for $$j$$ a half-odd integer.
• $$d$$ is a real matrix. $$D$$ is an special unitary matrix.
• Complex conjugation: $D^*(\alpha,\beta,\gamma) = Y D(\alpha,\beta,\gamma) Y^{-1}.$
• $Y = D[R_2(\pi)],\quad Y^{m'}{_m} = \delta^{m'}_{-m}(-1)^{j-m}.$
• Symmetry $d^j(\beta){^{m'}}{_m} = d^j(\beta){^{-m'}}{_{-m}}(-1)^{m' - m}.$
• Characters: $\chi^j(\psi) = \frac{\sin(j+1/2)\psi}{\sin (\psi/2)}.$
##### Spherical Harmonics
• $Y_{lm}(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}} \qty[D^l(\phi,\theta,0){^m}{_0}]^*.$
##### Conjugate Basis Vectors to SO(3)
###### Position Space
• $$\ket{r\hat{\vb*{z}}}$$ expanded as $\bra{r\hat{\vb*{z}}}\ket{E,l,m'} = \delta_{m' 0} \tilde{\psi}_{El}(r).$
• $\ket{r\hat{\vb*{n}}} = e^{-i\phi J_z} e^{-i\theta J_y} \ket{r\hat{\vb*{z}}}.$
• Wavefunction $\psi_{Elm}(r,\theta,\phi) = \tilde{\psi}_{El}(r)[D^l(\phi,\theta,0){^m}{_0}]^* = \psi_{El}(r) Y_{lm}(\theta,\phi).$
###### Momentum Space
• Basis transformation $\ket{p,l,m} = \int \dd{\Omega} \ket{p,\theta,\phi}Y_{lm}(\theta,\phi) \Leftrightarrow \ket{p,\theta,\phi} = \sum_{l,m} \ket{p,l,m} Y^*_{lm}(\theta,\phi).$

#### Clebsch-Gordan Coefficients

##### Clebsch-Gordan Coefficients of SO(3)
• Direct product representation: $U(R) \ket{m,m'} = \ket{n,n'} D^j(R){^n}{_m} D^{j'}(R){^{n'}}{_{m'}}.$
• Single-valued if $$j+j'$$ is an integer.
• Double-valued if $$j+j'$$ is a half-odd integer.
• Direct product representation of the Lie algebra: $J^{j\times j'}_n = J_n^j \otimes \mathbb{1}^{j'} + \mathbb{1}^j \otimes J_n^{j'}.$
• Eigenvectors labeled by $\qty{J^2, J_z}.$
• Ket of max $$j$$ and $$m$$ matches: $\ket{J = j+j', M = j + j'} = \ket{m=j, m'=j'},$
• $$\ket{J,M}$$ generated by lowering: $J_- \ket{J,M} = \sqrt{J(J+1) - M(M-1)} \ket{J, M-1}.$
• There are only two states of $$M=J-1$$. One of them is already obtained, and the other is $\ket{J-1, J-1}.$
• By repeating the process, we obtain all the $$\ket{JM}$$ kets.
• CG coefficients: \begin{align*} \ket{JM} &= \ket{mm'}\langle mm'(jj')JM\rangle, \\ \ket{mm'} &= \ket{JM}\langle JM(jj')mm'\rangle. \end{align*}{}
• Convention: \begin{align*} \langle mm'(jj') JM\rangle &\in \mathbb{R}, \\ \langle j,J-j(jj')JJ\rangle &> 0. \end{align*}{}
• Properties of CG coefficients:
• Orthogonality: $\sum_{mm'} \langle JM(jj')mm'\rangle \langle mm' (jj')J'M'\rangle = \delta^{J}_{J'}\delta^M_{M'}.$
• Completeness: $\sum_{JM} \langle mm'(jj')JM\rangle \langle JM(jj')\rangle nn'\rangle = \delta^m_n \delta^{m'}_{n'}.$
• Symmetry: $\langle mm'(jj')JM\rangle = (-1)^{j-J+m'}\langle M,-m'(Jj')jm\rangle \sqrt{(2J+1)(2j+1)}.$
• Reduction of products: \begin{align*} D^j(R){^m}{_n} D^{j'}(R){^{m'}}{_{n'}} &= \sum_{J,M,N} \langle mm'(jj')JM\rangle D^J(R){^M}{_N} \langle JN(jj')nn'\rangle, \\ \delta^J_{J'}D^J(R)^M{_{M'}} &= \sum_{mm'nn'} \langle JM(jj')mm'\rangle D^j(R){^m}{_n} D^{j'}(R)^{m'}{_{n'}} \langle nn'(jj')J'M'\rangle. \end{align*}{}

#### Applications

##### Partial Wave Expansion
• T-matrix: $\bra{\vb*{p}_f} T \ket{\vb*{p}_i} = \bra{p,\theta,\phi} T \ket{p,0,0}.$
• $$T$$ is invariant under rotation: $\bra{p,l,m} T \ket{p,l',m'} = \delta_{ll'} \delta_{mm'} T_l(p).$
• Partial wave expansion: $\bra{\vb*{p}_f} T \ket{\vb*{p}_i} = \sum_l \frac{2l+1}{4\pi} T_l(E) P_l(\cos\theta).$
##### Transformation of Wavefunctions
• $\psi'(\vb*{x}) = \psi(R^{-1} \vb*{x}).$
###### Transformation of Wavefunctions under T
• $\psi'(\vb*{x}) = \psi(R^{-1} \vb*{x}) = e^{i\vb*{p}\cdot R^{-1}\vb*{x}} = e^{i(R\vb*{p})\cdot \vb*{x}}.$
###### Transformation of Wavefunctions under SO(3)
• $\psi'(x) = \psi_{El}(r) Y_{lm'}(\hat{\vb*{x}}) D^l[R]{^{m'}}{_m}.$
###### Pauli Spinor Wavefunctions
• Expansion of wavefunctions: $\ket{\psi} = \sum_\sigma \int \ket{\vb*{x},\sigma} \psi^\sigma(\vb*{x}) \dd{^3 x}.$
• Transformation of basis under rotation: $U[R]\ket{\vb*{x},\sigma} = \ket{R\vb*{x},\lambda} D^{1/2}[R]{^\lambda}{_\sigma}.$
• Transformation of wavefunction under rotation: $\psi'{^\lambda}(\vb*{x}) = D^{1/2}[R]^{\lambda}{_\sigma} \psi^\sigma(R^{-1} \vb*{x}).$
##### Irreducible Wavefunctions, Irreducible Tensors, and Wigner-Eckart Theorem
• Irreducible wavefunctions and fields: a set of multi-component functions $\set{\phi^m(\vb*{x}) | m = -j,\cdots,j}$ that transform under rotation as $\phi \xrightarrow{R} \phi';\quad \phi'^m(\vb*{x}) = D^j[R]{^m}{_n} \phi^n(R^{-1} x).$
• Irreducible tensors: a set of multi-component operators $\set{O^s_\lambda | \lambda = -s,\cdots,s}$ that transform under rotation as $U(R) O^s_\lambda U(R)^{-1} = \sum_{\lambda'} O^s_{\lambda'} D^s(R){^{\lambda'}}{_\lambda}.$
• Lie brackets of irreducible tensors: \begin{align*} [J^2,O^s_\lambda] &= s(s+1) O^s_{\lambda}, \\ [J_z,O^s_\lambda] &= \lambda O^s_\lambda, \\ [J_\pm,O^s_\lambda] &= \sqrt{s(s+1) - \lambda(\lambda\pm 1)}O^s_{\lambda\pm 1}. \end{align*}
• Vector operators: multi-component operators that transforms under rotation as $U[R]X_i U[R]^{-1} = X_j R{^j}{_i}.$
• Lie brackets of vector operators: $[J_k, A_l] = i\epsilon^{klm} A_m.$
• $$n$$-th rank tensor: $[J_k, T_{l_1\cdots l_n}] = i\qty{\epsilon^{kl_1 m}T_{ml_2\cdots l_n} + \cdots + \epsilon^{kl_n m}T_{l_1\cdots l_{n-1}m}}.$
###### Pauli Spinor Field
• $$1/2$$-representation: $U[R] \Psi^\sigma(\vb*{x}) U[R]^{-1} = \Psi^\lambda(R\vb*{x}) D^{1/2}(R^{-1}){^\sigma}{_\lambda}.$

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## Topology in Physics (I)

#### Typical Lie Groups

##### SO(3)
• Homeomorphism between:
• $$\mathrm{SO}(3)$$;
• Unit solid ball in $$\mathbb{R}^3$$ with antipodal points identified.

#### Fundamental Groups

##### Fundamental Group of SO(3)
• Two kinds of loops:
• Loops lies in the interior of the unit ball.
• Loops connects two antipodal points.
• Fundamental group $$\mathbb{Z}^2$$.
##### Fundamental Group of Torus

The following code demonstrates why the fundamental group of a torus is abelian.

torus[t_, p_] := {2*Cos[t] + Cos[t]*Cos[p], 2*Sin[t] + Sin[t]*Cos[p],
Sin[p]};
torusStroke[t_, p_] := {2*Cos[t] + 1.01 Cos[t]*Cos[p],
2*Sin[t] + 1.01 Sin[t]*Cos[p], 1.01 Sin[p]};

Manipulate[
Show[ParametricPlot3D[
torus[2 $Pi] t, 2 \[Pi] p], {t, 0, 1}, {p, 0, 1}], ParametricPlot3D[ torusStroke[2 \[Pi] (t - 0.125), 2 \[Pi] (Piecewise[{{u/(1 - u) t, t < 1 - u}, {(1 - u)/u (t - 1) + 1, t > 1 - u}}])], {t, 0, 1}, PlotStyle -> {Thickness -> 0.01}, PlotPoints -> 200, ColorFunction -> (Hue[ Piecewise[{{#4/(1 - u)*0.5, #4 < 1 - u}, {0.5 + (#4 - (1 - u))/u*0.5, #4 > 1 - u}}]] &)]], {u, 0.01, 0.99}]  #### Fibre Bundles • Tangent bundles are not necessarily trivial: $$TS^2$$ is not $$S^2 \times \mathbb{R}^2$$, since vector fields on $$S^2$$ should vanish somewhere and therefore $$S^2\times \vb{v}$$ for $$\vb{v}\neq 0$$ could not be mapped to a section. Expand ## Index Theorems #### Elliptic Operators and Fredholm Operators • Operators: maps to sections \[ D: \Gamma(M,E) \rightarrow \Gamma(M,F).$
##### Elliptic Operators
• Differential operator: with $$E$$ and $$F$$ being complex vector bundles over $$M$$, $D: \Gamma(M,E) \rightarrow \Gamma(M,F).$
• Notation: multi-index, \begin{align*} M &= (\mu_1, \cdots, \mu_m),\quad \text{where}\quad \mu_j\in \mathbb{Z},\quad \mu_j \ge 0, \\ \abs{M} &= \mu_1 + \cdots + \mu_m, \\ D_M &= \pdv{^{\abs{M}}}{x^M}. \end{align*}
• Operator of order $$N$$: $[Ds(x)]^\alpha = \sum_{\substack{\abs{M}\le N \\ 1\le a\le k}} A^{M\alpha}{_a}(x) D_M s^a(x).$
• $$N=1$$: e.g. the Dirac operator, $[D\psi(x)]^\alpha = i(\gamma^\mu){^\alpha}{_\beta} \partial_\mu \psi^\beta(x) + m\psi^\alpha(x).$
• $$N=2$$: e.g. the Laplacian operator.
• Symbol of $$D$$: a $$k\times k'$$ matrix, given by $\sigma(D,\xi) = \sum_{\abs{M} = N} A^{M\alpha}{_a}(x) \xi_M.$
• Independently of coordinates: $\sigma(D,\xi) s = \frac{1}{N!} D(f^N \tilde{s})\vert_p.$
• Section $$\tilde{s}$$ satisfies $\tilde{s}(p) = s,$
• $$f$$ satisfies $$f(p) = 0$$,
• $\dd{f(p)} = \xi.$
• Elliptic operator: $$\sigma(D,\xi)$$ invertible for each $$x\in M$$ and $$\xi \neq 0$$.
• Composites, powers and roots of elliptic operators are elliptic.
• Let $\sigma(D,\xi) = A_{11}\xi^1 \xi^1 + 2A_{12} \xi^1 \xi^2 + A_{22}\xi^2 \xi^2.$ Then $$D$$ is elliptic if and only if $\sigma(D,\xi) = 1$ is an ellipse in $$\xi$$-space.
• $$x^\mu \in \mathbb{R}^m$$. $$E$$ and $$F$$ be real line bundles over $$\mathbb{R}^m$$.
• Laplacian: $\bigtriangleup = \pdv{^2}{(x^1)^2} + \cdots + \pdv{^2}{(x^m)^2}.$
• Symbol: $\sigma(\Delta,\xi) = \sum_\mu (\xi_\mu)^2.$
• The d'Alembertian is not invertible.
##### Fredholm Operators
• Adjoint: with inner products defined on $$E$$ and $$F$$, $D^\dagger: \Gamma(M,F) \rightarrow \Gamma(M,E),$ defined by $\langle s', Ds \rangle_F = \langle D^\dagger s', s\rangle_E.$
• Cokernel: $\operatorname{coker} D = \Gamma(M,F)/\operatorname{im} D.$
• Fredholm operator: an elliptic operator $$D$$ whose kernels and cokernels are finite dimensional.
• Analytical index: \begin{align*} \operatorname{ind} D &= \dim \ker D - \dim \operatorname{coker} D \\ &= \dim \ker D - \dim \ker D^\dagger. \end{align*}{}
##### Elliptic Complexes
• Elliptic complex: a sequence of Fredholm operators $\cdots \rightarrow \Gamma(M,E_{i-1}) \xrightarrow{D_{i-1}} \Gamma(M,E_i) \xrightarrow{D_i} \Gamma(M,E_{i+1}) \xrightarrow{D_{i+1}} \cdots$ that are nilpotent, i.e. $D_i \circ D_{i-1} = 0.$
• Laplacian: $\bigtriangleup = D_{i-1} D^\dagger_{i-1} + D^\dagger_i D_i.$
• Hodge decomposition: $s_i = D_{i-1} s_{i-1} + D^\dagger_i s_{i+1} + h_i,$ where $$h_i$$ is in the kernel of $$\Delta_i$$.
• De Rham cohomology groups: $H^i(E,D) = \ker D_i / \operatorname{im} D_{i-1}.$
• $\dim H^i(E,D) = \dim \mathrm{Harm}^i(E,D).$
• Index of elliptic complex: $\operatorname{ind} D = \sum_{i=0}^m (-1)^i \dim H^i(E,D) = \sum_{i=0}^m (-1)^i \dim \ker \bigtriangleup_i.$
• Special case: for $0 \overset{i}{\hookrightarrow} \Gamma(M,E) \xrightarrow{D} \Gamma(M,F) \xrightarrow{\varphi} 0,$ the index is given by $\dim D = \dim\ker D - \dim\operatorname{coker} D^\dagger.$
• Even bundle and odd bundle: $E_+ = \oplus_r E_{2r};\quad E_- = \oplus_r E_{2r+1}.$
• Laplacians: \begin{align*} \Delta_+ &= A^\dagger A = \oplus_r \bigtriangleup_{2r}, \\ \Delta_- &= A A^\dagger = \oplus_r \bigtriangleup_{2r+1}. \end{align*}
• Index: $\operatorname{ind}(E_\pm, A) = \dim\ker \bigtriangleup_+ - \dim\ker\bigtriangleup_- = \sum_r (-1)^r \dim\ker\bigtriangleup_r = \operatorname{ind}(E,D).$
• De Rham complex $$\Omega(M)$$ given by $0 \xrightarrow{i} \Omega^0(M) \xrightarrow{\dd{}} \Omega^1(M) \xrightarrow{\dd{}} \cdots \xrightarrow{\dd{}} \Omega^m(M) \xrightarrow{\dd{}} 0.$
• Index identified with Euler characteristic: \begin{align*} \mathrm{ind}(\Omega^*(M), \dd{}) &= \sum_{r=0}^m (-1)^r \dim H^r(M;\mathbb{R}) \\ &= \chi(M) = \int_M e(TM) \\ &= \sum_{r=0}^m (-1)^r \dim \ker \bigtriangleup_r. \end{align*}{}

#### Atiyah-Singer Index Theorem

• $$(E,D)$$ be an elliptic complex over an $$m$$-dimensional compact manifold $$M$$ without a boundary. $\operatorname{ind}(E,D) = (-1)^{m(m+1)/2}\int_M \left.\operatorname{ch}\qty({\oplus_r (-1)^r E_r})\frac{\operatorname{Td}(TM^{\mathbb{C}})}{e(TM)}\right\vert_{\mathrm{vol}}.$
• The division by $$e(TM)$$ is carries out at the formal level.
• Only $$m$$-forms are picked up in the integration.
• Let $\Gamma(M,E) \xrightarrow{D} \Gamma(M,F)$ be a two-term elliptic complex. The index of $$D$$ is given by \begin{align*} \operatorname{ind} D &= \dim\ker D - \dim\ker D^\dagger \\ &= (-1)^{m(m+1)/2} \int_M \left.(\operatorname{E} - \operatorname{F})\frac{\operatorname{Td}(TM^{\mathbb{C}})}{e(TM)}\right\vert_{\mathrm{vol}}. \end{align*}{}

Expand
 Index Index

## Representation of Finite Groups

ねぇ、あなたは何色になりたい？

#### Group Representation

##### Definitions
• Representation of a group: a homomorphism from a hroup $$G$$ to a group of operators $$U(G)$$ on a linear vector space $$V$$.
• Dimension: $$\dim V$$.
• Faithful: $$U$$ is a isomorphism.
• Degenerate: not faithful.
• Equivalent representation: two representations related by a similarity transformation, i.e. $U'(G) = S U(G) S^{-1}.$
• Characters of representation: $$\chi(g)$$ defined by $\chi(g) = \Tr U(g) = \sum_i D(g){^i}{_i}.$
• $$D(g)$$: the matrix of $$U(g)$$.
• Invariant subapce: $$V_1 \subset V$$ such that $U(g)V_1 \in V_1,\quad \forall g.$
• Minimal/Proper: $$V_1$$ doesn't contain any non-trivial invariant subspace.
• Irreducible representation: $$U(G)$$ that doesn't admit non-trivial invariant in $$V$$.
• Reducible: not irreducible.
• In matrix form: $D(g) = \begin{pmatrix} D_1(g) & * \\ 0 & D_2(g) \end{pmatrix}.$
• Fully reducible/Decomposible: (if $$V$$ is Hermitian) orthogonal complement of the invariant subspace also invariant.
• Unitary representation:
• $$V$$ is an Hermitian space, and
• $$U(g)$$ are unitary for all $$g$$.
• Direct sum representation: $$V = V_1 \oplus V_2$$ is a sum of two invariant subspaces.
• Notation:
• $$\abs{G}$$: order of $$G$$.
• $$\mu,\nu$$: labels for inequivalent IR of $$G$$.
• $$d_\mu$$: dimension of $$\mu$$-IR.
• $$D^\mu(g)$$: the matrix of $$g$$ in the $$\mu$$-IR, with respect to an orthonormal basis.
• $$\chi^\mu_i$$: character of class $$i$$ in the $$\mu$$-IR.
• $$\abs{C_i}$$: number of elements in the class $$\xi_i$$.
• $$n_C$$: number of conjugacy classes.
• $D{^i}{_j} = \bra{e^i}D\ket{e_j}.$
##### Theorems
• Reducible unitary representations are fully reducible.
• Let $$V$$ be the invariant subspace, and $\Pi^W = U^\dagger \Pi U.$ be the projection onto $$U^{-1} V$$. From $U\Pi^V = \Pi^V U \Pi^V.$ we find $\dim (V^\perp\cap W) = \tr ((1 - \Pi^V) \Pi^W) = 0.$
• Every representation on a Hermitian space is equivalent to a unitary representation.
• We redefine the inner product by $(x,y) = \sum_g \bra{D(g)x}\ket{D(g)y}$ and note that $$D(g)$$ is unitary under the new orthonormal basis.
• Reducible $$\Rightarrow$$ fully reducible.
• Schur's Lemma (I): if $$U(G)$$ is an IR and $AU(g) = U(g)A,\quad \forall A,$ then $$A = \lambda \mathbb{1}$$.
• We prove only for Hermitian $$A$$. It's easily seen that the eigenspace of $$A$$ is an invariant subspace of $$U(g)$$.
• Schur's Lemma (II): if $$U(G)$$ and $$U'(G)$$ are two IRs on $$V$$ and $$V'$$ respectively, and $A: V'\rightarrow V$ satisfies $AU'(g) = U(g)A,\quad \forall g\in G,$ then either
• $$A=0$$, or
• $$V$$ is isomorphic to $$V'$$ and $$U(G)$$ is equivalent to $$U'(G)$$.
• Proof:
• $$\operatorname{ker} A$$ is invariant under $$U'$$ and therefore $$A$$ is injective.
• $$\operatorname{im} A$$ is invariant under $$U$$ and therefore $$A$$ is surjective.
• IR of any abelian group is of dimension one.
• Orthonormality of IR: for unitary $$D_\mu$$, $\frac{d_\mu}{\abs{G}}\sum_g D^\dagger_\mu(g){^k}{_i}D^\nu(g){^j}{_l} = \delta^\nu_\mu \delta^j_i \delta^k_l.$
• $D^\dagger_\mu(g){^k}{_i} = \qty[D^\mu(g){^i}{_k}]^*.$
• Proof:
• Let $X_i^j = \sum_g D_\mu^\dagger(g) \ket{e_i}\bra{e^j} D^\nu(g).$
• We have $X_i^j D^\nu(g) = D_\mu^\dagger(g) X_i^j.$ Therefore, either $$X^j_i$$ is $$0$$ or $$D_\mu$$ is equivalent to $$D_\nu$$.
• If $$D_\mu$$ is equivalent to $$D_\nu$$, we have $X_i^j = \lambda \mathbb{1}.$
• Taking trace on both sides we get the desired result.
• The vectors $\ket{\mu,i,j} = \begin{pmatrix} D_\mu(g_1){^i}{_j} & \cdots & D_\mu(g_{\abs{G}}){^i}{_j} \end{pmatrix}{}$ are orthogonal. Each IR contribute $$d_\mu^2$$ such vectors.
• The vectors are complete: $\sum_\mu d_\mu^2 = \abs{G}.$
• The number of IRs equal to the number of conjugacy classes.
• Completeness of IR: $\sum_{\mu,l,k} \frac{d_\mu}{\abs{G}} D^\mu(g){^l}{_k} D^\dagger_\mu(g'){^k}{_l} = \delta_{gg'}.$
• Orthonormality of group characters: $\sum_i \frac{\abs{C_i}}{\abs{G}} \chi^{\dagger i}_\mu \chi^\nu_i = \delta^\nu_\mu.$
• $\chi^{\dagger i}_{\mu} = (\chi^\mu_i)^*.$
• Completeness of group characters: $\sum_\mu \frac{\abs{C_i}}{\abs{G}} \chi^\mu_i \chi^{\dagger j}_\mu = \delta^j_i.$
• Combining the completeness relation of IR with the following lemma:
• Let $$U^\mu(G)$$ be an IR. $\sum_{h\in C_i} U^\mu(h) = \frac{\abs{C_i}}{d_\mu} \chi^\mu_i \mathbb{1}.$
• LHS commutes with any $$U^h(g)$$.
• In the decomposition $U(G) = \bigoplus_{\mu} a_\mu U^\mu(G),$ we have $a_\nu = \sum_i \frac{\abs{C_i}}{\abs{G}} \chi^{\dagger i}_\nu \chi_i.$
• $\mathrm{IR} \quad \Leftrightarrow \quad \sum_i \abs{C_i} \abs{\chi_i}^2 = \abs{G}.$
##### Real, Complex, and Quaternionic Representations
• Real representation: $$\rho$$ equivalent to a real representation.
• In this case, we always have $$\rho \cong \overline{\rho}$$.
• Quaternionic representation: $$\rho \cong \overline{\rho}$$ but $$\rho$$ is not a real representation.
• Complex representation: $$\rho\ncong \overline{\rho}$$.
• Frobenius-Schur indicator: $\frac{1}{\abs{G}} \sum_{g\in G} \chi(g^2) = \begin{cases} 1, & \text{if } g \text{ is real}, \\ 0, & \text{if } g \text{ is complex}, \\ -1, & \text{if } g \text{ is quaternionic}. \end{cases}{}$

For the quaternion group $Q_8 = \langle \overline{e},i,j,k \vert \overline{e}^2 = e, i^2=j^2=k^2=ijk=\overline{e} \rangle$ that consists of eight elements $\qty{e,i,j,k,\overline{e},\overline{i},\overline{j},\overline{k}},$ the representation given by $\rho(i) = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix},\quad \rho(j) = w = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is quternionic since $w\rho(g) w^{-1} = \overline{\rho}(g)$ for all $$g$$ but $$\rho$$ could not be sent to a real representation in that way.

#### Typical Representations

##### The Regular Representation
• $$V$$: the group ring.
• Basis: $$\qty{g_j}$$.
• $\mathrm{R}(g_i){^m}_{j} = \delta^{g_m}_{g_i g_j}.$
• $g_i g_j = g_m \Delta^m_{ij}.$
• $\chi^{\mathrm{R}}_i = \begin{cases} \abs{G}, & C_i = \qty{e}, \\ 0, & \text{otherwise}. \end{cases}$
• $\mathrm{R}(G) = \bigoplus_\mu d_\mu U^\mu(G).$
##### Direct Product Representations
• Direct product of vector spaces: $W = U \otimes V$ consists of $\set{\hat{\vb*{w}}_k | k = (i,j); i = 1,\cdots,d_U; j = 1,\cdots,d_V}.$
• $$\hat{\vb*{w}}_k$$ is the formal product $\hat{\vb*{w}}_k = \hat{\vb*{u}}_i \cdot \hat{\vb*{v}}_j.$
• $\bra{w^{k'}}\ket{w_{k}} = \delta^{k'}_{k} = \delta^{i'}_i \delta^{j'}_j.$
• $W = \qty{x^k \ket{w_k}}.$
• $\bra{x}\ket{y} = x^\dagger_k y^k, \quad x^\dagger_k = (x^k)^*.$
• Direct product of operator: $D = A \otimes B.$
• $D\ket{w_k} = \ket{w_{k'}} D^{k'}{_k}.$
• $D^{k'}{_k} = A^{i'}{_i} B^{j'}{_j}.$
• Direct product of representation: $D^{\mu\times \nu}(g) = D^{\mu}(g)\otimes D^{\nu}(g).$
• Character of direct product: $\chi^{\mu\times \nu} = \chi^\mu \chi^\nu.$
• Decomposition $D^{\mu\times \nu}(G) = \bigoplus_{\lambda} a_\lambda D^\lambda.$
• Invariant subspace: $W^\lambda_\alpha,$
• $$\lambda$$: which representation acts on this invariant subspace?
• $$\alpha$$: if there are multiple invariant subspace, distinguish them with this index.
• Basis: new orthonormal basis, $\ket{w^\lambda_{\alpha l}} = \sum_{ij} \ket{w_{i,j}}\langle i,j(\mu\nu)\alpha,\lambda,l \rangle.$
• $$(i,j)$$ as row index.
• $$(\alpha,\lambda,l)$$ as column index.
• The new basis is demanded to be orthornormal (as the old basis is). We have large freedom to mix the basis in each invariant subspace. Therefore, the new basis (and hence the CG coefficients) are not uniquely determined.
• The transformation is unitary.
• Action on old basis: $U(g)\ket{w_{i,j}} = \ket{w_{i',j'}} D^\mu(g)^{i'}{_i} D^\nu(g)^{j'}{_j}.$
• Action on new basis: $U(g)\ket{w^\lambda_{\alpha l}} = \ket{w^\lambda_{\alpha l'}} D^\lambda(g)^{l'}{_l}.$
• Clebsch-Gordan coefficients: $\langle i,j(\mu\nu)\alpha,\lambda,l \rangle.$
• Orthonormality of CG coefficients: $\sum_{\alpha,\lambda,l} \langle i',j'(\mu,\nu)\alpha,\lambda,l\rangle \langle \alpha,\lambda,l (\mu,\nu) i,j\rangle = \delta^{i'}_i \delta^{j'}_j.$
• $\langle \alpha,\lambda,l(\mu\nu) i,j \rangle = \langle i,j(\mu,\nu)\alpha,\lambda,l\rangle^*.$
• Completeness of CG coefficients: $\sum_{i,j} \langle \alpha',\lambda',l'(\mu,\nu) i,j\rangle \langle i,j(\mu,\nu)\alpha,\lambda,l\rangle = \delta^{\alpha'}_{\alpha}\delta^{\lambda'}_{\lambda}$
• Reduction of product representation: \begin{align*} D^\mu(g)^{i'}{_i} D^\nu(g)^{j'}{_j} &= \bra{i'j'}\ket{\alpha,\lambda,l'} D^\lambda(g)^{l'}{_l}\bra{\alpha,\lambda,l}\ket{i,j}, \\ \delta^{\alpha'}_{\alpha}\delta^{\lambda}_{\lambda'} D^\lambda(g)^{l'}{_l} &= \bra{\alpha',\lambda',l'}\ket{i',j'} D^\mu(g)^{i'}{_i} D^\nu(g)^{j'}{_j} \bra{i,j}\ket{\alpha,\lambda,l}. \end{align*}{}
##### Subduced Representations
• If $$T$$ is a representation of a group $$G$$, and
• $$G_1\subset G$$ is a subgroup, then
• $$T$$ subduces the representation $$T_1$$ of $$G_1$$ according to $\color{orange} T_1(g) = T(g),\quad \forall g\in G_1.$
• When this is done for all the IRs of $$G$$, we have a subduction table.
• Subduction of IRs may be reducible.
• Subduced representation denoted $\color{orange} \rho_{G_1} = \rho_G \downarrow G_1.$
##### Induced Representations

Let $$H$$ be a subgroup of $$G$$ and $$\rho: H \rightarrow \mathrm{GL}(V)$$ be a representation of $$H$$. The induced representation $$\mathrm{Ind}^G_H(\pi)$$ acts on $W = \bigoplus_{i=1}^n g_i V,$ where $$g_i \in G$$ are representative elements of the quotient $$G/H$$, by $g\cdot g_i v_i = g_{j(i)} h_i v_i.$ The induced representation is denoted $\color{orange} \rho_G = \rho\uparrow G.$

Let $G = D_{12} = \bra{\rho,\sigma}\ket{\rho^6 = \sigma^2 = 1, \rho\sigma = \sigma \rho^{-1}},$ and $H = \qty{1,\sigma,\rho^3, \sigma\rho^3}.$ A representation of $$H$$ is given by $\rho(\sigma) = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \rho(\rho^3) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \rho(\sigma\rho^3) = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}.$ Note that $$[G:H] = 3$$ and therefore the induced representation consists of $$3\times 3$$ matrices of $$2\times 2$$ blocks. $\rho_G(\rho) = \begin{pmatrix} 0 & 0 & \rho(\rho^3) \\ \mathbb{1} & 0 & 0 \\ 0 & \mathbb{1} & 0 \end{pmatrix},\quad \rho_G(\sigma) = \begin{pmatrix} \rho(\sigma) & 0 & 0 \\ 0 & 0 & \rho(\sigma\rho^3) \\ 0 & \rho(\sigma\rho^3) & 0 \end{pmatrix}.$ We pick representative elements $g_1 = e,\quad g_2 = \rho,\quad g_3 = \rho^2.$ Let's figure out how $$g\in G$$ acts on $$\bigoplus g_i V_i$$, i.e. $g g_i v_i = g_? h_? v_i.$ That is to seek a $$g_?$$ such that $g_?^{-1} g_i v_i = h_? \in H.$ We take $$g = \rho$$ for example, and find that \begin{align*} g_2^{-1} g g_1 &= 1 \in H, \\ g_3^{-1} g g_2 &= 1 \in H, \\ g_1^{-1} g g_3 &= \rho^3 \in H. \end{align*} Therefore, we are now ready to answer the following questions:

1. How does $$g = \rho$$ act on $$g_1 V_1$$?
• $g_1\cdot v_1 \rightarrow g_2\cdot \mathbb{1} v_1.$
2. How does $$g = \rho$$ act on $$g_2 V_2$$?
• $g_2 \cdot v_2 \rightarrow g_3\cdot \mathbb{1} v_2.$
3. How does $$g = \rho$$ act on $$g_3 V_3$$?
• $g_3 \cdot v_3 \rightarrow g_1\cdot \rho(\rho^3) v_3.$

Now we obtain the $$\rho_G$$ above.

This could be written as $[\rho_G(g)]_{\alpha i,\beta j} = [\tilde{\rho}(g_\alpha^{-1} g g_\beta)]_{ij},$ where $[\tilde{\rho}(g)]_{ij} = \begin{cases} [\rho(g)]_{ij}, & \text{if } g\in H, \\ 0, & \text{else}. \end{cases}$

• The induced representation is always in block form, i.e. $$\rho(g)$$ is always a matrix of blocks.
• Big indices, labelling $$g_i\in G/H$$: $$g$$ sends $$g_i V$$ to which $$g_? V$$?
• Small indices, labelling $$v\in V$$: how $$h = g_i^{-1}g$$ acts on $$V$$?
##### Tensor Representations: Symmetry Dictates Selection Rule
• $$U(G)$$ be a representation not necessarily reducible.
• Acting on $V = \bigoplus_{\mu,\alpha} V^\alpha_\mu.$
• Irreducible tensors: a set of operators $\set{O^\mu_i | i = 1,\cdots,d_\mu}$ acting on $$V$$ and transforming as $U(g) O^\mu_i U(g)^{-1} = O^\mu_j D^\mu(g){^j}{_i}.$
• $$O^\mu_i \ket{e^\nu_j}$$ transforms like direct product: $U(g) O^\mu_i \ket{e^\nu_j} = O^\mu_k \ket{e^\nu_j} D^\mu(g){^k}{_i} D^\nu(g){^l}{_j}.$
• New basis $O^\mu_i \ket{e^\nu_j} = \sum_{\alpha,\lambda,l} \langle \alpha,\lambda,l(\mu,\nu)i,j \rangle.$
• Wigner-Eckart theorem: $\bra{e^l_\lambda} O^\mu_i \ket{e^\nu_j} = \sum_\alpha \langle \alpha,\lambda,l(\mu,\nu)i,j \rangle \bra{\lambda} O^\mu \ket{\nu}_\alpha.$
• Reduced matrix element: $\bra{\lambda} O^\mu \ket{\nu}_\alpha = \frac{1}{\abs{d_\lambda}} \sum_k \bra{e^k_\lambda}\ket{w^\lambda_{\alpha k}}.$

#### Obtaining Irreducible Representations

##### Irreducible Vectors
• $$U(G)$$ be a representation not necessarily reducible.
• Acting on $V = \bigoplus_{\mu,\alpha} V^\alpha_\mu.$
• Irreducible set: a set of vectors $\set{\hat{\vb*{e}}^\mu_i | i = 1,\cdots,d_\mu}$ transforming according to $U(g)\ket{e^\mu_i} = \ket{e^\mu_j} D^\mu(g){^j}{_i}.$
• If
• $$\qty{\hat{\vb*{u}}^\mu_i}$$ and $$\qty{\hat{\vb*{v}}^\nu_j}$$ are two sets of irreducible basis, and
• $$\mu$$ and $$\nu$$ labels different IR, then
• the two invariant subspaces spanned by these bases are orthogonal to each other.
• Proof: prove that $\bra{v^j_\nu}\ket{u^\mu_i} = \frac{1}{\abs{G}} \sum_g \bra{v^j_\nu} U^\dagger(g) U(g) \ket{u^\mu_i}$ vanished.

States of different $$l$$ (living in different spaces of the IRs of $$\mathrm{SO}(3)$$) doesn't overlap.

• Rotates one; kills the rest. $P^j_{\mu i} = \frac{d_\mu}{\abs{G}} \sum_g D^{-1}_\mu(g){^j}{_i} U(g).$
• For any $$\ket{x} \in V$$, the sets of vectors $\set{\ket{e^\mu_i} = P^j_{\mu i}\ket{x}, i = 1,\cdots,d_\mu}.$ transform irreducibly according to $$D^\mu$$ (as long as they are not all null). $U(g)P^j_{\mu i}\ket{x} = P^j_{\mu k}\ket{x} D^\mu(g){^k}{_i}.$
• $$P^j_{\mu i}$$ kills and rotates in the invariant subspace. $P^j_{\mu i} \ket{e^\nu_{\alpha k}} = \ket{e^\nu_{\alpha i}} \delta^\nu_\mu \delta^j_k.$
• Consecutive kill-and-rotate: $P^j_{\mu i} P^l_{\nu k} = \delta^\nu_\mu \delta^j_k P^l_{\mu i}.$
• Inversion $U(g) = \sum_{\mu,i,j} P^{j}_{\mu i} D^\mu(g){^i}{_j}.$
• $U(g) P^l_{\nu k} = \sum_i P^l_{\nu i} D^\nu(g){^i}{_k}.$
• Projection operators:
• Onto the basis vector $$\hat{\vb*{e}}^\mu_i$$ $P_{\mu i} = P^{i}_{\mu i}.$
• Onto the irreducible invariant space $$V_\mu$$ spanned by $$\qty{\hat{\vb*{e}}^\mu_i}$$ $P_\mu = \sum P_{\mu i}.$
• Completeness: $\sum_\mu P_\mu = \mathbb{1}.$

The IRs of one-dimensional discrete translation group are given by $$\qty{e^{ikna}}$$. From a local (Wannier) state $$\phi$$ we generate a one-dimensional invariant (Bloch) basis $\ket{\psi_k} = \sum_n T_n\ket{\phi} e^{ikna}.$

#### Representations of the Permutation Groups

##### One-Dimensional Representations
• Trivial representation: $$E$$.
• Generated by the symmetrizer $$s$$, essentially idempotent and primitive.
• Parity representation: $D(p) = (-1)^{\operatorname{sign} p}.$
• Generated by the anti-symmetrizer $$a$$, essentially idempotent and primitive.
• These are the only one-dimensional representations.
##### Partitions and Young Diagrams
• Partition:
• Partition $\lambda = \qty{\lambda_1,\cdots,\lambda_r}$ of $$n$$:
• $\lambda_i \ge \lambda_{i+1}.$
• $\sum \lambda_i = n.$
• Partition $$\lambda = \mu$$ if $\lambda_i = \mu_i,\quad \forall i.$
• $$\lambda>\mu$$ or $$\lambda<\mu$$ by lexicographic order.
• Parition to Young diagram:
• $4 = 2 + 1 + 1$ represented as follows.
• $4 = 3 + 1$ as follows.
• Partitions of $$n$$ $$\leftrightarrow$$ conjugacy classes of $$S_n$$ $$\leftrightarrow$$ IRs of $$S_n$$.
• Young tableau, normal tableau, and standard tableau:
• A tableau is obtained by filling the squares of a Young diagram with $$1,\cdots,n$$.
• Normal Young tableau: $$1,\cdots,n$$ appear in order from left to right and from the top row to the bottom row.
• Denoted by $$\Theta_\lambda$$.
• Examples as follows.
 1 2 3 4
and
 1 2 3 4
• An arbitrary tableau obtained from $$\Theta_\lambda$$ by permutation $\Theta^p_\lambda = p\Theta_\lambda.$
• Standard Young tableau: number of each row appear increasing to the right, and in each column appear to the bottom.
##### Symmetrizers and Anti-symmetrizers of Young Tableaux

Let $$\Theta_2$$ be given as follows.

 1 2 3

The irreducible symmetrizer is given by $e_2 = \frac{[e+(12)][e-(31)]}{4}.$ Let $r_2 = (23) e_2.$ Then $$\qty{e_2,r_2}$$ span an invariant subspace of the group ring, giving rise to the two-dimensional IR of $$S_3$$.

##### Symmetry Classes of Tensors
• Horizontal and vertical permutations: $$\Theta^p_\lambda$$ given,
• Horizontal permutations: $$\qty{h^p_\lambda}$$ leave invariant the sets of numbers appearing in the same row of $$\Theta^p_\lambda$$.
• $$\qty{h_\lambda}$$ form a subgroup of $$S_n$$.
• Vertical permutations: $$\qty{v^p_\lambda}$$ leave invariant the sets of numbers appearing in the same column of $$\Theta^p_\lambda$$.
• $$\qty{v_\lambda}$$ form a subgroup of $$S_n$$.
• Symmetrizer, anti-symmetrizer, and irreducible/Young symmetrizer: associated with the Young tableau $$\Theta^p_\lambda$$,
• Symmetrizer: summation over horizontal permutations, $s_\lambda^p = \sum_{h} h^p_\lambda.$
• Idempotent.
• Anti-symmetrizer: summation over vertical permutations, $a^p_\lambda = \sum_v (-1)^{v_\lambda} v_\lambda^p.$
• Idempotent.
• Irreducible/Young symmetrizer: sum over horinzontal and vertical permutations: $e^p_\lambda = \sum_v (-1)^{v_\lambda} h^p_\lambda v^p_\lambda.$
• Primitive idempotent.
• Properties of symmetrizers: superscript $$p$$ omitted.
• For $$\lambda$$ in the group ring $$\mathbb{C}S_n$$, these exists $$\xi\in\mathbb{C}$$ such that $s_\lambda r a_\lambda = \xi e_\lambda.$
• There exists $$\xi \in \mathbb{\eta}$$ such that $e_\lambda^2 = \eta e_\lambda.$
• $$e^p_\lambda$$ associated with $$\Theta_\lambda^p$$ is a primitive idempotent. It generates an IR of $$S_n$$ on the group ring.
• The representations generated by $$e^p_\lambda$$ and $$e_\lambda$$ are equivalent.
• Two irreducible symmetrizers $$e_\lambda$$ and $$e_\mu$$ generate inequivalent IR if $$\lambda \neq \mu$$.
• If $$\lambda \neq \mu$$, then $e^p_\mu e^q_\lambda=0$ for all $$p,q\in S_n$$.
• The left ideals generated by irreducible symmetrizers $$\qty{e_\lambda}$$ associated with normal Young tableaux $$\qty{\Theta_\lambda}$$ are all IRs of $$S_n$$.
• The direct sum (yes they are lineraly-independent) of the left ideals generated by all standard tableaux is the group ring $$\mathbb{C}S_n$$.
##### Irreducible Representations of the Permutation Groups
• $$\qty{i}$$ denotes a collection of indices. We work on the space $$V^n_m$$, i.e. $$n$$ copies of a $$m$$-dimensional space $$V$$.

• Direct product of operator acting on direct product of spaces: $g\ket{\qty{i}} = \ket{\qty{j}} D(g)^{\qty{j}}{_{\qty{i}}}.$

• $(gx)^{\qty{j}} = D(g)^{\qty{j}}{_{\qty{i}}} x^{\qty{i}}.$
• $(Dg)^{\qty{j}}{_{\qty{i}}} = g^{{j_1}}{_{i_1}} \cdots g^{{j_n}}{_{i_n}}.$
• Action of permutation: $p\ket{\qty{i}} = \ket{p^{-1}\qty{i}}.$

• $(px)^{\qty{i}} = x^{p\qty{i}}.$
• $D(p)^{\qty{j}}{_{\qty{i}}} = \delta^{p\qty{j}}{_{\qty{i}}}.$
• The two set of matrices $$\qty{D(p)}$$ of where $$p\in S_n$$ and $$\qty{D(g)}$$ where $$g\in \mathrm{GL}(m)$$ commute with each other.

• Tensors of symmetry class $$\lambda$$: $T_\lambda(\alpha) = \set{re_\lambda \ket{\alpha} | r\in \mathbb{C}S_n},$

• $$\lambda$$ the Young diagram given.
• $$T_\lambda$$ is an irreducible invariant subspace with respect to $$S_n$$.
• If $$T_\lambda(\alpha)$$ is not empty, then the realization of $$S_n$$ on $$T_\lambda(\alpha)$$ coincides with the IR generated by $$e_\lambda$$.
• $T'_\lambda = \set{e_\lambda\ket{\alpha} | \ket{\alpha} \in V^n_m}.$
• With Young symmetrizers, tensors may be completely decomposed into irreducible tensors $\ket{\lambda, \alpha, a},$ where
• $$\lambda$$: the symmetry class (Young diagram),
• $$\alpha$$: distinct set of tensors $$T_\lambda(\alpha)$$ invariant under $$S_n$$,
• $$a$$: basis element whtin each $$T_\lambda(\alpha)$$.
• Indices of basis chosen in a way such that $p\ket{\lambda,\alpha,a} = \ket{\lambda,\alpha,b} D_\lambda(p){^{b}}{_{a}}.$
• Tensors of symmetry $$\Theta^p_\lambda$$: $\set{e^p_\lambda\ket{\alpha}}.$

• $$\Theta^p_\lambda$$ the Young tableau given.
• Two tensor subspaces irreducibly invariant with respect to $$S_n$$ and belonging to the same symmetry class either overlap completely or are disjoint.

• Two irreducible invariant tensor subspaces corresponding to two distinct symmetry classes are necessarily disjoint.

• If $$g\in \mathrm{GL}_m$$ and $$\qty{\ket{\lambda,\alpha,a}}$$ is a set of basis tensors, then the subspaces $$T'\lambda(a)$$ spanned by $$\qty{\ket{\lambda,\alpha,a}}$$ with fixed $$\lambda$$ and $$a$$ are invariant under $$\mathrm{GL}_m$$. The representations of $$\mathrm{GL}_m$$ on $$T'_\lambda(a)$$ are independent of $$a$$. $g\ket{\lambda,\alpha,a} = \ket{\lambda,\beta,a} D_\lambda(g){^\beta}{_\alpha}.$

• The representations of $$\mathrm{GL}_m$$ on the subspace $$T'_\lambda(a)$$ of $$V^n_m$$ are irreducible.

Let $$\Theta_{\lambda=s}$$ be the Young tableau associated with the total symmetrizer, then $$T_\lambda(\alpha)$$ is a totally symmetric tensor, $$S_n$$ acts on which irreducibly (and trivially).

There is one and only one independent totally anti-symmetric tensor of rank $$n$$ in $$n$$-dimensional space.

For second-rank tensors on $$m$$-dimensional $$V$$, we have $e_s\ket{ij} = \frac{\ket{ij} + \ket{ji}}{2},$ while $e_a\ket{ij} = \frac{\ket{ij} - \ket{ji}}{2}.$

Let $$\Theta_m$$ be given as follows.

 1 2 3

The irreducible symmetrizer is given by $e_m = \frac{[e+(12)][e-(31)]}{4}.$ With \begin{align*} \ket{\alpha} &= \ket{++-}, \\ \ket{\beta} &= \ket{--+}. \end{align*} we obtain \begin{align*} e_m\ket{\alpha} = \ket{m,\alpha,1} &= \frac{2\ket{++-} - \ket{-++} - \ket{+-+}}{4}, \\ (23) e_m\ket{\alpha} = \ket{m,\alpha,2} &= \frac{2\ket{+-+} - \ket{-++} - \ket{++-}}{4}; \\ e_m\ket{\beta} = \ket{m,\beta,1} &= \frac{2\ket{--+} - \ket{+--} - \ket{-+-}}{4}, \\ (23) e_m\ket{\beta} = \ket{m,\beta,2} &= \frac{2\ket{-+-} - \ket{+--} - \ket{--+}}{4}. \end{align*}{} The action of $$S_3$$ on the subspace generated by these two vectors gives rise to its two-dimensional representation.

#### Application: Degeneracy in Central Potential

An Example of Symmetry Group Acting on Wavefunctions

• The states $\set{\ket{nl,m}|m = -l,\cdots,l}$ could be related by operators in $$\mathrm{O(3)}$$.

#### Application: Crystal Field Theory

An Example of Compatibility Relations

• My free atom has five $$d$$-orbitals. Without an external potential, the atom enjoys the full $$\mathrm{O}(3)$$ symmetry. Therefore, its five $$d$$-orbitals are degenerate since $$\mathrm{O}(3)$$ connects them.
• Now my atom is placed in a crystal of $$\mathrm{T_d}$$ symmetry. The symmetries that $$d$$-orbitals enjoy are reduced. Therefore, they are not guaranteed to be degenerate since two orbitals may not be be connected by an operation in $$\mathrm{T_d}$$.
• In the worst case, when there is no symmetry present, we no longer have any degeneracy.
• The case has not come to the worst. We still have a non-trivial symmetry group and we wanna know if there could be any degenracy. We note the following one-to-one correspondence: $\text{degeneracy} \Leftrightarrow \text{invariant subspace} \Leftrightarrow \text{IR}.$
• We take the following procedures to determine the possible degeneracies:
1. Determine how $$\mathrm{T_d}$$ acts on our $$d$$-orbitals. Find out the character. From this we get a representation of $$\mathrm{T_d}$$.
2. With the character of IRs of $$\mathrm{T_d}$$, we could easily decompose the above representation into IRs and know the invariant subspaces, hence the degeneracy.
• We know how $$\mathrm{O}(3)$$ acts on $$d$$-orbitals. The action of $$\mathrm{T_d}$$ is obtained directly by subduction, and the characters are obtained just as if we are treating $$\mathrm{O}(3)$$ operations.
• The only question is, to each $$l$$ we have two IRs of $$\mathrm{O}(3)$$ of dimension $$2l+1$$. Which one should we use?
• Since $$d$$-orbitals are invariant under parity, we should therefore use the IR where parity has positive character.
• Character table of $$\mathrm{T_d}$$: where $$6S_4$$ denotes the rotoreflections by $$90^\circ$$ and $$6\sigma_{\mathrm{d}}$$ the reflections.
$$T_d$$ $$E$$ $$8C_3$$ $$3C'_2$$ $$6S_4$$ $$6\sigma_{\mathrm{d}}$$
$$A_1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$A_2$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$E$$ $$2$$ $$-1$$ $$2$$ $$0$$ $$0$$
$$T_2$$ $$3$$ $$0$$ $$-1$$ $$1$$ $$-1$$
$$T_1$$ $$3$$ $$0$$ $$-1$$ $$-1$$ $$1$$
• We determine the characters of $$\mathrm{O}(3)$$ on the above conjugacy classes with ($$j=2$$) $\chi = \frac{\sin (j+1/2)\psi}{\sin (\psi/2)}.$
• Character table of $$\mathrm{O}(3)$$.
$$\mathrm{O}(3)$$ $$E$$ $$8C_3$$ $$3C'_2$$ $$6S_4$$ $$6\sigma_{\mathrm{d}}$$
$$D_2^+$$ $$5$$ $$-1$$ $$1$$ $$-1$$ $$1$$
• The decomposition is given by $D_2^+ = E\oplus T_1.$
• The five $$d$$-orbitals splits into a two-fold level and a three-fold level.
###### Wrap Up: Compatibility Relations
• Originally, the atom has a higher symmetry $$\mathrm{O}(3)$$.
• As we move the atom into a crystal, the symmetry is reduced and therefore the originally irreducible representation ($$d$$-orbitals) $$D_2^+$$ becomes reducible.
• The IR $$D_2^+$$ is decomposed into a direct sum. The possible decompositions are those decompositions into IRs of subgroups of $$\mathrm{O}_3$$. The decomposition will be different if the reduced symmetry is given by $$\mathrm{O}_h$$. These possible decompositions are compatibility relations.
• Compatibility relations $$\Leftrightarrow$$ patterns of crystal field splitting.

#### Application: Band Representations

##### Point Group Action on Bloch Functions
• The actions that carry a point $$\vb*{k}$$ in the reciprocal space to its equivalent $\vb*{k}' = \vb*{k} + \vb*{K}$ form the little group at point $$\vb*{k}$$.
• If $$g$$ is an element of the point group, then $g \psi_{\vb*{k}}(\vb*{r}) = \psi_{\vb*{k}}(g^{-1}\vb*{r}) = \psi_{g\vb*{k}}(\vb*{r}).$
• If, in particular, $$h$$ is an element of the little group of $$\vb*{k}$$, then the set of wavefunctions $\set{h \psi_{\vb*{k}}(\vb*{r})}$ are degenerate, just like what we have in a central potential.
##### Compatibility Relations: Moving Away From Points of High Symmetry
• The symmetry group of points on the line $$XY$$ joining two points of high symmetry $$X$$ and $$Y$$ is a subgroup of the symmetry group thereof. Therefore, the representations decomposes along the line into direct sums. The decomposition should satisfy the compatibility relations, just like what we have in crystal field splitting.

#### Application: Stark Effect

• We ignore the spin-orbit interaction.
• When $$E=0$$, the $$l$$-orbitals ($$2l+1$$-fold degenerate) transforms according to $$D^l$$.
• For $$E=E\hat{\vb*{z}}$$, the $$l$$-orbitals split, each of which transforms according to the $$m$$-th representation of $$\mathrm{SO}(2)$$. Therefore, the eigenkets are of the form $$\ket{m}$$.
• That's why we could directly apply the perturbation theory to $$\ket{lm}$$.
 Index Index

Expand

## Fibre Bundles

ゲージのばの庭

### Fibre Bundles

#### Fibre Bundles

##### Example: Tangent Bundles
• Tangent Bundle $$TM$$: $TM = \bigcup_{p\in M} T_p M.$
• $$M$$ is called the base space.
• $$TU_i$$ is a $$2m$$-dimensional differentiable manifold.
• Projection: $\pi: (p, V) \in TU_i \rightarrow u\in U_i.$
• Fibre at $$p$$: $$T_p M$$.
• Structure group of $$TM$$: $\mathrm{GL}(n,\mathbb{R}) \ni (G^\nu_\mu) = \pdv{y^\nu}{x^\mu}.$
• Section: $s: M\rightarrow TM$ such that $\pi \circ s = \mathrm{id}_M.$
• Local section: $s_i: U_i \rightarrow TU_i.$
##### Definition of Fibre Bundle
• Differentiable fibre bundle $$(E,\pi,M,F,G)$$ consists of
• Total space: a differentiable manifold $$E$$,
• Base space: a differentiable manifold $$M$$,
• (Typical) fibre: a differentiable manifold $$F$$,
• Projection: $\pi: E\rightarrow M.$
• The inverse image $\pi^{-1}(p) = F_p \cong F$ is called the fibre at $$p$$.
• Structure group: a Lie group $$G$$ which acts $$F$$ on the left.
• A set of open covering $$\qty{U_i}$$ of $$M$$ with a diffeomorphism $\phi_i: U_i \times F \rightarrow \pi^{-1}(U_i).$
• $$\phi_i$$ is called the local trivialization.
• Let $\phi_{i,p}(f) = \phi_i(p,f),$ the map $\phi_{i,p}: F \rightarrow F_p$ is a diffeomorphism. On $$U_i\cap U_j \neq \emptyset$$, we require $t_{ij}(p) = \phi^{-1}_{i,p} \circ \phi_{j,p}: F \rightarrow F$ be an element of $$G$$. $$t_{ij}$$ are called the transition functions. $\phi_j(p,f) = \phi_i(p,t_{ij}(p)f).$
• The transition functions should satisfy (assuming intersection of $$U_i$$, $$U_j$$, $$U_k$$):
• $t_{ii}(p) = \mathrm{id}.$
• $t_{ij}(p) = t_{ji}(p)^{-1}.$
• $t_{ij}(p) \cdot t_{jk}(p) = t_{ik}(p).$
• If all the transition functions can be take to be $$\mathrm{id}$$, then the fibre bundle is called a trivial bundle, i.e. $$M\times F$$.
• Section: a smooth map $s: M\rightarrow E$ which satisfies $\pi \circ s = \mathrm{id}_M.$
• $$\Gamma(M,F)$$: the set of sections on $$M$$.
• Local section: defined on $$U$$.
• $$\Gamma(U,F)$$: the set of local sections on $$U$$.
• Not all fibre bundles admit global sections.

Let $E \xrightarrow{\pi} S^1,$ and the typical fibre $F = [-1,1].$ Let $U_1 = (0,2\pi), \quad B = (-\pi,\pi),$ and $$A = (0,\pi)$$ and $$B = (\pi,2\pi)$$ be the intersection of $$U_1\cap U_2$$.

• Local trivialization:
• On $$A$$: $\phi_1^{-1}(u) = (\theta, t),\quad \phi_2^{-1}(u) = (\theta, t).$
• Transition function: $t_{12}(\theta) = \mathrm{id}.$
• On $$B$$: two choices,
• $\phi_1^{-1}(u) = (\theta,t),\quad \phi_2^{-1}(u) = (\theta,t),$
• Transition function: $t_{12}(\theta) = \mathrm{id}.$
• $$G=\qty{e}$$, cylinder, trivial bundle $$S^1\times F$$.
• $\phi_1^{-1}(u) = (\theta,t),\quad \phi_2^{-1}(u) = (\theta,-t).$
• Transition function: $t_{12}(\theta): t\mapsto -t.$
• $$G = \mathbb{Z}_2$$, Möbius strip.
##### Reconstruction of Fibre Bundles

For given $$M$$, $$\qty{U_i}$$, $$t_{ij}(p)$$, $$F$$ and $$G$$, we can reconstruct the fibre bundle $$(E,\pi,M,F,G)$$.

• $\color{red} E = X/\sim.$
• $X = \bigcup_i U_i \times F.$
• Equivalence relation $$\sim$$: $(p,f)\in U_i \times F \sim (q,f') \in U_j \times F$ if and only if $p = q\quad \text{and} \quad f' = t_{ji}(p) f.$
• Projection $\pi: [(p,f)] \mapsto p.$
• Local trivialization $\phi_i: (p,f) \mapsto [(p,f)].$
##### Bundle Maps
• Bundle map: a smooth map $\overline{f}: E'\rightarrow E$ that
• maps each fibre $$F'_p$$ of $$E'$$ onto $$F_q$$ on $$E$$.
• $$\overline{f}$$ induces a smooth map $f: M' \rightarrow M$ such that $$f(p) = q$$.
• The diagram $\begin{CD} E' @>\overline{f}>> E \\ @VV\pi' V @VV\pi V \\ M' @>f>> M \end{CD}$ commutes.
##### Equivalent Bundles
• Two bundles are equivalent if
• there exists a bundle map $\overline{f}: E'\rightarrow E$
• such that $$f: M\rightarrow M$$ is the identity map and
• $$\overline{f}$$ is a diffeomorphism,
• i.e. the diagram $\begin{CD} E' @>\overline{f}>> E \\ @VV\pi' V @VV\pi V \\ M @>\mathrm{id}_M>> M \end{CD}$ commutes.
##### Pullback Bundles
• Pullback of $$E$$ by $f: N\rightarrow M,$
• $f^*E = \qty{(p,u)\in N\times E | f(p) = \pi(u)}.$
• The diagram $\begin{CD} f^*E @>\pi_2>> E \\ @VV\pi_1V @VV\pi V\\ N @>f>> M \end{CD}$ commutes.
• $t^*_{ij}(p) = t_{ij}(f(p)).$

Let $$M$$ and $$N$$ be differentiable manifolds with $\dim M = \dim N = m.$ Let $$f: N\rightarrow M$$ be a smooth map. $$f$$ induces $$\pi_2$$ such the the diagram $\begin{CD} TN @>\pi_2>> TM \\ @VV\pi_1V @VV\pi V\\ N @>f>> M. \end{CD}$ If $$TN$$ is a pullback bundle $$f^*(TM)$$, then $$\pi_2$$ maps $$T_p N$$ to $$T_{f(p)}M$$ diffeomorphically.

##### Homotopy Axiom
• $$f,g: M'\rightarrow M$$ are homotopic if
• there exists a smooth map $F: M'\times [0,1] \rightarrow M$ such that $$F(p,0) = f(p)$$ and $$F(p,1) = g(p)$$ for any $$p\in M'$$.
• Let $$E\xrightarrow{\pi}M$$ be a fibre bundle and $$f,g$$ be homotopic maps from $$N$$ to $$M$$. Then $$f^* E$$ and $$g^* E$$ are equivalent bundles over $$N$$.
• Let $$E\xrightarrow{\pi}M$$ be a fibre bundle. $$E$$ is trivial if $$M$$ is contractible to a point.

#### Vector Bundles

##### Definition and Examples of Fibre Bundles
• Vector bundle: $$E\xrightarrow{\pi} M$$,
• a fibre bundle,
• whose fibre is a vector space, i.e. $$F = \mathbb{R}^k$$.
• $$k$$ is called the fibre dimension, denoted by $k = \dim E.$
• Transition functions belong to $$\mathbb{GK}(k)$$.
• Line bundle: a vector bundle whose fibre is one-dimensional.
• Canonical line bundle $$L$$: $L = \set{(p,v)\in \mathbb{C}P^n \times \mathbb{C}^{n+1} | v=ap,a\in \mathbb{C}}.$

Let $$U_{\mathrm{N}}$$ be $$S^2$$ without south pole and $$U_{\mathrm{S}}$$ be $$S^2$$ without north pole. Let $$(X,Y)$$ and $$(U,V)$$ be the respective stereographic coordinates. Putting $$X = r\cos\theta$$ and $$Y = r\sin\theta$$ we obtain the transition $t_{\mathrm{SN}} = \pdv{(U,V)}{(X,Y)} = \frac{1}{r^2} \begin{pmatrix} -\cos 2\theta & -\sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}.$

• Let $$M$$ be an $$m$$-dimensional manifold embedded in $$\mathbb{R}^{m+k}$$.
• Let $$N_p M$$ be the vector space normal to $$T_p M$$ in $$\mathbb{R}^{m+k}$$.
• Normal bundle: $NM = \bigcup_{p\in M} N_p M.$
• Fibre in $$\mathbb{R}^k$$.

For example, the normal bundle of $$S^2$$ embedded in $$\mathbb{R}^3$$ is a trivial bundle $$S^2 \times \mathbb{R}$$.

A wavefunction is a section of the trivial complex line bundle $$L = \mathbb{R}^3 \times \mathbb{C}$$.

##### Frames
• Frame: linearly independent sections $\qty{e_1(p),\cdots,e_k(p)}$ defined on each $$U_i$$.
• Local trivialization $\phi_i^{-1}(V) = (p,\qty{V^\alpha(p)}).$
• Change of frame: \begin{align*} \tilde{e}_\beta(p) &= e_\alpha(p) G(p){^\alpha}{_\beta}, \\ \tilde{V}^\beta &= G^{-1}(p){^\beta}{_\alpha} V^\alpha. \end{align*}
##### Cotangent Bundles and Dual Bundles
• Contangent bundle: $T^* M = \bigcup_{p\in M} T^*_p M.$
• Change of frame: \begin{align*} \dd{y^\mu} &= \dd{x^\nu} \qty(\pdv{y^\mu}{x^\nu})_p, \\ \tilde{\omega}_\mu ^= G{_\mu}{^\nu}(p) \omega_\nu. \end{align*}
• Dual bundle $$E^* \xrightarrow{\pi} M$$: fibre $$F^*$$ of which is the set of linear maps from $$F$$ to $$\mathbb{R}$$ or $$\mathbb{C}$$.
• Dual basis on $$F^*_p$$ by $\langle \theta^\alpha(p), e_\beta(p)\rangle = \delta{^\alpha}{_\beta}.$
##### Sections of Vector Bundles
• $(s+s')(p) = s(p) + s'(p)$.
• For $$f\in \mathcal{F}(M)$$, $(fs)(p) = f(p) s(p).$
• Null section: $$s_0 \in \Gamma(M,E)$$ such that $\phi_i^{-1}(s_0(p)) = (p,0).$

Let $$L$$ be the canonical line bundle $$L$$ over $$\mathbb{C}P^n$$. Let The local section $$s_\mu$$ over $$U_\mu$$ is defined by $s_\mu = \qty{\xi{^0}{_{(\mu)}}, \cdots, 1, \cdots, \xi{^n}{_{(\mu)}}} \in \mathbb{C}^{n+1}.$ The transition is given by $s_\nu = \frac{z^\mu}{z^\nu} s_\mu.$ The dual section is defined by $s^*_\mu(s_\mu) = 1.$ The transition is given by $s^*_\nu = \frac{z^\nu}{z^\mu} s_\mu^*.$ A fibre metric $$h_{\mu\nu}(p)$$ is defined by $(s,s')_p = h_{\mu\nu}(p) \overline{s^\mu(p)} s'^\nu(p).$

##### The Product Bundle and Whitney Sum Bundle
• Product bundle: $E\times E' \xrightarrow{\pi\times \pi'} M\times M'.$
• Curvature on product bundle: $\mathcal{F}_{E\otimes F} = \mathcal{F}_E \otimes \mathbb{1} + \mathbb{1} \otimes \mathcal{F}_F.$
• Typical fibre $$F \oplus F'$$.
• Whitney sum bundle $$E\oplus E'$$: pullback of $$E\times E'$$ by $f: p\in M\rightarrow (p,p) \in M\times M,$
• denoted by $$E\oplus E'$$.
• $$E\xrightarrow{\pi} M$$ and $$E' \xrightarrow{\pi'} M$$ are vector bundles.
• Commutative diagram: $\begin{CD} E \oplus E' @>\pi_2>> E\times E' \\ @VV\pi' V @VV\pi \times \pi'V \\ M @>f>> M \times M \end{CD}$
• Curvature on Whitney sum bundle: $\mathcal{F}_{E\oplus F} = \mathcal{F}_E \oplus \mathcal{F}_F.$
• Transition $T_{ij}(p) = \begin{pmatrix} t^E_{ij}(p) & 0 \\ 0 & t^{E'}_{ij}(0) \end{pmatrix}.$

Let $$E = TS^2$$ and $$E' = NS^2$$ defined in $$\mathbb{R}^3$$. The trivialization $$\Phi_{i,j}$$ is given by $\Phi^{-1}_{i,j}(u,v) = (p,q; V,W) = (p; V, W),$ where $\phi_i^{-1} = (p,V),\quad \psi_i^{-1} = (q,W).$ The Whitney sum $TS^2 \oplus NS^2$ is therefore a trivial bundle $$S^2 \times \mathbb{R}^3$$.

##### Tensor Product Bundles
• Tensor product bundle: denoted by $$E\otimes E'$$.
• $$E\xrightarrow{\pi} M$$ and $$E\xrightarrow{\pi'} M$$.
• Fibres are given by $F_p \otimes F'_p.$
• $$\bigwedge^r E$$ is spanned by $\qty{e_{\alpha_1} \wedge \cdots \wedge e_{\alpha_r}}.$
• $\Omega^r(M) = \Gamma(M, \bigwedge^r(T^* M)).$

#### Principal Bundles

##### Definition and Examples of Principal Bundles
• Principal bundle:
• Fibre $$F$$ is identical to the structure group $$G$$.
• Denoted by $$P(M,G)$$.
• Often called $$G$$-bundle over $$M$$.
• Action on the right: for $$u\in P$$ and $$a\in G$$, $ua = \phi_i(p, g_i a),$
• $$\phi_i^{-1}(u) = (p, g_i)$$.
• The action is transitive.
• The action is free, i.e. if $$ua=u$$ for some $$u$$ then $$a = e$$.
• Canonical local trivialization:
• Section $$s_i(p)$$ defined over $$U_i$$.
• For $$u\in \pi^{-1}(p)$$, there is a unique $$g_u$$ such that $u = s_i(p) g_u.$
• Define $\phi_i^{-1}(u) = (p, g_u).$
• In this local trivialization, the section $$s_i(p)$$ is expressed as $s_i(p) = \phi_i(p,e).$
• $\phi_i(p,g) = s_i(p)g.$
• Two sections on $$U_i\cap U_j$$ are related by $s_i(p) = s_j(p) t_{ji}(p).$

Let $$P$$ be a principal bundle with fibre $$\mathrm{U}(1) = S^1$$ and base space $$S^2$$. Let \begin{align*} U_{\mathrm{N}} &= \set{(\theta,\phi)| 0 \le \theta \le \pi/2+\epsilon}, \\ U_{\mathrm{S}} &= \set{(\theta,\phi)| \pi/2-\epsilon \le \theta\le \pi}. \end{align*}{} $$U_{\mathrm{N}} \cap U_{\mathrm{S}}$$ is the equator. Let $$\phi_{\mathrm{N}}$$ and $$\phi_{\mathrm{S}}$$ be the local trivializations such that $\phi_{\mathrm{N}}^{-1}(u) = (p, e^{i\alpha_{\mathrm{N}}}),\quad \phi_{\mathrm{S}}^{-1}(u) = (p, e^{i\alpha_{\mathrm{S}}}).$ Fibre coordinates related on the equator as $e^{i\alpha_{\mathrm{N}}} = e^{in\phi} e^{i\alpha_{\mathrm{S}}},$ characterized by $$\pi_1(\mathrm{U}(1)) = \mathbb{Z}$$. Trivial bundle for $$n=0$$, and twisted otherwise. Right action $$g = e^{i\Lambda}$$ is a $$\mathrm{U}(1)$$-gauge transformation.

With $S^4 = \mathbb{R}^4 \cup \qty{\infty},$ and the open covering \begin{align*} U_{\mathrm{N}} &= \set{(x,y,z,t) | x^2 + y^2 + z^2 + t^2 \le R^2 + \epsilon}, \\ U_{\mathrm{S}} &= \set{(x,y,z,t) | x^2 + y^2 + z^2 + t^2 \ge R^2 - \epsilon}. \end{align*}{} We have $U_{\mathrm{N}}\cap U_{\mathrm{S}} \cong S^3.$ For an $$\mathrm{SU}(2)$$ bundle $$P$$ over $$S^4$$, the transition function $t_{\mathrm{NS}}: S^3 \rightarrow \mathrm{SU}(2)$ is characterized by $$\pi_3(\mathrm{SU}(2)) = \mathbb{Z}$$. The integer is called the instanton number. Nontrivial bundles are given by $t_{\mathrm{NS}}(p) = \frac{1}{\sqrt{x^2+y^2+z^2+t^2}^n}\qty(t\mathbb{1} + i\sum_i x^i \sigma_i)^n.$

$$S^3$$ as a $$\mathrm{U}(1)$$ bundle over $$S^2$$.

• Hopf map $$\pi:S^3\rightarrow S^2$$ \begin{align*} \xi^1 &= 2(x^1 x^3 + x^2 x^4), \\ \xi^2 &= 2(x^2 x^3 - x^1 x^4), \\ \xi^3 &= (x^1)^2 + (x^2)^2 - (x^3)^2 - (x^4)^2. \end{align*}{}
• Denote \begin{align*} z^0 &= x^0 + ix^1, \\ z^1 &= x^2 + ix^3. \end{align*}{}
• Stereographic projection of $$S^2$$ given by $Z = \frac{z^0}{z^1},\quad W = \frac{z^1}{z^0},$ are both invariant under $$\mathrm{U}(1)$$ transformation $(z^0,z^1) \mapsto (\lambda z^0, \lambda z^1).$
• Local trivializations: $\phi_{\mathrm{S}}^{-1}: \pi^{-1}(U_{\mathrm{S}}) \rightarrow U_{\mathrm{S}} \times \mathrm{U}(1)$ by $(z^0,z^1) \mapsto (\frac{z^0}{z^1}, \frac{z^1}{\abs{z^1}}),$ and $\phi_{\mathrm{N}}^{-1}: \pi^{-1}(U_{\mathrm{N}}) \rightarrow U_{\mathrm{N}} \times \mathrm{U}(1)$ by $(z^0,z^1) \mapsto (\frac{z^1}{z^0}, \frac{z^0}{\abs{z^0}}).$
• On the equator $t_{\mathrm{NS}}(\xi) = \xi^1 + i\xi^2 \in \mathrm{U}(1).$
• $$t_{\mathrm{NS}}(\xi)$$ traverses the unit circle once as we circumnavigate the equator. The $$\mathrm{U}(1)$$ bundle $S^3 \xrightarrow{\pi} S^2$ is therefore characterized by the homotopy class $$1$$ of $\pi_1(\mathrm{U}(1)) = \mathbb{Z}.$
• This Hopf map describes a magnetic monopole of unit strength.
• Unit quaternion represents $$S^3$$.
• Let $S^1_{\mathbb{H}} = \set{(q^0,q^1)\in \mathbb{H}^2 | \abs{q^0}^2 + \abs{q^1}^2 = 1},$ which represents $$S^7$$.
• Let $\mathbb{H}P^1 = \qty{\eta(q^0,q^1)\in \mathbb{H}^2 | \eta \in \mathbb{H} - \qty{0}},$ which represents $$S^4$$.
• We have the Hopf map $$\pi: S^7 \rightarrow S^4$$. The transition function is defined by the class $$1$$ of $\pi_3(\mathrm{SU}(2)) \cong \mathbb{Z}.$
• This Hopf map describes a instanton of unit strength.
• Let $$H$$ be a closed Lie subgroup of a Lie group $$G$$. Then $$G$$ is a principal bundle with fibre $$H$$ and base space $$M = G/H$$.
• Local trivializations defined by $\phi_i^{-1}(g) = ([g], f_i(g)).$
• $f_i(g) = s([g])^{-1}g.$
• $$s$$ is a section over $$U_i$$.
• Examples: \begin{align*} \mathrm{O}(n) / \mathrm{O}(n-1) &= \mathrm{SO}(n) / \mathrm{SO} (n-1) = S^{n-1}, \\ \mathrm{U}(n) / \mathrm{U}(n-1) &= \mathrm{SU}(n) / \mathrm{SU}(n-1) = S^{2n-1}. \end{align*}{}
##### Associated Bundles
• Principal bundle $$P(M,G)$$ given.
• Let $$F$$ be a new fibre.
• Action of $$g\in G$$ on $$P\times F$$ given by $(u,f) \rightarrow (ug, g^{-1}f).$
• Associated fibre bundle: $(E,\pi,M,G,F,P).$
• An equivalence class $$(P\times F)/G$$ in which two point $(u,f) \sim (ug, g^{-1}f).$
• Associated vector bundle:
• $(u,v) \sim (ug, \rho(g)^{-1} v).$
• $$\rho$$ is a $$\dim V$$-dimensional representation of $$G$$.
• A vector bundle naturally induces a principal bundle associated with it, by $P(E) = P(M,G).$

Frame bundle $LM = \bigcup_{p\in M} L_p M,$ where $$L_p M$$ is the set of frames at $$p$$. A frame $u = \qty{X_1,\cdots,X_m}$ is expressed as $X_\alpha = X{^\mu}{_\alpha} \partial_\mu.$ The local trivialization may be given by $\phi_i^{-1}(u) = (p, (X{^\mu}{_\alpha})) \in U_i \times \mathrm{GL}(m, \mathbb{R}).$

• Action of $$\mathrm{GL}(m,\mathbb{R})$$ given by $Y_\beta = X_\alpha a{^\alpha}{_\beta}.$
• Transition function given by ${X}{^\mu}{_\alpha} = \pdv{x^\mu}{y^\nu} \tilde{X}{^\nu}{_\alpha}.$

The universal covering group of $$\mathrm{O}^+_{\uparrow}(3,1)$$ is $$\mathrm{SL}(2,\mathbb{C})$$. The kernel of the homomorphism $\varphi: \mathrm{SL}(2,\mathbb{C}) \rightarrow \mathrm{O}^+_{\uparrow}(3,1)$ is $$\qty{I_2,-I_2}$$.

• Weyl spinor: a section of $(W,\pi,M,\mathbb{C}^2,\mathrm{SL}(2,\mathbb{C})).$
• Dirac spinor: a section of $(D,\pi,M,\mathbb{C}^4,\mathrm{SL}(2,\mathbb{C})\oplus \overline{\mathrm{SL}(2,\mathbb{C})}).$
##### Triviality of Bundles
• A principal bundle is trivial if and only if it admits a global section.
• A vector bundle $$E$$ is trivial if and only if its associated principal bundle $$P(E)$$ admits a global section.

### Connections on Fibre Bundles

#### Connections on Principal Bundles

##### Definitions of Connection
• Vertical subspace $$V_u P$$:
• a subspace of $$T_u P$$,
• tangent to $$G_p$$ at $$u$$,
• where $$P(M,G)$$ is a principal bundle given, $$u$$ an element thereof, and $$G_p$$ a fibre at $$p=\pi(u)$$.
• Right action defined by $R_{\exp(tA)} u = u\exp(tA).$
• $$A\in\frak{g}$$.
• Fundamental vector field $A^{\#} f(u) = \dv{}{t} f(u\exp(tA))\vert_{t=0}.$
• $$f:P\rightarrow\mathbb{R}$$.
• $$A^{\#}$$ is tangent to $$G_p$$ at $$u$$.
• Isomorphism $\#: A\in{\frak{g}} \rightarrow A^{\#}\in V_u P.$
• Horizontal subspace $$H_u P$$: complement of $$V_u P$$ in $$T_u P$$.
• Ehresmann Connection on $$P(M,G)$$:
• A unique separation of the tangent space $$T_u P$$ such that $T_u P = H_u P \oplus V_u P,$
• a smooth vector field $$X$$ on $$P$$ is separated into smooth vector fields $X = X^H + X^V,$
• for arbitrary $$u\in P$$ and $$g\in G$$, $H_{ug} P = R_{g*} H_u P.$
##### The Connection One-Form
• Ehresmann connection one-form $$\omega$$:
• A projection of $$T_u P$$ onto the vertical component $V_u P \cong {\frak{g}}.$
• $\omega \in {\frak{g}} \otimes T^* P.$
• The projection should satisfy
• $\omega(A^{\#}) = A,\quad \text{for}\quad A\in{\frak{g}}.$
• $R^*_g \omega = \mathrm{Ad}_{g^{-1}}\omega,$ i.e. for $$X\in T_u P$$, $R^*_g \omega_{ug}(X) = \omega_{ug}(R_{g*} X) = g^{-1} \omega_u (X) g.$
• Ehresmann connection: $\color{orange} H_u P = \qty{X\in T_u P | \omega(X) = 0}.$
• In harmony with the previous definition.
##### The Local Connection Form and Gauge Potential
• Let $\mathcal{A}_i = \sigma_i^* \omega \in {\frak{g}}\otimes \Omega^1(U_i).$
• $$A_i$$ is a Lie-algebra-valued one-form on $$U_i$$.
• $$\sigma_i$$ is a given section of $$P$$ on $$U_i$$.
• With $$\mathcal{A}_i$$ given we could also recontruct $$\sigma_i$$.
• Given a $$\frak{g}$$-valued one-form $$\mathcal{A}_i$$ on $$U_i$$ and a local section $$\sigma_i$$, there exists a connection one-form $$\omega$$ such that $$\mathcal{A}_i = \sigma_i^* \omega$$.
• $$\omega$$ may be given by $\omega_i = g_i^{-1} \pi^* \mathcal{A}_i g_i + g_i^{-1} \dd{_P} g_i.$
• $$\dd{_P}$$ is the exterior derivative on $$P$$.
• $$g_i$$ is the canonical local trivialization defined by $\phi_i^{-1}(u) = (p, g_i),$ for $u = \sigma_i(p) g_i.$
• For $$\omega$$ to be uniquely (consistently) defined, we demand the compatibility relation $\color{red} \mathcal{A}_j = t_{ij}^{-1} \mathcal{A}_i t_{ij} + t_{ij}^{-1} \dd{t_{ij}}.$
• Since $$t_{ij}\in G$$, the first term maps elements in $$\frak{g}$$ to another element in $$\frak{g}$$.
• In the second term, the $$\dd{}$$ takes with respect to $$M$$. Therefore, the second term yields an element in $$\frak{g}$$ by acting on $$X\in T_p M$$.
• In particular, on the chart $$U$$ we could perform the gauge transform by using a new trivialization $$\sigma_2(p) = \sigma_1(p) g(p)$$. $\mathcal{A}_{2\mu} = g^{-1}(p) \mathcal{A}_{1\mu} g(p) + g^{-1}(p) \partial_\mu g(p).$
• $$\mathcal{A}_i$$ is identified with the gauge potential or Yang-Mills potential.
• The physical field is encoded in $$\omega$$.
• By picking a specific trivialization $$\sigma$$ (coordinate system of the bundle), we slice $$\omega$$ and pull it back to $$M$$, yielding the potential.
• By the equation for $$\omega$$ above, its value on $$P$$ is determined by its value on a single slice (thanks to the projection properties or the invariance property of Ehresmann connection).
• Due to the presence of $$g_i$$ in the equation for $$\omega$$, it's seemingly that $$\omega$$ depends on the trivialization we choose. We don't want that to happen. We demand $$\omega$$ stays the same even if we switch to another trivialization.
• Therefore, we simultaneously (1) go from one trivialization to another, and (2) transform $$\mathcal{A}_i$$ to $$\mathcal{A}_j$$, i.e. a gauge transformation.
• The transformation may be as well understood as slicing $$\omega$$ in a different way.

For a $$\mathrm{U}(1)$$-bumdle, we define a new trivialization by $t_{ij}(p) = \exp[i\Lambda(p)].$ The potentials are therefore related as $\mathcal{A}_j(p) = \mathcal{A}_i(p) + i\partial_\mu \Lambda.$ The connection $$\mathcal{A}_\mu$$ differs from the standard vector potential by the Lie algebra factor, i.e. $\mathcal{A}_\mu = iA_\mu.$

##### Horizontal Lift and Parallel Transport
• Horizontal lift of $$\gamma:[0,1] \rightarrow M$$:
• a curve $$\tilde{\gamma}: [0,1] \rightarrow P$$,
• $$P(M,G)$$,
• $$\pi\circ\tilde{\gamma} = \gamma$$,
• the tangent vector to $$\tilde{\gamma}(t)$$ belongs to $$H_{\tilde{\gamma}(t)}P$$.
• For $$\gamma:[0,1]\rightarrow M$$ and $$u_0\in \pi^{-1}(\gamma(0))$$, there exists a unique horizontal lift $$\tilde{\gamma}(t)$$ such that $$\tilde{\gamma}(0)$$ = u_0.
• Let $$\tilde{\gamma}'$$ be another horizontal lift of $$\gamma$$, such that $$\tilde{\gamma}'(0) = \gamma(0)g$$. Then $$\tilde{\gamma}'(t) = \tilde{\gamma}(t)g$$.
• Explicitly, for $$\omega$$ and $$\sigma_i$$ given, $g_i(\gamma(t)) = \mathcal{P} \exp\qty({-\int_{\gamma(0)}^{\gamma(t)} \mathcal{A}_{i\mu}(\gamma(t))\dd{x^\mu}}).$
• $$\mathcal{P}$$ denotes path ordering. $\mathcal{P}[A(\gamma(t))B(\gamma(s))] = \begin{cases} A(\gamma(t))B(\gamma(s)), & \text{if } t>s, \\ B(\gamma(s))A(\gamma(t)), & \text{if } s>t. \end{cases}$
• As a differential equation: $\dv{g_i(t)}{t} = -\mathcal{A}_i(X) g_i(t).$
• Parallel transport:
• $$\gamma: [0,1] \rightarrow M$$.
• $$u_0 \in \pi^{-1}(\gamma(0))$$.
• There is a unique horizontal lift $$\tilde{\gamma(t)}$$ of $$\gamma(t)$$ through $$u_0$$,
• and hence a unique point $u_1 = \tilde{\gamma}(1) \in \pi^{-1}(\gamma(1)).$
• $$u_1$$ is called the parallel transport of $$u_0$$ along $$\gamma$$.
• This defines a map $\Gamma(\tilde{\gamma}): u_0 \in \pi^{-1}(\gamma(0)) \rightarrow u_1\in \pi^{-1}(\gamma(1)).$
• $$\Gamma$$ commutes with $$R_g$$, i.e. $R_g \Gamma(\tilde{\gamma}) = \Gamma(\tilde{\gamma}) R_g.$
• Explicitly, $u_1 = \sigma_i(1) \mathcal{P} \exp\qty(-\int_0^1 \mathcal{A}_{i\mu} \dv{x^\mu(\gamma(t))}{t} \dd{t}).$
• $$P(M,\mathbb{R}) \cong M\times \mathbb{R}$$ where $$M = \mathbb{R}^2 - \qty{0}$$.
• The local trivialization be given by $\phi: ((x,y),f) \mapsto u.$
• $\omega = \frac{y\dd{x} - x\dd{y}}{x^2 + y^2} + \dd{f}.$
• $\gamma: t\in [0,1] \rightarrow (\cos 2\pi t, \sin 2\pi t)\in M.$
• $X = \dv{}{t} = \dv{x}{t} \pdv{}{x} + \dd{y}{t} \pdv{}{y} + \dv{f}{t} \dd{}{f}.$
• $0 = \omega(X) = \dv{x}{t}\frac{y}{r^2} - \dv{y}{t}\frac{x}{r^2} + \dv{f}{t} = -2\pi + \dv{f}{t}.$
• Horizontal lift $\tilde{\gamma}(t) = ((\cos 2\pi t, \sin 2\pi t), 2\pi t).$

#### Holonomy

##### Definition of Holonomy
• A loop $$\gamma: [0,1] \rightarrow M$$ gives rise to a transformation $\tau_\gamma: \pi^{-1}(p) \rightarrow \pi^{-1}(p)$ by parallel transport.
• $$\tau_\gamma$$ is compatible with the right action of the group, $\tau_\gamma(ug) = \tau_\gamma(u)g.$
• Holonomy group $$\Phi_u \subset G$$: $\Phi_u = \set{g\in G | \tau_\gamma(u) = ug,\gamma\in C_p(M)}.$
• $$p = \pi(u)$$.
• $C_p(M) = \set{\gamma:[0,1] \rightarrow M | \gamma(0) = \gamma(1) = p}.$
• Restricted holonomy group: $\Phi_u^0 = \set{g\in G | \tau_\gamma(u) = ug, \gamma\in C_p^0(M)}.$
• $$C_p^0$$ denotes the set of loops at $$p$$ homotopic to the constant loop at $$p$$.

In the previous example, $\tau_\gamma(g) = g + 2\pi.$

#### Curvature

##### Covariant Derivatives in Principal Bundles
• Covariant derivative of $$\phi$$: $\color{orange} \mathrm{D}\phi(X_1,\cdots,X_{r+1}) = \dd{_P \phi}(X_1^H,\cdots,X_{r+1}^H).$
• $\phi\in \Omega^r(P)\otimes V.$
• $X_1,\cdots,X_{r+1}\in T_u P.$
• $\dd{_P \phi} = \dd{_P \phi^\alpha} \otimes e_\alpha.$
##### Curvature
• $$\omega$$ is $$\frak{g}$$-valued. Therefore $\omega\wedge \omega \neq 0.$
• $[\omega,\omega](X,Y) = 2[\omega(X), \omega(Y)].$
• Curvature two-form $$\Omega$$: $\color{orange} \Omega = \mathrm{D}\omega \in \Omega^2(P)\otimes {\frak{g}}.$
• $R^*_a \Omega = a^{-1}\Omega a.$
• For $$\xi = \xi^\alpha\otimes T_\alpha$$, $$\eta = \eta^\alpha \otimes T_\alpha$$ where $\zeta^\alpha \in \Omega^p(P),\quad \eta^\alpha\in \Omega^q(P)$ and $$\qty{T_\alpha}$$ a basis of $$\frak{g}$$, \begin{align*} \color{orange} [\zeta,\eta] &\color{orange}= \zeta\wedge \eta - (-1)^{pq} \eta\wedge \zeta \\ &\color{orange}= [T_\alpha,T_\beta] \otimes \zeta^\alpha \wedge \eta^\beta. \end{align*}
• $[\zeta,\zeta] = 2\zeta\wedge \zeta.$
• If $$X\in H_u p$$ and $$Y\in V_u P$$ then $[X,Y] \in H_uP.$
• Cartan's structure equation: for $$X,Y\in T_u P$$, $\color{red} \Omega(X,Y) = \dd{_P \omega}(X,Y) + [\omega(X), \omega(Y)].$
• Equivalently, $\color{red} \Omega = \dd{_P \omega} + \omega \wedge \omega.$
##### Geometrical Meaning of the Curvature
• Ambrose-Singer theorem. Let $$P(M,G)$$ be a $$G$$ bundle over a connected manifold $$M$$. The Lie algebra of the holonomy group $$\Phi_{u_0}$$ of a point $$u_0\in P$$ agrees with the subalgebra of $$\frak{g}$$ spanned by the elements of the form $\Omega_u(X,Y),\quad X,Y\in H_u P,$ where $$u\in P$$ is a point on the same horizontal lift as $$u_0$$.
##### Local Form of the Curvature
• Yang-Mills field strength (local form) $\color{orange} \mathcal{F} = \sigma^* \Omega.$
• $$\sigma$$ is a local section defined on a chart $$U\subset M$$.
• In terms of gauge field $\color{red} \mathcal{F} = \dd{\mathcal{A}} + \mathcal{A}\wedge \mathcal{A}.$
• $$\mathcal{A} = \sigma^* \omega$$.
• $$\dd{}$$ is the exterior derivative on $$M$$.
• $\color{red} \mathcal{F}(X,Y) = \dd{\mathcal{A}}(X,Y) + [\mathcal{A}(X), \mathcal{A}(Y)].$
• Component form $\color{red} \mathcal{F}_{\mu\nu} = \partial_\mu \mathcal{A}_\nu - \partial_\nu \mathcal{A}_\mu + [\mathcal{A}_\mu, \mathcal{A}_\nu].$
• $\mathcal{F} = \frac{1}{2}\mathcal{F}_{\mu\nu} \dd{x^\mu} \wedge \dd{x^\nu}.$
• Let $$\qty{T_\alpha}$$ be a basis of the Lie algebra, $\color{red} F_{\mu\nu}{^\alpha} = \partial_\mu A{_\nu}{^\alpha} - \partial_\nu A{_\mu}{^\alpha} + f_{\beta\gamma}{^\alpha} A{_\mu}{^\beta} A{_\nu}{^\gamma}.$
• $[T_\beta, T_\gamma] = f_{\beta\gamma}{^\alpha} T_\alpha.$
• Compatibility condition: $\color{red} \mathcal{F}_j = t_{ij}^{-1} \mathcal{F}_i t_{ij}.$
##### The Bianchi Identity
• Bianchi identity: $\color{red} \mathrm{D}\Omega = 0.$
• $\color{red} \mathcal{D}\mathcal{F} = \dd{\mathcal{F}} + [\mathcal{A}, \mathcal{F}] = \dd{\mathcal{F}} + \mathcal{A} \wedge \mathcal{F} - \mathcal{F} \wedge \mathcal{A} = 0.$
• $\color{orange} \mathcal{D}\eta = \dd{\eta} + [\mathcal{A},\eta].$

#### The Covariant Derivative on Associated Vector Bundles

##### The Covariant Derivative on Associated Bundles
• $$P(M,G)$$ be a $$G$$ bundle.
• $$E = P\times_\rho V$$ be a associated bundle thereof.
• Representatives as $s(p) = [(\sigma_i(p), \xi(p))].$
• $$\gamma$$ a curve in $$M$$. $$\tilde{\gamma}$$ the horizontal lift thereof.
• Parallel transported section: $\color{orange} s(\gamma(t)) = [(\tilde{\gamma}(t), \eta(\gamma(t)))]$ where $$\eta$$ is constant along $$\gamma(t)$$.
• Covariant derivative $\color{orange} \grad_X s = \qty[\qty({\tilde{\gamma}(0), \left.\dv{}{t}\eta(\gamma(t))\right\vert_{t=0} })].$
• $$X$$ the tangent vector of $$\gamma(t)$$ at $$t=0$$.
• $\grad_X: \Gamma(M,E) \rightarrow \Gamma(M,E).$
$\grad: \Gamma(M,E) \rightarrow \Gamma(M,E)\otimes \Omega^1(M)$ defined by $\grad s(X) = \grad_X s.$
• Properties of covariant derivatives are satisfied.
• Linearity in $$s$$: $\grad(a_1 s_1 + a_2 s_2) = a_1 \grad s_1 + a_2 \grad s_2.$
• Linearity in $$X$$: $\grad_{f_1 X_1 + f_2 X_2} s = f_1 \grad_{X_1} s + f_2 \grad_{X_2} s.$
• Leibniz rule: $\grad(fs) = (\dd{f}) s + f\grad s.$
• Independent of
• horizontal lift, and
• local section (trivialization).
• $$\mathcal{A}_i$$ transforms under the change of local trivialization in a way such that $$\grad_X s$$ is well-defined.
##### Local Expression for Covariant Derivative
• $$\tilde{\gamma}$$, the horizontal shift of $$\gamma$$, is written as $\tilde{\gamma}(t) = \sigma_i(t) g_i(t).$
• $\qty(\mathcal{A}_{i\mu}){^\beta}{_\alpha} = \mathcal{A}_{i\mu}{^\gamma}(T_\gamma){^\beta}{_\alpha}.$
• $$e{_\alpha}{^0}$$ denotes the $$\alpha$$th basis vector of $$V$$.
• $e_\alpha(t) = [(\tilde{\gamma}(t) g_i(t)^{-1}, e{_\alpha}{^0})] = [(\tilde{\gamma}(t), g_i(t)^{-1} e{_\alpha}{^0})].$
• Covariant derivative of basis: $\grad_X e_\alpha = [(\tilde{\gamma}(0) g_i(0)^{-1}, \mathcal{A}_i(X) e{_\alpha}{^0})].$
• Local expression: \begin{align*} \color{red} \grad e_\alpha &\color{red}= \mathcal{A}{_i}{^\beta}{_\alpha} e_\beta, \\ \color{red} \grad_{\partial/\partial x^\mu} e_\alpha &\color{red}= \mathcal{A}_{i\mu}{^\beta}{_\alpha} e_\beta. \end{align*}{}
• For a general vector field, $\color{red} \grad_X s = X^\mu \qty{\pdv{\xi{_i}{^\alpha}}{x^\mu} + \mathcal{A}_{i\mu}{^\alpha}{_\beta}\xi{_i}{^\beta}} e_\alpha.$
• $$FM$$ be frame bundles over $$M$$.
• $FM = P(M,\mathrm{GL}(m,\mathbb{R})).$
• $$TM$$ be its associated bundle.
• $TM = FM \times_\rho \mathbb{R}^m.$
• $$m = \dim M$$.
• $\mathcal{A}_i = \Gamma{^\alpha}_{\mu\beta} \dd{x^\mu}.$
• $\grad_{\partial/\partial x^\mu} e_\alpha = [(\sigma_i(0), \Gamma_\mu e{_\alpha}{^0})] = \Gamma{^\beta}_{\mu\alpha} e_\beta.$
• $\grad_{\partial/\partial x^\mu} s = \qty(\frac{\partial}{\partial x^\mu} X{_i}{^\alpha} + \Gamma{^\alpha}_{\mu\beta} X^\beta) e_\alpha.$
• $$\mu$$ is the $$\Omega^1(M)$$ index while $$\alpha$$ and $$\beta$$ are $$\frak{gl}(m,\mathbb{R})$$ indices.
• $$P(M,\mathrm{U}(1))$$.
• $$E = P\times_\rho \mathbb{C}$$.
• $$\sigma_i$$ be a local section. $$\tilde{\gamma}$$ be a horizontal lift defined by $\tilde{\gamma}(t) = \sigma_i(t) e^{i\varphi(t)}.$
• Basis section $e = [(\sigma_i(p), 1)].$
• My field (section) $\phi(p) = [(\sigma_i(p), \Phi(p))] = \Phi(p)e = \Phi(t)[(\tilde{\gamma}(t), e^{-i\varphi(t)})].$
• Covariant derivative $\grad_X \phi = X^\mu \qty(\pdv{\Phi}{x^\mu} + \mathcal{A}_{i\mu} \Phi)e.$
• $$P(M,\mathrm{SU}(2))$$.
• $$E = P \times_\rho \mathbb{C}^2$$.
• Gauge potential
• $\mathcal{A}_i = \mathcal{A}_{i\mu} \dd{x^\mu} = A_{i\mu}{^\alpha}\qty(\frac{\sigma_\alpha}{2i})\dd{x^\mu}.$
• $$e{_\alpha}{^0}$$ where $$\alpha=1,2$$ be basis vectors of $$\mathbb{C}^2$$.
• Basis section $e_\alpha(p) = [(\sigma_i(p), e{_\alpha}{^0})].$
• My field (section) $\phi(t) = [(\tilde{\gamma}(t), U(t)^{-1} \Phi^\alpha(t) e{_\alpha}{^0})].$
• $\tilde{\gamma}(t) = \sigma_i(p) U(t),\quad U(t) \in \mathrm{SU}(2).$
• Covariant derivative $\grad_X \phi = X^\mu \qty(\pdv{\Phi^\alpha}{x^\mu} + \mathcal{A}_{i\mu}{^\alpha}{_\beta}) e_\alpha.$
##### Curvature Rederived
• $$\grad$$ on vector-valued $$p$$-form: $\color{orange} \grad(s\otimes \eta) = (\grad s) \wedge \eta + s\otimes \dd{\eta}.$
• $$\eta \in \Omega^p(M)$$.
• $\color{red} \grad \grad e_\alpha = e_\beta \otimes \mathcal{F}{_i}{^\beta}{_\alpha}.$
##### A Connection Which Preserves the Inner Product
• Riemann structure: a positive-definite symmetric inner product $g_p: \pi^{-1}(p) \otimes \pi^{-1}(p) \rightarrow \mathbb{R}.$
• $$E\xrightarrow{\pi} M$$ a vector bundle.
• Metric connection: $$\grad$$ such that $\color{orange} \dd{[g(s,s')]} = g(\grad s, s') + g(s, \grad s').$
• $\dd{g_{\alpha\beta}} = \mathcal{A}{_i}{^\gamma}{_\alpha} g_{\gamma\beta} + \mathcal{A}{_i}{^\gamma}{_\beta} g_{\alpha\gamma}.$
• Torsion-free: $\Gamma{^\gamma}_{\rho\sigma} \dd{x^\rho} = \mathcal{A}{^\gamma}{_\alpha}.$
• $$\grad$$ associated with the orthonormal frame is a metric connection.
• Orthonormal frame: $$\qty{\hat{e}_\alpha}$$ such that $g(\hat{e}_\alpha, \hat{e}_\beta) = \delta_{\alpha\beta}.$
• Structure group $$O(k)$$. Lie algebra $${\frak{o}}(k)$$ spanned by skew symmetric matrices. $\omega{^\alpha}{_\beta} = -\omega{^\beta}{_\alpha}.$
##### Holomorphic Vector Bundles and Hermitian Inner Products
• Holomorphic vector bundle $$E$$:
• $$\pi: E\rightarrow M$$ is a holomorphic surjection, where $$E$$ and $$M$$ are complex manifolds.
• The typical fibre is $$\mathbb{C}^k$$ and the structure group is $$\mathrm{GL}(k,\mathbb{C})$$.
• The local trivialization $\phi_i: U_i \times \mathbb{C}^k \rightarrow \pi^{-1}(U_i)$ is a biholomorphism.
• The transition function $t_{ij}: U_i \cap U_j \rightarrow G = \mathrm{GL}(k,\mathbb{C})$ is a holomorphic map.
• Hermitian structure: $h_p: \pi^{-1}(p) \times \pi^{-1}(p) \rightarrow \mathbb{C}.$
• $h_p(u,av + bw) = ah_p(u,v) + bh_p(u,w).$
• $h_p(u,v) = \overline{h_p(v,w)}.$
• $h_p(u,u) \ge 0.$
• Equality holds only if $u = \phi_i(p,0).$
• $h(s_1, s_2) \in \mathcal{F}(M)^\mathcal{C}.$
• Unitary frame: $h(\hat{e}_i, \hat{e}_j) = \delta_{ij}.$
• The unitary frame bundle is not a holomorphic vector bundle, because $$\mathrm{U}(m)$$ is not a complex manifold.
• Hermitian connection: $\grad: \Gamma(M,E) \rightarrow \Gamma(M,E\otimes T^* M^{\mathbb{C}}).$
• $\grad(fs) = (\dd{f})s + f\grad s.$
• $\dd{h(s_1,s_2)} = h(\grad s_1, s_2) + h(s_1, \grad s_2).$
• $\grad s = \mathrm{D}s + \overline{\mathrm{D}}s.$
• $$\mathrm{D}$$ and $$\overline{\mathrm{D}}$$ being a $$(1,0)$$-form and $$(0,1)$$-form valued section, respectively.
• We demand $\overline{\mathrm{D}} = \overline{\partial}.$
• For a given $$E$$ and Hermitian metric $$h$$, there exists a unique Hermitian connection $$\grad$$.
• $$\mathcal{A}$$ and $$\mathcal{F}$$ under a unitary frame is skew Hermitian: $\mathcal{A}{^\alpha}{_\beta} = -\overline{\mathcal{A}}{^\beta}{_\alpha},\quad \mathcal{F}{^\alpha}{_\beta} = -\overline{\mathcal{F}}{^\beta}{_\alpha}.$
• $$\mathcal{F}$$ is a $$(1,1)$$-form (bidegree).

The holomorphic tangent bundle $$TM^+$$ is a holomorphic vector bundle. The typical fibre is $$\mathbb{C}^m$$ and the local basis is $$\qty{\partial/\partial z^\mu}$$.

#### Gauge Theories

##### U(1) Gauge Symmetry
• $$P = \mathbb{R}^4 \times \mathrm{U}(1)$$.
• Gauge potential $\mathcal{A} = \mathcal{A}_\mu \dd{x^\mu}.$
• Field strength $\mathcal{F} = \dd{\mathcal{A}}.$
• Component form $\mathcal{F}_{\mu\nu} = \pdv{\mathcal{A}_\nu}{x^\mu} - \pdv{\mathcal{A}_\mu}{x^\nu}.$
• Bianchi identity $\dd{\mathcal{F}} = \mathcal{F}\wedge \mathcal{A} - \mathcal{A} \wedge \mathcal{F} = 0.$
• Component form $\partial_\lambda \mathcal{F}_{\mu\nu} + \partial_\nu \mathcal{F}_{\lambda\mu} + \partial_\mu \mathcal{F}_{\nu\lambda} = 0.$
• The two homogeneous Maxwell equations.
• Maxwell action: $S_{\mathrm{M}}[\mathcal{A}] = \frac{1}{4}\int \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu} \dd{^4 x} = -\frac{1}{4} \int F_{\mu\nu} F^{\mu\nu} \dd{^4 x}.$
##### Dirac Monopole
• On $$U_{\mathrm{N}}$$ and $$U_{\mathrm{S}}$$, \begin{align*} \mathcal{A}_{\mathrm{N}} &= \sigma_{\mathrm{N}}^* \omega = ig(1-\cos\theta)\dd{\phi}, \\ \mathcal{A}_{\mathrm{S}} &= \sigma_{\mathrm{S}}^* \omega = -ig(1+\cos\theta)\dd{\phi}. \end{align*}{}
• $t_{\mathrm{NS}}\in\mathrm{U}(1)$ satisfies $\mathcal{A}_{\mathrm{N}} = t^{-1}_{\mathrm{NS}} \mathcal{A}_{|mathrm{S}} t^{\mathrm{NS}} + t^{-1}_{\mathrm{NS}}\dd{t_{\mathrm{NS}}} = \mathcal{A}_{\mathrm{S}} + i\dd{\varphi}.$
• $t_{NS}(\phi) = e^{i\varphi(\phi)}.$
• Quantization condition: $2g\in \mathbb{Z}.$
• $$2g$$ represents the homotopy class to which the bundle belongs.
##### The Aharonov-Bohm Effect
• $$M = \mathbb{R}^2 - \qty{0}$$.
• $$P(M,\mathrm{U}(1))$$.
• $$E = P\times_\rho \mathbb{C}$$.
• On $$S^1$$, $\vb*{A}(\vb*{r}) = \qty(-\frac{y\Phi}{2\pi r^2}, \frac{x\Phi}{2\pi r^2}),\quad \mathcal{A} = i\frac{\Phi}{2\pi}\dd{\theta}.$
• $$\psi(\theta)$$ be parallel transported: $\mathcal{D}\psi(\theta) = \qty{\dd{} + i\frac{\Phi}{2\pi} \dd{\theta}} \psi(\theta) = 0.$
• Holonomy $\Gamma: \psi(0) \mapsto e^{-i\Phi}\psi(0).$
##### Yang-Mills Theory
• $$P(\mathbb{R}^4, \mathrm{SU}(2))$$.
• Connection one-form $\mathcal{A} = A{_\mu}{^\alpha} T_\alpha \dd{x^\mu}.$
• $T_\alpha = \frac{\sigma_\alpha}{2i}.$
• Field strength $\mathcal{F} = \dd{\mathcal{A}} + \mathcal{A} \wedge \mathcal{A} = \frac{1}{2} \mathcal{F}_{\mu\nu} \dd{x^\mu} \wedge \dd{x^\nu}.$
• $\mathcal{F}_{\mu\nu} = F_{\mu\nu}{^\alpha} T_\alpha.$
• $F_{\mu\nu}{^\alpha} = \partial_\mu A_{\nu\alpha} - \partial_\nu A_{\mu\alpha} + \epsilon_{\alpha\beta\gamma} A_{\mu\beta} A_{\nu\gamma}.$
• Bianchi identity $\mathcal{D}\mathcal{F} = \dd{\mathcal{F}} + [\mathcal{A}, \mathcal{F}] = 0.$
• Yang-Mills action $\mathcal{S}_{\mathrm{YM}}[\mathcal{A}] = -\frac{1}{4}\int \tr{\mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu}} = \frac{1}{2} \int_M \tr(\mathcal{F} \wedge * \mathcal{F}).$
• Equations of motion: $\mathcal{D}_\mu \mathcal{F}^{\mu\nu} = \mathcal{D}*\mathcal{F} = 0.$
##### Instantons
• $$P(\mathbb{R}^4, \mathrm{SU}(2))$$, Euclidean metric.
• $$\mathbb{R}^4$$ compatified to $$S^4$$.
• Euclidean action $S^{\mathrm{E}}_{\mathrm{YM}} = \frac{1}{4}\int \tr{\mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu}} = -\frac{1}{2} \int_M \tr(\mathcal{F} \wedge * \mathcal{F}) = \mp \frac{1}{2} \int_M \tr(\mathcal{F} \wedge \mathcal{F}).$
• Self-dual condition: $\mathcal{F}_{\mu\nu} = \pm * \mathcal{F}_{\mu\nu}.$
• Asymptotic behavior: for a large $$L$$, $\mathcal{A}_\mu(x) \rightarrow g(x)^{-1} \partial_\mu g(x),\quad \text{as}\quad \abs{x}\rightarrow L.$
• $U_{\mathrm{N}} = \qty{x\in \mathbb{R}^4 | \abs{x} \le L+\epsilon},\quad U_{\mathrm{S}} = \qty{x\in \mathbb{R}^4 | \abs{x} \ge L - \epsilon}.$
• No twist on $$U_{\mathrm{S}}$$: $\mathcal{A}_S(x) = 0.$
• Transition $\mathcal{A}_{\mathrm{N}} = t_{\mathrm{NS}}^{-1} \dd{t_{\mathrm{NS}}}.$
• Transition $$S^3 \rightarrow \mathrm{SU}(2)$$ classified by $\pi_3(\mathrm{SU}(2)) \cong \mathbb{Z}.$
• Class $$0$$: constant map $g_0: x\mapsto e.$
• Class $$1$$: identity map $g_1: x_1\mapsto \frac{1}{r}\qty[x^4 \mathbb{1} + x^i \sigma_i].$
• Class $$n$$: $g_n: x \mapsto \frac{1}{r^n}\qty[x^4 \mathbb{1} + x^i \sigma_i]^n.$
• $\mathcal{F}^2 = \mathcal{F} \wedge \mathcal{F} = \dd{K}.$
• $K = \tr\qty[\mathcal{A} \dd{\mathcal{A}} + \frac{2}{3} \dd{\mathcal{A}}^3].$
• $\int_{S^4} \tr \mathcal{F}^2 = -\frac{1}{3} \int_{S^3} \tr{\mathcal{A}^3}.$
• Degree of mapping $n = \frac{1}{24\pi^2} \int_{S^3} \tr(g^{-1} \dd{g})^3 = \frac{1}{2} \int_{S^4} \tr\qty(\frac{i\mathcal{F}}{2\pi})^2.$

#### Berry's Phase

##### Definition of Berry's Phase
• $$H(\vb*{R}(t))$$ where $$\vb*{R}$$ collects all parameters of $$H$$.
• For $$\ket{\psi(0)} = \ket{n}$$, $\ket{\psi(t)} = \exp\qty[i\eta(t) - i\int_0^t E_n(\vb*{R}(s))\dd{s}]\ket{n,\vb*{R}(t)}.$
• $\eta_n(T) = i\int_{\vb*{R}(0)}^{\vb*{R}(T)} \bra{n,\vb*{R}} \grad_{\vb*{R}} \ket{n,\vb*{R}} \dd{\vb*{R}}.$
##### Berry's Phase, Berry's Connection, and Berry's Curvature
• $$M$$ a manifold describing the parameter space.
• $$\vb*{R}$$ the local coordinate.
• $$P(M,\mathrm{U}(1))$$.
• Kets are in equivalence classes $[\ket{\vb*{R}}] = \set{g\ket{\vb*{R}} | g\in \mathrm{U}(1)}.$
• Index $$n$$ omitted.
• Projection $\pi(g\ket{\vb*{R}}) = \ket{\vb*{R}}.$
• Fixing the phase $$\Leftrightarrow$$ choosing a section.
• Local section $\sigma(\vb*{R}) = \ket{\vb*{R}}.$
• Local trivialization $\phi^{-1}(\ket{\vb*{R}}) = (\vb*{R},e).$
• Right action $\phi^{-1}(\ket{\vb*{R}}\cdot g) = (\vb*{R}, g).$
• Berry's connection: $\mathcal{A} = \mathcal{A}_\mu \dd{R}^\mu = \bra{\vb*{R}}(\dd{\ket{\vb*{R}}}) = -(\dd{\bra{\vb*{R}}})\ket{\vb*{R}}.$
• $$\mathcal{A}$$ is anti-Hermitian.
• For $\ket{\vb*{R}}_j = \ket{\vb*{R}}_i t_{ij}(\vb*{R}),$ we have the transition $\mathcal{A}_j(\vb*{R}) = {_j}\bra{\vb*{R}}\dd{\ket{\vb*{R}}}_j = \mathcal{A}_i(\vb*{R}) + t_{ij}(\vb*{R})^{-1} \dd{t_{ij}(\vb*{R})}.$
• Berry's curvature: $\mathcal{F} = \dd{\mathcal{A}} = \qty(\pdv{\bra{\vb*{R}}}{R^\mu})\qty(\pdv{\ket{\vb*{R}}}{R^\nu}) \dd{R^\mu} \wedge \dd{R^\nu}.$
• Biased Hamiltonian $\mathcal{H}(\vb*{R})\ket{\vb*{R}(t),t} = i \dv{}{t}\ket{\vb*{R}(t),t}.$
• $\ket{\vb*{R}(t),t} = e^{i\eta(t)}\ket{\vb*{R}(t)}.$
• Berry's phase as holonomy: $\tilde{\vb*{R}}(1) = \exp\qty(-\oint \bra{\vb*{R}}\dd{\ket{\vb*{R}}}) \cdot \ket{\vb*{R}(0)}.$
• Electron in a magnetic field: $H(\vb*{B}) = \vb*{B}\cdot \vb*{\sigma}.$
• Stick with the positive energy state $$\ket{\vb*{R}}_{\mathrm{N}}$$.
• $\mathcal{A}_{\mathrm{N}} = {_\mathrm{N}}\bra{\vb*{R}} \dd{\ket{\vb*{R}}_\mathrm{N}}.$
• Field strength $\mathcal{F} = \dd{\mathcal{A}} = \frac{i}{2} \frac{R_1 \dd{R_2} \wedge R_3 + R_2 \dd{R_3} \wedge \dd{R_1} + R_1 \dd{R_1}\wedge \dd{R_2}}{R^3}.$
• With $$\vb*{B}$$ given in spherical coordinates we have $\mathcal{A}_{\mathrm{S}} = \mathcal{A}_{\mathrm{N}} + e^{i\phi} \dd{e^{-i\phi}}.$
• Similarity to the Dirac monopole due to the local $$\mathrm{U}(1)$$ invariance here.

### Characteristic Classes

#### Invariant Polynomials and Chern-Weil Homomorphism

##### Invariant Polynomials
• $$S^r(M(k,\mathbb{C}))$$ denote the vector space of symmetric $$r$$-linear $$\mathbb{C}$$-valued functions on $$M(k,\mathbb{C})$$.
• $S^*(M(k,\mathbb{C})) = \bigoplus_{r=0}^\infty S^r(M(k,\mathbb{C})).$
• Product in $$S^*$$: $\tilde{P}\tilde{Q}(X_1,\cdots,X_{p+q}) = \frac{1}{(p+q)!}\sum_{\sigma} \tilde{P}(X_{\sigma(1)}, \cdots,X_{\sigma(p)})\tilde{Q}(X_{\sigma(p+1)},\cdots,X_{\sigma(p+q)}).$
• $$\tilde{P}$$ invariant if $\tilde{P}(gA_1g^{-1},\cdots,gA_r g^{-1}) = \tilde{P}(A_1,\cdots,A_r).$
• Denoted $$\tilde{P}\in I^r$$.
• $I^*({\frak g}) = \bigoplus_{r\ge 0} I^r({\frak g}).$
• An algebra under the induced multiplication $I^p({\frak g}) \oplus I^q({\frak g}) \rightarrow I^{p+q}({\frak g}).$
• Invariant polynomial: $P(A) = \tilde{P}(A,\cdots,A).$
• $$P\in I^r({\frak{g}})$$.
• $$A\in{\frak{g}}$$.
• Any invariant polynomial may be written as a sum of $$\tr(A^r)$$.
• Acting on $$\frak{g}$$-valued $$p$$-forms by extension: $\tilde{P}(A_1 \eta_1,\cdots,A_r \eta_r) = \eta_1 \wedge \cdots \wedge \eta_r \tilde{P}(A_1,\cdots,A_r).$
• Polarization: from any invariant polynomial $$P$$ be define $$\tilde{P}\in I^r$$ by expanding $P(t_1A_1 + \cdots + t_r A_r)$ and extract the coefficient of $t_1 t_2 \cdots t_r,$ and multiply it by $$1/r!$$.
• Chern-Weil theorem: $$P$$ an invariant polynomial, then
• $\dd{P(\mathcal{F})} = 0.$
• Let $$\mathcal{F}$$ and $$\mathcal{F}'$$ be curvature two-forms corresponding to different connections $$\mathcal{A}$$ and $$\mathcal{A}'$$. Then $P(\mathcal{F}') - P(\mathcal{F})$ is exact.
• Notes:
• We should clarity how $$\dd{}$$ is acting: if $\mathcal{F} = {\frak{g}}_i \eta^{(i)}_{\mu\nu}(x) \dd{x^\mu} \wedge \dd{x^\nu},$ then $$\dd{}$$ acts on $\eta^{(i)}_{\mu\nu}(x) \dd{x^\mu} \wedge \dd{x^\nu}$ and $${\frak{g}}_i$$ are deemed as constants.
• A useful identity is (for $$\Omega_i$$ being $${\frak{g}}_i$$-valued forms) $\dd{\tilde{P}_r}(\Omega_1,\cdots,\Omega_r) = \sum_{i=1}^r (-1)^{p_1 + \cdots + p_{i-1}}\tilde{P}_r(\Omega_1, \cdots, \dd{\Omega}_i,\cdots,\Omega_r).$
• With $\mathcal{A}_t = \mathcal{A} + t\theta;\quad \theta = (\mathcal{A}' - \mathcal{A}),$ and $\mathcal{F}_t = \dd{\mathcal{A}}_t + \mathcal{A}_t \wedge \mathcal{A}_t = \mathcal{F} + t\mathcal{D}\theta + t^2 \theta^2,$ we have $P_r(\mathcal{F}') - P_r(\mathcal{F}) = \dd{\qty[r\int_0^1 \tilde{P}_r(\mathcal{A}' - \mathcal{A}, \mathcal{F}_t,\cdots,\mathcal{F}_t) \dd{t}]}.$
• Transgression: $TP_r(\mathcal{A}',\mathcal{A}) = r\int_0^1 \dd{t} \tilde{P}_r(\mathcal{A}' - \mathcal{A}, \mathcal{F}_t,\cdots,\mathcal{F}_t).$
• Invariant polynomial is closed and (in the sense of cohomology) independent of the gauge potential chosen.
• $$P$$ stays in the same cohomology class after changing the gauge potential $$\mathcal{A}$$.
• The cohomology class is called the characteristic class, denoted by $\chi_E(P),$ where $$E$$ is the fibre bundle over which connections and curvatures are defined.

The symmetrized trace $\tilde{P}(A_1,\cdots,A_r) = \operatorname{str}(A_1,\cdots,A_r) = \frac{1}{r!} \sum_\sigma \tr(A_{\sigma(1)} \cdots A_{\sigma(r)})$ is symmetric, $$r$$-linear and invariant. $P(A) = \tr(A^r)$ is an invariant polynomial. Acting on $$p$$-forms $$P$$ becomes $\operatorname{str}(A_1 \eta_1,\cdots,A_r \eta_r) = \eta_1 \wedge \cdots \wedge \eta_r \operatorname{str}(A_1,\cdots,A_r).$ The diagonal combination becomes $P(A\eta) = \eta\wedge \cdots \wedge \eta P(A).$

From $P(A) = \tr(A^3)$ we define the polarization $\tilde{P}(A_1,A_2,A_3) = \frac{1}{6!}\cdot 3\tr(A_1A_2A_3 + A_2A_1A_3) = \operatorname{str}(A_1,A_2,A_3).$

• Theorem. Let $$P$$ be an invariant polynomial in $$I^*(G)$$ and $$E$$ a fibre bundle over $$M$$ with structure group $$G$$.
• The map $\chi_E: P\in I^*(G) \rightarrow \chi_E(P)\in H^*(M)$ is a homomorphism (the Weil homomorphism), i.e. $(P_r P_s)(\mathcal{F}) = P_r(\mathcal{F}) \wedge P_s(\mathcal{F})$ is in the cohomology class defined by $\chi_E(P_r) \wedge \chi_E(P_s).$
• If $$f:N\rightarrow M$$ is a differentiable map, and $$f^* E$$ is the pullback bundle, then we have the naturality $\chi_{f^* E} = f^* \chi_E,$ i.e. $f^* \chi_E(P) = \chi_{f^* E}(P).$
• It could be verified that $$f^*\mathcal{A}$$ is a connection in $$f^* E$$. Without this it could not be guaranteed that $$\chi_{f^* E}(P)$$ makes sense.
• Characteristic classes of trivial bundles are trivial.

#### Chern Class

##### Definition of Chern Classes
• Total Chern class: for a complex vector bundle whose fibre is $$\mathbb{C}^k$$, \begin{align*} c(\mathcal{F}) &= \det\qty(\mathbb{1} + \frac{i\mathcal{F}}{2\pi}) \\ &= 1 + c_1(\mathcal{F}) + c_2(\mathcal{F}) + \cdots. \end{align*}{}
• $$j$$th Chern class: $c_j(\mathcal{F}) = \Omega^{2j}(M).$
• Diagonalization: at each point the field is diagonalized as $g^{-1}\qty(\frac{i\mathcal{F}}{2\pi}) g = \operatorname{diag}(x_1,\cdots,x_k).$
• $$x_k$$ are two-forms (scalar-valued).
• Total Chern class $\det\qty(\mathbb{1} + \frac{i\mathcal{F}}{2\pi}) = S_0(x) + S_1(x) + \cdots + S_k(x).$
• $$S_i$$ are elementary symmetric functions: \begin{align*} c_0 = S_0(x) &= 1, \\ c_1 = S_1(x) &= \sum x_j = \frac{i}{2\pi} \tr \mathcal{F}, \\ c_2 = S_2(x) &= \sum_{i<j} x_i \wedge x_j = \frac{1}{2}\qty(\frac{i}{2\pi})^2 \qty[\tr \mathcal{F} \wedge \tr \mathcal{F} - \tr(\mathcal{F} \wedge \mathcal{F})], \\ &\vdots \\ c_k = S_k(x) &= x_1 \wedge \cdots \wedge x_k = \qty(\frac{i}{2\pi})^k \det \mathcal{F}. \end{align*}{}
• $$F$$ a complex vector bundle $$\mathbb{C}^2$$ over $$M$$.
• $$G = \mathrm{SU}(2)$$.
• $$\dim M = 4$$.
• Field $\mathcal{F} = \mathcal{F}^\alpha \frac{\sigma_\alpha}{2i},$ where $\mathcal{F}^\alpha = \frac{1}{2} \mathcal{F}{^\alpha}_{\mu\nu} \dd{x^\mu} \wedge \dd{x^\nu}.$
• Total Chern class \begin{align*} c(\mathcal{F}) &= \det\qty(\mathbb{1} + \frac{i}{2\pi} \mathcal{F}^\alpha \frac{\sigma_\alpha}{2i}) \\ &= \det \begin{pmatrix} 1 + (i/2\pi)\mathcal{F}^3/(2i) & (i/2\pi) (\mathcal{F}^1 - i\mathcal{F}^2)/(2i) \\ (i/2\pi) (\mathcal{F}^1 + i\mathcal{F}^2)/(2i) & 1-(i/2\pi)(\mathcal{F}^3)/(2i) \end{pmatrix} \\ &= 1 + \frac{1}{4}\qty(\frac{i}{2\pi})^2\qty{\mathcal{F}^3 \wedge \mathcal{F}^3 + \mathcal{F}^1 \wedge \mathcal{F}^1 + \mathcal{F}^2 \wedge \mathcal{F}^2}. \end{align*}{}
• Individual Chern class: \begin{align*} c_0(\mathcal{F}) &= 1, \\ c_1(\mathcal{F}) &= 0, \\ c_2(\mathcal{F}) &= \qty(\frac{i}{2\pi})^2 \sum_\alpha \frac{\mathcal{F}^\alpha \wedge \mathcal{F}^\alpha}{4}. \end{align*}{}
##### Properties of Chern Classes
• $$E$$ a vector bundle with $$G = \mathrm{GL}(k,\mathbb{C})$$ and $$F = \mathbb{C}^k$$.
• Naturality of total Chern class: if $$f:N\rightarrow M$$ is a smooth map, then $c(f^* E) = f^* c(E).$
• Let $$E'$$ be another vector bundle and $$E\oplus E'$$ be their Whitney sum bundle. Then $c(E\oplus E') = c(E) \wedge c(E').$
• Note that the curvature of a Whitney sum bundle is block diagonal: $\mathcal{F}_{E\oplus E'} = \operatorname{diag}(\mathcal{F}_E, \mathcal{F}_{E'}).$
##### Splitting Principle
• Let $$E\rightarrow M$$ be a vector bundle of rank $$n$$ over a paracompact space $$M$$. There exists a space $$N$$ called the flag bundle associated to $$E$$ and a map $$p:N\rightarrow M$$ such that
• the induced cohomology homomorphism $p^*: H^*(M) \rightarrow H^*(N)$ is injective, and
• the pullback bundle $$p^* E$$ breaks up as a direct sum of line bundles $p^*(E) = L_1 \oplus \cdots \oplus L_n.$
##### Universal Bundles and Classifying Spaces
• Let $$M$$ be a manifold with $$\dim M = m$$ and let $$E\rightarrow M$$ be a complex vector bundle with the fibre $$\mathbb{C}^k$$. Then there exists a natural number $$N$$ such that for all $$n>N$$,
• there exists a map $$f:M\rightarrow G_{k,n}(\mathbb{C})$$ such that $E \cong f^* L_{k,n}(\mathbb{C});$ where $$L_{k,n}(\mathbb{C})$$ is the canonical $$k$$-plane byndle $$L_{k,n}$$ over $$G_{k,n}(\mathbb{C})$$ with fibre $$\mathbb{C}^k$$;
• $$f,g: M\rightarrow G_{k,n}(\mathbb{C})$$ are homotopic if and only if $f^* L_{k,n}(\mathbb{C}) \cong g^* L_{k,n}(\mathbb{C}).$
• Classifying space $$G_k(\mathbb{C})$$: $G_k(\mathbb{C}) = \bigcup_{n=k}^\infty G_{k,n}(\mathbb{C}).$
• Natural inclusions: $G_{k,k}(\mathbb{C}) \hookrightarrow G_{k,k+1}(\mathbb{C}) \hookrightarrow \cdots \hookrightarrow G_{k}(\mathbb{C}).$
• Universal bundle: $L_k \rightarrow G_k(\mathbb{C})$ with fibre $$\mathbb{C}^k$$.
• For any fibre bundle $$E$$ with fibre $$\mathbb{C}^k$$ we have a map $$f$$ such that $E = f^* L_k(\mathbb{C}).$
• A character class is a map $\chi: E\rightarrow \chi(E) \in H^*(M)$ such that
• Naturality: $\chi(f^* E) = f^* \chi(E).$
• If $$E$$ is equivalent to $$E'$$: $\chi(E) = \chi(E').$
• Chern class $$c(E)$$ defined axiomatically by
• $c(f^* E) = f^* c(E).$
• $c(E) = c_0(E) + c_1(E) + \cdots + c_k(E),$ where $c_i(E) \in H^{2i}(M)$ and $$c_i(E) = 0$$ for $$i>k$$.
• $c(E\oplus F) = c(E) \wedge c(F).$
• Normalization $c(L) = 1 + x$ where $$L$$ is the canonical line bundle over $$\mathbb{C}P^n$$.

#### Chern Characters

##### Definition of Chern Characters
• Total Chern character: $\operatorname{ch}(\mathcal{F}) = \tr \exp \qty(\frac{i\mathcal{F}}{2\pi}) = \sum_{j=0} \frac{1}{j!} \tr\qty(\frac{i\mathcal{F}}{2\pi})^j.$
• $$j$$th Chern character: $\mathrm{ch}_j(\mathcal{F}) = \frac{1}{j!} \mathrm{tr}\qty(\frac{i\mathcal{F}}{2\pi})^j.$
• Diagonalized: $\operatorname{ch}(\mathcal{F}) = \sum_{j=1}^k \exp(x_j).$
• In terms of Chern classes: \begin{align*} \mathrm{ch}_0(\mathcal{F}) &= k, \\ \mathrm{ch}_1(\mathcal{F}) &= c_1(\mathcal{F}), \\ \mathrm{ch}_2(\mathcal{F}) &= \frac{1}{2}\qty[c_1(\mathcal{F})^2 - 2c_2(\mathcal{F})]. \end{align*}{}
• $$P$$ a $$\mathrm{U}(1)$$ bundle over $$S^2$$.
• $$\mathcal{A}_{\mathrm{N}}$$ and $$\mathcal{A}_{\mathrm{S}}$$ are local connections.
• $$\mathcal{F}^n = 0$$ for $$n \ge 2$$.
• Chern character $\mathrm{ch}(\mathcal{F}) = \mathbb{1} + \frac{i\mathcal{F}}{2\pi}.$
• Magnetic charge $2g = N = \frac{i}{2\pi} \int_{S^2} \mathcal{F} = \int_{S^2} \mathrm{ch}_1(\mathcal{F}).$
• $$P$$ a $$\mathrm{SU}(2)$$ bundle over $$S^4$$.
• Total Chern character $\mathrm{ch}(\mathcal{F}) = 2 + \cancelto{0}{\tr\qty(\frac{i \mathcal{F}}{2\pi})} + \frac{1}{2} \tr\qty(\frac{i\mathcal{F}}{2\pi})^2.$
• Instanton number $\frac{1}{2} \int_{S^4} \tr\qty(\frac{i\mathcal{F}}{2\pi})^2 = \int_{S^4} \mathrm{ch}_2(\mathcal{F}).$
• $$P$$ a $$\mathrm{U}(1)$$ bundle over $$M$$ a $$2m$$-dimensional manifold.
• The $$m$$th Chern character $\frac{1}{m!} \tr\qty(\frac{i\mathcal{F}}{2\pi})^m = \qty(\frac{i}{4\pi})^m \epsilon^{\mu_1\nu_1 \cdots \mu_m \nu_m} F_{\mu_1\nu_1 \cdots \mu_m \nu_m} \dd{x^1} \wedge \cdots \wedge \dd{x^{2m}}.$
• $$L$$ a complex line bundle.
• Curvature under the Fubini-Study metric $\mathcal{F} = -\partial \overline{\partial} \ln(1+\abs{z}^2) = -\frac{\dd{z}\wedge \dd{\overline{z}}}{(1+z\overline{z})^2} = 2i \frac{r\dd{r}\wedge \dd{\theta}}{(1+r^2)^2}.$
• The first Chern character $\mathrm{ch}_1(\mathcal{F}) = -\frac{1}{\pi} \frac{r\dd{r}\wedge \dd{\theta}}{(1+r^2)^2}.$
• Integral of the first Chern character $\mathrm{Ch}_1(L) = -\frac{1}{\pi} \int \frac{r\dd{r} \dd{\theta}}{(1+r^2)^2} = -1.$
##### Properties of the Chern Characters
• Naturality: $\mathrm{ch}(f^* E) = f^* \mathrm{ch}(E).$
• Let $$E$$ and $$F$$ be vector bundles over $$M$$. The Chern classes of their direct product and Whitney sum are given by, respectively, \begin{align*} \mathrm{ch}(E\otimes F) &= \mathrm{ch}(E) \wedge \mathrm{ch}(F), \\ \mathrm{ch}(E\oplus F) &= \mathrm{ch}(E) + \mathrm{ch}(F). \end{align*}{}
##### Todd Classes
• Todd class: \begin{align*} \mathrm{Td}(\mathcal{F}) &= \prod_j \frac{x_j}{1-e^{-x_j}} \\ &= \prod_j \qty({1 + \frac{1}{2} x_j + \sum_{k\ge 1} (-1)^{k-1} \frac{B_{k}}{(2k)!} x^{2k}_j}) \\ &= 1 + \frac{1}{2} c_1(\mathcal{F}) + \frac{1}{12}\qty[c_1(\mathcal{F})^2 + c_2(\mathcal{F})] + \cdots. \end{align*}{}

#### Pontrjagin and Euler Classes

##### Pontrjagin Classes
• $$E$$ a real vector bundle.
• Orthonormal frames assigned.
• Structure group $$\mathrm{O}(k)$$.
• Lie algebra: not diagonalizable, but reducible to $\begin{pmatrix} 0 & \lambda_1 & & & \\ -\lambda_1 & 0 & & & \\ & & 0 & \lambda_2 & \\ & & -\lambda_2 & 0 & \\ & & & & \ddots \end{pmatrix}.$
• Total Pontrjagin class: $p(\mathcal{F}) = \det\qty(\mathbb{1} + \frac{\mathcal{F}}{2\pi}).$
• For skew-symmetric $$\mathcal{F}^T = -\mathcal{F}$$, $p(\mathcal{F}) = p(\mathcal{F}^T).$
• Expansion $p(\mathcal{F}) = 1 + p_1(\mathcal{F}) + p_2(\mathcal{F}) + \cdots.$
• $$p_j$$ is a polynomial of order $$2j$$ and is an element of $$H^{4j}(M;\mathbb{R})$$.
• Diagonalized $p(\mathcal{F}) = \prod_{i=1}^{\lfloor k/2 \rfloor} (1+x_i^2),$ where $\frac{\mathcal{F}}{2\pi} \rightarrow \begin{pmatrix} -ix_1 & & & & \\ & ix_1 & & & \\ & & -ix_2 & & \\ & & & ix_2 & \\ & & & & \ddots \end{pmatrix}.$
• Pontrjagin class $p_j(\mathcal{F}) = \sum_{i_1<\cdots<i_j} x_{i_1}^2 \cdots x_{i_j}^2.$
• In terms of $$\mathcal{F}$$, \begin{align*} p_1(\mathcal{F}) &= -\frac{1}{2}\qty(\frac{1}{2\pi})^2 \tr \mathcal{F}^2, \\ p_2(\mathcal{F}) &= \frac{1}{8}\qty(\frac{1}{2\pi})^4 \qty[(\tr \mathcal{F}^2)^2 - 2\tr \mathcal{F}^4], \\ & \vdots \\ p_{\lfloor k/2 \rfloor}(\mathcal{F}) &= \qty(\frac{1}{2\pi})^k \det \mathcal{F},\quad \text{if } k \text{ is even}. \end{align*}{}
• Pontrjagin class of Whitney sum: $p(E\oplus F) = p(E) \wedge p(F).$
• Relation to Chern classes: $p_j(E) = (-1)^j c_{2j}(E^{\mathbb{C}}).$
• $$M$$ a four-dimensional Riemannian manifold.
• Orthonormal frame employed.
• Structure group $$\mathrm{O}(4)$$.
• Curvature two-form $\mathcal{R} = \frac{1}{2} \mathcal{R}_{\alpha\beta} \theta^\alpha \wedge \theta^\beta.$
• Pontrjagin class: $\det\qty(\mathbb{1} + \frac{\mathcal{R}}{2\pi}) = \underbrace{1}_{p_0} \underbrace{\mathbin{-}\frac{1}{8\pi}^2 \tr \mathcal{R}^2}_{p_1} + \underbrace{\frac{1}{128\pi^4}\qty[(\tr \mathcal{R}^2)^2 - 2\tr \mathcal{R}^4]}_{p_2}.$
##### Euler Classes
• $$M$$ a $$2l$$-dimensional orientable Riemannian manifold.
• Euler class: square root of $$p_l$$, defined by $e(A)e(A) = p_l(A).$
• Both sides understood as functions of a $$2l\times 2l$$ matrix $$A$$.
• We solve for $$e(A)$$ first and then substitute $$\mathcal{R}$$ for $$A$$.
• If $$M$$ is odd-dimensional we define $e(M) = 0.$
• Pfaffian: for $$P$$ a skew-symmetric $$2l\times 2l$$-matrix only, $\operatorname{Pf}(A) = \frac{(-1)^l}{2^l l!} \sum_\sigma \operatorname{sign}(\sigma) A_{\sigma(1)\sigma(2)} \cdots A_{\sigma(2l-1) \sigma(2l)}.$
• $\det A = (\operatorname{Pf}(A))^2.$
• $\operatorname{Pf}(X^t A X) = \operatorname{Pf}(A) \det A.$
• Since the determinant of an odd-dimensional skew-symmetric matrix vanishes, we put $$e(M) = 0$$ for an odd-dimensional manifold.
• Euler class in terms of $$\mathcal{R}$$: $e(M) = \operatorname{Pf}\qty(\frac{\mathcal{R}}{2\pi}) = \frac{(-1)^l}{(4\pi)^l l!} \sum_\sigma \operatorname{sign}(\sigma) \mathcal{R}_{\sigma(1)\sigma(2)} \cdots \mathcal{R}_{\sigma(2l-1)\sigma(2l)}.$
• Generating function $e(x) = x_1 \cdots x_l,$ where $x_j = -\frac{\lambda_j}{2\pi}.$
• $$M = S^2$$. Consider $$TS^2$$.
• Curvature two-form $\mathcal{R}_{\theta\phi} = \sin\theta \dd{\theta} \wedge \dd{\phi}.$
• $$p_1(S^2) = 0$$ as a differential form: \begin{align*} p_1(S^2) = -\frac{1}{8\pi^2} \tr \mathcal{R}^2 = \qty(\frac{1}{2\pi} \sin\theta \dd{\theta} \wedge \dd{\phi})^2. \end{align*}
• Reading off the Euler form: $e(S^2) = \frac{1}{2\pi} \sin\theta \dd{\theta} \wedge \dd{\phi}.$
• Euler characteristic: $\int_{S^2} e(S^2) = 2.$
• $$M$$ a four-dimensional orientable manifold.
• The structure group of $$TM$$ is $$\mathrm{O}(4)$$.
• Euler class $e(M) = \frac{1}{2(4\pi)^2} \epsilon^{ijkl} \mathcal{R}_{ij} \wedge \mathcal{R}_{kl}.$
• Pontrjagin class: $p_2(M) = \frac{1}{128\pi^4}\qty[(\tr \mathcal{R}^2)^2 - 2\tr \mathcal{R}^4] = x_1^2 x_2^2.$
##### Hirzebruch L-Polynomial and Â-Genus
• Hirzebruch $$L$$-Polynomial: \begin{align*} L(x) &= \prod_{j=1}^k \frac{x_j}{\tanh x_j} \\ &= \prod_{j=1}^k \qty(1 + \sum_{n\ge 1} (-1)^{n-1} \frac{2^{2n}}{(2n)!}B_n x^{2n}_j). \end{align*}{}
• In terms of Pontrjagin classes: $L(\mathcal{F}) = 1 + \frac{1}{3} p_1 + \frac{1}{45}(-p_1^2 + 7p_2) + \cdots.$
• $L(E\oplus F) = L(E) \wedge L(F).$
• $$\hat{A}$$ genus, Dirac genus: \begin{align*} \hat{A}(\mathcal{F}) &= \prod_{j=1}^k \frac{x_j/2}{\sinh(x_j / 2)} \\ &= \prod_{j=1}^k \qty({1 + \sum_{n\ge 1} (-1)^n \frac{(2^{2n} - 2)}{(2n)!} B_n x^{2n}_j}). \end{align*}{}
• $\hat{A}(E\oplus F) = \hat{A}(E) \wedge \hat{A}(F).$
• In terms of Pontrjagin classes: $\hat{A}(\mathcal{F}) = 1 - \frac{1}{24}p_1 + \frac{1}{5760}(7p_1^2 - 4p_2) + \cdots.$
• $$M$$ a compact connected and orientable four-dimensional manifold.
• Define $\sigma: H^2(M,\mathbb{R}) \times H^2(M,\mathbb{R}) \rightarrow \mathbb{R}$ by $\sigma([\alpha], [\beta]) = \int_M \alpha \wedge \beta.$
• $$\sigma$$ is a $$b^2 \times b^2$$ symmetric matrix, where $b^2 = \dim H^2(M;\mathbb{R}).$
• Hirzebruch signature: $\tau(M) = p - q.$ - $$p$$ and $$q$$ are the number of positive and negative eigenvalues of $$\sigma$$, respectively.
• Hirzebruch signature theorem: $\tau(M) = \int_{M} L_1(M) = \frac{1}{3} \int_M p_1(M).$

#### Chern-Simons Forms

##### Definition of Chern-Simons Forms
• Chern-Simons form of $$P_j(\mathcal{F})$$, where $$P_j(\mathcal{F})$$ is an arbitrary $$2j$$-form characteristic class: $Q_{2j-1}(\mathcal{A},\mathcal{F}) = TP_j(\mathcal{A},0) = j\int_0^1 \tilde{P}_j(\mathcal{A}, \mathcal{F}_t,\cdots,\mathcal{F}_t) \dd{t}.$
• We set $$\tilde{A}' = 0$$ and $$\tilde{F}' = 0$$.
• $Q$ exists only locally..text-danger
• $$Q_{2j-1}$$ is an $$(2j-1)$$-form.
• $$Q_{2j-1}$$ defines the topology of the boundary.
##### The Chern-Simons Form of the Chern Character
• $\mathcal{A}_t = t\mathcal{A}.$
• $\mathcal{F}_t = t\mathcal{F} + (t^2 - t)\mathcal{A}^2.$
• $Q_{2j-1}(\mathcal{A},\mathcal{F}) = \frac{1}{(j-1)!}\qty(\frac{i}{2\pi})^j \int_0^1 \dd{t} \operatorname{str}(\mathcal{A}, \mathcal{F}_t^{j-1}).$
• Examples: \begin{align*} Q_1(\mathcal{A},\mathcal{F}) &= \frac{i}{2\pi} \tr \mathcal{A}, \\ Q_3(\mathcal{A},\mathcal{F}) &= \frac{1}{2} \qty(\frac{i}{2\pi})^2 \tr \qty(\mathcal{A} \dd{\mathcal{A}} + \frac{2}{3}\mathcal{A}^3), \\ Q_5(\mathcal{A},\mathcal{F}) &= \frac{1}{6} \qty(\frac{i}{2\pi})^3 \tr \qty[\mathcal{A}(\dd{\mathcal{A}})^2 + \frac{3}{2}\mathcal{A}^3 \dd{\mathcal{A}} + \frac{3}{5}\mathcal{A}^5]. \end{align*}{}
##### Cartan's Homotopy Operator and Applications
• $$S(\mathcal{A},\mathcal{F})$$ a polynomial.
• Homotopy operator: $k_{01} S(\mathcal{A}_t,\mathcal{F}_t) = \int_0^1 \delta t l_t S(\mathcal{A}_t, \mathcal{F}_t).$
• Cartan's homotopy formula: $S(\mathcal{A}_1,\mathcal{F}_1) - S(\mathcal{A}_0,\mathcal{F}_0) = (\dd{k_{01}} + k_{01}\dd{})S(\mathcal{A_t},\mathcal{F}_t).$
• $$l_t$$: an anti-derivative operator, \begin{align*} l_t \mathcal{A}_t &= 0, \\ l_t \mathcal{F}_t &= \delta t(\mathcal{A}_1 - \mathcal{A}_0), \\ l_t(\eta_p \omega_q) &= (l_t \eta_p)\omega_q + (-1)^p \eta_p l_t(\omega_q). \end{align*}

With $\mathrm{ch}_{j+1}(\mathcal{F}) = \dd{\qty[k_{01} \mathcal{ch}_{j+1}(\mathcal{F}_t)]} = \dd{Q_{2j+1}},$ we find $Q_{2j+1} = k_{01} \mathrm{ch}_{j+1}(\mathcal{F}_t) = \frac{1}{j!}\qty(\frac{i}{2\pi})^{j+1} \int_0^1 \delta t \mathrm{str}(\mathcal{A}, \mathcal{F}^j_t).$

• The Chern-Simons form is not necessarily gauge invariant: $Q_{2j+1}(\mathcal{A}^g,\mathcal{F}^g) - Q_{2j+1}(\mathcal{A},\mathcal{F}) = Q_{2j+1}(g^{-1} \dd{g}, 0) + \dd{\alpha_{2j}},$
• where \begin{align*} \mathcal{A}^g &= g^{-1}(\mathcal{A} + \dd{}) g, \\ \mathcal{F}^g &= g^{-1} \mathcal{F} g. \end{align*}{}
• The first term on the RHS $Q_{2j+1}(g^{-1}\dd{g},0)$ is closed.
• Lagrangian: $$\mathrm{U}(1)$$ bundle in three-dimensional spacetime, with $$\mathcal{A} = iA$$ and $$\mathcal{F} = iF$$, $\mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{1}{4} m \epsilon^{\lambda\mu\nu} F_{\lambda\mu} A_\nu.$
• The second term is the Chern-Simons forms of the second Chern character $$F^2$$.
• Field equation $\partial_\mu F^{\mu\nu} + m {*} F^\nu = 0.$
• Under $A_\mu \rightarrow A_\mu + \partial_\mu \theta,$ the Lagrangian changes by $\mathcal{L} \rightarrow \mathcal{L} + \frac{1}{4} m \epsilon^{\lambda\mu\nu} F_{\lambda\mu}(A_\nu + \partial_\nu \theta) = \mathcal{L} + \frac{1}{2} m \partial_\nu({*} F^\nu \theta).$

#### Stiefel-Whitney Classes

##### Spin Bundles
• $$LM$$ the frame bundle related to $$TM$$, where $$M$$ a $$m$$-dimensional manifold.
• Spin structure: $$\tilde{t}_{ij}\in \mathrm{Spin}(m)$$ such that $\varphi(t_{ij}) = t_{ij}$ where $$\varphi$$ is the projection from $$\mathrm{Spin}(m)$$ to $$\mathrm{SO}(m)$$.
• Not all manifolds admit spin structures.
##### &circC;ech Cohomology Groups
• &circC;ech $$r$$-cochain: totally symmetric function to $$\mathbb{Z}_2$$, i.e. for any permutation $$\sigma$$, $f(i_{\sigma(0)}, \cdots, i_{\sigma(r)}) = f(i_0,\cdots,i_r).$
• $$C^r(M;\mathbb{Z}_2)$$ be the multiplicative group of &circC;ech $$r$$-cochains.
• Coboundary operator $\delta: C^r(M;\mathbb{Z}_2) \rightarrow C^{r+1}(M;\mathbb{Z}_2)$ defined by $(\delta f)(i_0,\cdots,i_{r+1}) = \prod_{j=1}^{r+1}f(i_0,\cdots,\cancel{i}_j,\cdots,i_{r+1}).$
• $$\delta$$ is nilpotent: $\delta^2 f = 1.$
• Cocycle group $$Z^r(M;\mathbb{Z}_2)$$, coboundary group $$B^r(M;\mathbb{Z}_2)$$, and &circC;ech cohomology group $$H^r(M;\mathbb{Z}_2)$$: \begin{align*} Z^r(M;\mathbb{Z}_2) &= \set{f\in C^r(M;\mathbb{Z}_2) | \delta f = 1}, \\ B^r(M;\mathbb{Z}_2) &= \set{f\in C^r(M;\mathbb{Z}_2) | f = \delta f'}, \\ H^r &= \operatorname{ker} \delta_r / \operatorname{im} \delta_{r-1} = Z^r(M,\mathbb{Z}_2)/B^r(M,\mathbb{Z}_2). \end{align*}
##### Definition of Stiefel-Whitney Classes
• Stiefel-Whitney class $$w_r$$: a characteristic class which takes values in $$H^r(M;\mathbb{Z}_2)$$.
• $$\qty{U_i}$$ a simple open covering of $$M$$.
• The intersection of any number of charts is either empty of contractible.
• $$\qty{e_{i\alpha}}$$ be a local orthonormal frame of $$TM$$ over $$U_i$$.
• &circC;ech $$1$$-cochain defined by $f(i,j) = \det(t_{ij}) = \pm 1.$
• $$\delta f = 1$$, i.e. $f\in Z^1(M;\mathbb{Z}_2).$
• First Stiefel-Whitney class: $w_1(M) = \qty[f] \in H^1(M;\mathbb{Z}_2).$
• $$f$$ on a new frame $\tilde{e}_{i\alpha} = h_i e_{i\alpha}$ still defines the same cohomology class.
• $$M$$ is orientable if and only if $$w_1(M)$$ is trivial.
• Second Stiefel-Whitney class: $w_2(M) \in H^2(M;\mathbb{Z}_2)$ defined by $$[f]$$ where $\tilde{t}_{ij}\tilde{t}_{jk}\tilde{t}_{ki} = f(i,j,k) \mathbb{1}.$
• $$f$$ on a new lift $$-\tilde{t}_{ij}$$ still defines the same cohomology class.
• If $$M$$ is orientable, then a spin bundle over $$M$$ exists if and only if $$w_2(M)$$ is trivial.
• Examples:
• $$x$$ denotes the generator of $$H^2(\mathbb{C}P^m;\mathbb{Z}_2)$$, $w_1(\mathbb{C}P^m) = 1,\quad w_2(\mathbb{C}^m) = \begin{cases} 1, & m \text{ odd}, \\ x, & m \text{ even}. \end{cases}$
• $w_1(S^m) = w_2(S^m) = 1.$
• $$\Sigma_g$$ denotes the Riemann surface of genus $$g$$, $w_1(\Sigma_g) = w_2(\Sigma_g) = 1.$

Expand

## Japanese Phonology and Phonetics

#### Hiraganas

##### ん
• [m] when the following sound is [m], [p], [b].
• えんぴつ
• なん (Nambu)
• [ŋ] when the following sound is [k], [g].
• てん
• [N] when at the end of utterances. Touch the root of your tongue with your uvula.
• ほん
• [n] elsewhere. Touch the roof of your mouth with your tongue just a little farther back than your upper gums.

#### Phonology

##### 大和言葉

See 大和言葉.

• There should be no 濁音/半濁音 at the beginning of a word. However, exceptions do exist in modern Japanese.
• く (c. 999) from いだく (c. 850), where the beginning い was dropped, the latter from うだく (c. 810), ultimately from むだく (c. ) from old Japanese.
• むだく (c. 759) from + く, where was shifted from , itself cognate with , while く was probably derived from .
• 薔薇ばら from いばら, ultimately from うばら which shifted also to むばら.
• from にぱ, cognate with にわ.
• This rule does not apply to オノマトペ (e.g. ピカピカ), 動植物名 (e.g. ブリ, ブナ), alternation from voiceless consonants (e.g. ばば, じじ), and words that emphasizes negative meaning (e.g. がにまた, ずるい).
• There should be no ら行音 at the beginning of a word, a property that shared with Altaic languages.
• Upon forming compounds, 連濁 may occur, and the consonants in the first word may alter, e.g. ち from + ち, and さかだる from さけ + たる.

On the etymology of 大和やまと: 諸説ある

• やまふもと
• やまかど
• かど itself from か + , the latter of which is to be distinguished from .
• か was probably prepended the same way as in かご, where ご from こ appears in む/める.
• Why not やま?
• やまあと
• あと from + .
• Why not やま?
##### Apophony
• acts as an unbound nominalizing suffix (emphatic nominal particle), e.g.
• (standalone form) from (bounded form, only in compounds) + い.
• from + い.
• たけ from たか + い.
• from + い.
• from + い.
• しろ from しら + い.
• くち from くつ + い.
• さけ from さか + い.
• あめ from あま + い.
• あめ from あま + い.
##### Rendaku
• Lyman's Law states that there can be no more than one voiced obstruent (a consonant sound formed by obstructing airflow) within a morpheme.
• やまかど
• ひとたび
• やま
• For some lexical items, rendaku does not manifest if there is a voiced obstruent near the morphemic boundary, including preceding the boundary.
• Rendaku usually only applies to native Japanese words, but it also occurs infrequently in Sino-Japanese words (Japanese words of Chinese origin) especially where the element undergoing rendaku is well integrated ("vulgarized").
• Rendaku also tends not to manifest in compounds which have the semantic value of "X and Y" (so-called dvandva or copulative compounds).
• Rendaku is also blocked by what is called a "branching constraint". In a right-branching compound, the process is blocked in the left-branching elements.
• See here for more rules.

#### Vocabulary

##### Verbs
• 言う as いう or ゆう?
• 終止形/連体形 pronounced as ゆう.
• Otherwise pronounced as いって/いえば.
• Always written as う.
• える is possibly cognate with える.

#### Names

##### Surnames
• 小鳥遊たかなし from たか + し, possibly from たかなし

Expand

## English Vocabulary

#### Daily Life

• Glovebox
• A sealed container that is designed to allow one to manipulate objects where a separate atmosphere is desired.
• A compartment built into the dashboard of an automobile, located over the front-seat passenger's footwell, and often used for miscellaneous storage.
• Vouch (for)
• To assert or confirm as a result of one's own experience that something is true or accurately described.
• Subversion
• A systematic attempt to overthrow a government.
• A revision considered similar to the preceding subversions than a revision deemed a new version is to preceding versions.
• To let (something) lapse
• To let slip.
• Giveaway
• A thing that is given free, often for promotional purposes.
• A thing that makes an inadvertent revelation.
• No-go
• Something which should not or cannot be done.
• Totaled
• (For cars) when the cost to repair the vehicle is higher than the actual cash value (ACV) of the vehicle.
• Thrice
• Three times.
• Punchline
• オチ.
• Humblebrag
• 凡尔赛.

#### Linguistics and Typography

• Diphthong (ˈdifˌTHäNG)
• A sound formed by the combination of two vowels in a single syllable.
• Apostrophe (əˈpästrəfē)
• '
• Double entendre
• A figure of speech or a particular way of wording that is devised to have a double meaning.
• Apophasis
• A rhetorical device wherein the speaker or writer brings up a subject by either denying it, or denying that it should be brought up. 阴阳怪气.
• Glottal stop
• e.g. 鸭 /ʔaʔ/ in 闽南语.
• No audible release
• e.g. /p̚/, /t̚/ and /k̚/ in 粤语.
• Voiceless velar plosive
• /k/, aspirated or not.
• Voiceless dental and alveolar plosives
• /t/, aspirated or not.
• Voiceless bilabial plosive
• /p/, aspirated or not.
• Garden-path sentence
• e.g. The old man the boat. The complex houses married and single soldiers and their families. The horse raced past the barn fell.

#### Mathematics

• Ergodic theory
• A branch of mathematics that studies statistical properties of deterministic dynamical systems.

#### Materials Science

• Castellate
• To build like a castle.
• Electron deficient/precise/rich compound
• The compound with insufficient/exact/more number of electrons than required to form covalent bonds.
• Norbornane
• Isostrutral
• Relating to or having a similar crystal structure in that the atoms correspond in position and function although there may not be a close chemical relationship.
• Pellet
• A small, rounded, compressed mass of a substance.
• Sinter
• Make (a powdered material) coalesce into a solid or porous mass by heating it (and usually also compressing it) without liquefaction.

#### Law

• Restraining order
• An order to protect a person, object, business, company, state, country, establishment, or entity, and the general public, in a situation involving alleged domestic violence, child abuse, assault, harassment, stalking, or sexual assault.

#### Latin

• Caveat lector

#### French

• Cri de cœur
• A passionate appeal, complaint, or protest.

#### Miscellaneous

• A godan word:
 gaggle a disorderly or noisy group of people giggle a light, silly laugh guggle to drink making a gurgling sound goggle to look with wide open eyes, typically in amazement
• Queer
• Sexual and gender identities other than straight and cisgender.
• Cattail
• /ˈkæt.teɪl/ A rare example of gemination in English.
• Uber-
• Denoting an outstanding or supreme example of a particular kind of person or thing.
• To a great or extreme degree.
• Thucydides Trap
• A term popularized by American political scientist Graham T. Allison to describe an apparent tendency towards war when an emerging power threatens to displace an existing great power as a regional or international hegemon. It was coined and is primarily used to describe a potential conflict between the United States and China.
• 修昔底德陷阱.
##### Phrases
• To tell on
• To tell someone in authority about the bad behavior or actions of someone else.
• To have a noticeable effect on someone or something.
• As a matter of course
• Used to say that something will or should happen because it is natural, usual, or logical.
• In due course
• At the appropriate time.
• Manners maketh man
• On the chopping block
• A person who's on the chopping block is in danger of losing their job or getting into serious trouble. A project that's on the chopping block is likely to be terminated. See on the chopping block.
• Dash someone's hopes
• To take with a grain of salt
• To take something with a "grain of salt" or "pinch of salt" is an English idiom that suggests to view something, specifically claims that may be misleading or unverified, with skepticism or to not interpret something literally. See grain of salt.
• Be grateful for what one has.
• On the cusp of
• At the point when something is about to change to something else.
• e.g. She is on the cusp of being a star.
• To keep an honest man honest.
• 防君子不防小人.

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#### Word Formation

• Mostly words for feeling.
• From old Japanese しく.
• 悲しい, 嬉しい, etc.
• May be appended after 未然形, e.g. 勇ましい, 忙しい, etc.
• Subjective.
• や for softness and か for visible.
• 鮮やか, 穏やか, 爽やか, etc.
• Similar to やか.
• 明らか, 柔らか, etc.
• Archaic.
• 連体形-たる, 連用形-と/として.
• 堂々たる, 茫然たる, etc.
• 単なる, 聖なる, etc.

#### Verb Suffixes

• めく: to show signs of, to have the appearance of, to behave like
• ときめく: とき as in どきどき
• ときめく
• うめ
• めかす: causative, to dress up oneself, to make like

#### Uncommon Verbs

• とも/とも: to burn, to be lit / to light, to turn on
• from とどまる or とどめる
• +
• 火を灯す.
• おお/しお, しわ, しな: to bend / to bend, to bully
• ぬくもる/ぬくめる: to become warm / to warm, to heat, to nurse, to renew
• from なつ +
• : to calm down, to die down

おいでになります

おいでになります

おいでになります
おこしになります

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test2

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test

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## Density Functional Theory (II)

Kohn-Sham Equations

#### Foundations

##### The Hohenberg-Kohn Theorems
• Hamiltonian: $H = -\frac{\hbar^2}{2m} \sum_i \grad_i^2 + \sum_i V_{\mathrm{ext}}(\vb*{r}_i) + \frac{1}{2}\sum_{i\neq j} \frac{e^2}{\abs{\vb*{r}_i - \vb*{r}_j}}.$
• Theorem. For any system of interacting particles in an external potential $$V_{\mathrm{ext}}(\vb*{r})$$, the potential $$V_{\mathrm{ext}}(\vb*{r})$$ is determined uniquely, up to a constant, by the ground state particle density $$n_0(\vb*{r})$$.
• Therefore, teh wavefunctions of all states are determined, given only $$n_0(\vb*{r})$$.
• Theorem. A universal functional $$E[n]$$ in terms of the density $$n(\vb*{r})$$ can be defined, valid for any external potential $$V_{\mathrm{ext}}(\vb*{r})$$. For any particular $$V_{\mathrm{ext}}(\vb*{r})$$, the exact ground state energy of the system is the global minimum of the functional.
• The functional $$E[n]$$ alone is sufficient to determined the exact ground state energy.
• Spin included: $E = E_{\mathrm{HK}}[n_\uparrow, n_\downarrow].$

#### The Kohn-Sham Equations

##### Framework
• Assumption. The exact ground state can be represented by the ground state density of an axuiliary system of non-interacting particles.
• The auxiliary Hamiltonian is chosen to have the usual kinetic operator and an effective local potential $$V^\sigma_{\mathrm{eff}}(\vb*{r})$$.
• Energy functional: $E_{\mathrm{KS}} = T_s[n] + \int \dd{\vb*{r}} V_{\mathrm{ext}}(\vb*{r}) n(\vb*{r}) + E_{\mathrm{Hatree}}[n] + E_{II} + E_{\mathrm{xc}}[n].$
• Density $n(\vb*{r}) = \sum_\sigma \sum_{i=1}^{N^\sigma} \abs{\psi_i^\sigma(\vb*{r})}^2.$
• Kinetic energy $T_s = \frac{1}{2}\sum_\sigma \sum_{i=1}^{N^\sigma} \int \dd{^3 \vb*{r}} \abs{\grad \psi_i^\sigma (\vb*{r})}^2.$
• Hatree energy $E_{\mathrm{Hatree}}[n] = \frac{1}{2} \iint \dd{^3 \vb*{r}} \dd{^3 \vb*{r}'} \frac{n(\vb*{r}) n(\vb*{r}')}{\abs{\vb*{r} - \vb*{r}'}}.$
• $$V_{\mathrm{ext}}$$ includes the potential due to the nuclei and any other external fields.
• $$E_{II}$$ is the interaction between the nuclei.
• $$E_{\mathrm{xc}}$$ is the exchange-correlation energy.
• Kohn-Sham equations: $\color{red} (H^\sigma_{\mathrm{KS}} - \epsilon^\sigma_i) \psi^\sigma_i = 0.$
• $H^\sigma_{\mathrm{KS}} = -\frac{1}{2}\grad^2 + V^\sigma_{\mathrm{KS}}(\vb*{r}).$
• $V^\sigma_{\mathrm{KS}}(\vb*{r}) = V_{\mathrm{ext}}(\vb*{r}) + V_{\mathrm{Hatree}}(\vb*{r}) + V^\sigma_{\mathrm{xc}}(\vb*{r}).$
• $V_{\mathrm{Hatree}}(\vb*{r}) = \frac{\delta E_{\mathrm{Hatree}}}{\delta n(\vb*{r},\sigma)}.$
• $V^\sigma_{\mathrm{xc}}(\vb*{r}) = \frac{\delta E_{\mathrm{xc}}}{\delta n(\vb*{r},\sigma)}.$
##### Exchange-Correlation
• Exchange-correlation functional $E_{\mathrm{xc}}[n] = \int \dd{\vb*{r}} n(\vb*{r}) \epsilon_{\mathrm{xc}}([n], \vb*{r}).$
• $$n(\vb*{r})$$ is the total density.
• $$\epsilon_{xc}$$ may depend on $$n_\uparrow$$ and $$n_\downarrow$$.
• Exchange-correlation potential $V^\sigma_{\mathrm{xc}}(\vb*{r}) = \epsilon_{\mathrm{xc}}([n],\vb*{r}) + n(\vb*{r}) \frac{\delta \epsilon_{\mathrm{xc}}([n],\vb*{r})}{\delta n(\vb*{r},\sigma)}.$
##### Meaning of the Eigenvalues
• The highest eigenvalue in a finite system is $$-I$$, where $$I$$ is the ionization energy,
• Slater-Janak theorem: $\epsilon_i = \dv{E_{\mathrm{total}}}{n_i}.$
• $$n_i$$ is the occupation of the state $$i$$.
##### Time-Dependent Density Functional Theory

TDDFT. Time-Dependent Density Functional Theory.

• $i\hbar \dv{\psi_i(t)}{t} = H(t)\psi_i(t).$
• $$V_{\mathrm{xc}}$$ depends on $$n(\vb*{r}, t')$$ for all $$t' \le t$$.
• Adiabatic LDA: approximation that $$V_{\mathrm{xc}}$$ depends on $$n$$ at the same $$t$$ only.

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## Complex Manifolds

#### Definition and Examples

##### Definitions
• Holomorphic $$f: \mathbb{C}^m \rightarrow \mathbb{C}{}$$:
• Cauchy-Riemann relations: $\pdv{f_1}{x^\mu} = \pdv{f_2}{y^\mu},\quad \pdv{f_2}{x^\mu} = -\pdv{f_1}{y^\mu}.$
• $$f: \mathbb{C}^m \rightarrow \mathbb{C}{}$$ is holomorphic if each $$f^\lambda$$ is holomorphic.
• Complex manifold $$M$$:
• $$M$$ is a topological space,
• $$M$$ is provided with an atlas $$\qty{(U_i,\varphi_i)}$$,
• each $$\varphi_i$$ maps the open set $$U_i$$ to an open subset $$U\subset \mathbb{C}^m$$, and
• for each pair of $$U_i\cap U_j \neq \emptyset$$, $\psi_{ji} = \varphi_j\circ \varphi_i^{-1}$ is holomorphic.
• $$\dim_{\mathbb{C}} M = m$$, while
• $$\dim M = 2m$$.
• If the union of two atlases is again an atlas which satisfy the definition of a complex manifold, they are said to define the same complex structure.
##### Examples

Stereographic projection $$S^2 \rightarrow \mathbb{C}$$ naturally promote $$S^2$$ to a complex manifold.

Torus:

• Let $$L$$ be a lattice $L(\omega_1,\omega_2) = \set{\omega_1 m + \omega_2 n|m,n\in \mathbb{Z}}.$
• $$\omega_1/\omega_2 \notin \mathbb{R}$$.
• Take $$\Im(\omega_2/\omega_1)>0$$.
• Homeomorphism $T^2 \cong \mathbb{C}/L(\omega_1,\omega_2).$
• The pair $$(\omega_1,\omega_2)$$ defined a complex structure on $$T^2$$.
• Two lattices $$L(\omega_1,\omega_2)$$ coincides if and only if there exists a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{PSL}(2,\mathbb{Z}) = \mathrm{SL}(2,\mathbb{Z})/\mathbb{Z}_2$ where $$\pm A$$ are identified such that $\begin{pmatrix} \omega_1' \\ \omega_2' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix}.$
• Take $$(1,\tau)$$ to be the generators of a lattice.
• $$\tau$$ and $$\tau'$$ define the same complex structure if $\tau = \frac{a\tau + b}{c\tau + d}$ where $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{PSL}(2,\mathbb{Z}).$
• Modular transform: $$\tau \rightarrow \tau'$$ generated by
• $$\tau \rightarrow \tau+1$$,
• $$\tau \rightarrow -1/\tau$$.

Complex projective space: denoted by $$\mathbb{C}P^n$$.

• $$z\in \mathbb{C}^{n+1}$$ identified with $$z'$$ if $$z=az'$$ for some complex number $$a\neq 0$$.
• The $$n+1$$ numbers $$z^0,\cdots,z^n$$ are called homogeneous coordinates.
• For $U_\mu = \set{z\in \mathbb{C}\backslash\qty{0} | z^\mu \neq 0},$ a chart could defined by $$\xi^\nu_{(\mu)} = z^\nu/z^\mu$$.

Complex Grassman manifolds:

• $G_{k,n}(\mathbb{C}) = M_{k,n}(\mathbb{C})/\mathrm{GL}(k,\mathbb{C}).$
• $$M_{k,n}$$ consists of all $$k\times n$$ matrices of rank $$k$$.
• In particular, $\mathbb{C}P^n = G_{1,n+1}(\mathbb{C}).$
• A chart $$U_\alpha$$ may be defined to be a subset of $$G_{k,n}$$ such that $$\det A_\alpha \neq 0$$.
• The $$k(n-k)$$ coordinates of $$U_\alpha$$ are given by the non-trivial entries of the matrix $$A_\alpha^{-1} A$$.

Algebraic variety:

• The common zeros of a set of homogeneous polynomials.
• A compact submanifold of $$\mathbb{C}P^n$$.
• $$U_\mu$$ defined as in a complex projective space.

#### Calculus on Complex Manifolds

##### Holomorphic Maps
• Holomorphic map: $$f:M\rightarrow N$$ is holomorphic if $\psi\circ f \circ \varphi^{-1}: \mathbb{C}^m \rightarrow \mathbb{C}^n$ is a holomorphic function.
• Independent of the coordinates choosen.
• Biholomorpism: $$M$$ is biholomorphic to $$N$$ if there exists a diffeomorphism $$f:M\rightarrow N$$ that is holomorphic.
• The map $$f$$ is called a biholomorpism.
• Holomorphic function: a holomorphic map $$f:M\rightarrow \mathbb{C}$$.
• Any holomorphic function on a compact complex manifold is constant.
• The set of holomorphic functions on $$U\subset M$$ is denoted by $$\mathcal{O}(U)$$.
##### Complexifications
• Complexification $$\mathcal{F}(M)^{\mathbb{C}}$$ of $$\mathcal{F}(M)$$: the set of complex-valued smooth functions on $$M$$.
• Complexified functions do not satisfy the Cauchy-Riemann relation in general.
• Complexification of a vector space $$V$$: consists of $$\qty{X+iY}$$ where $$X,Y\in V$$.
• Vectors in $$V$$ are said to be real.
• Linear operator or linear function $$A$$ on $$V$$ extended as $A(X+iY) = A(X) + i A(Y).$
• Complexification of tensors: $t = t_1 + i t_2.$
• A tensor $$t$$ is real if $\overline{t(X+iY)} = t(X-iY).$
• Complexified $$T_p M^{\mathbb{C}}$$:
• consists of $Z = X + iY,$ where $$X,Y\in T_p M$$.
• $$Z$$ acts on $$\mathcal{F}(M)^{\mathbb{C}}$$ as $Z[f] = X[f_1] - Y[f_2] + i\qty{X[f_2] + Y[f_1]}.$
• $$T_p^* M^{\mathbb{C}}$$ consists of $\zeta = \omega + i\eta.$
• $(T^*_p M)^{\mathbb{C}} = (T_p M^{\mathbb{C}})^*.$
• Complexified vector field $$\mathcal{X}(M)^\mathbb{C}$$ consists of $Z = X+iY.$
• Lie bracket as $[X+iY,U+iV] = \qty{[X,U] - [Y,V]} + i\qty{[X,V] + [Y,U]}.$
• Complexified one-form $$\Omega^1(M)^{\mathbb{C}}$$ consists of $\xi = \omega + i\eta.$
##### Almost Complex Structure
• Basis: \begin{align*} \pdv{}{z^\mu} &= \frac{1}{2}\qty{\pdv{}{x^\mu} - i\pdv{}{y^\mu}}, \\ \pdv{}{\overline{z}^\mu} &= \frac{1}{2}\qty{\pdv{}{x^\mu} + i\pdv{}{y^\mu}}. \end{align*}{}
• Dual basis: \begin{align*} \dd{z^\mu} &= \dd{x^\mu} + i \dd{y^\mu}, \\ \dd{\overline{z}^\mu} &= \dd{x^\mu} - i\dd{y^\mu}. \end{align*}{}
• Inner products: \begin{align*} \langle \dd{z^\mu}, \pdv{\overline{z}^\nu} \rangle &= \langle \dd{\overline{z}^\mu}, \pdv{}{z^\nu} \rangle = 0, \\ \langle \dd{z^\mu}, \pdv{{z}^\nu} \rangle &= \langle \dd{\overline{z}^\mu}, \pdv{}{\overline{z}^\nu} \rangle = \delta{^\mu}{_\nu}. \end{align*}{}
• $$J_p: T_p M \rightarrow T_p M$$ defined by $J_p\qty{\pdv{}{x^\mu}} = \pdv{}{y^\mu},\quad J_p\qty(\pdv{}{y^\mu}) = -\pdv{}{x^\mu}.$
• Real tensor.
• $J_p^2 = -\operatorname{id}_{T_p M}.$
• Corresponding to the multiplication by $$\pm i$$.
• Independent of charts: $J_p\qty{\pdv{}{x^\mu}} = \pdv{}{y^\mu} \Rightarrow J_p\qty{\pdv{}{u^\mu}} = \pdv{}{v^\mu}.$
• In matrix form: $J_p = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix}.$
• Almost complex structure: a smooth tensor field $$J$$ whose component at $$p$$ is $$J_p$$.
• $$J$$ may be patched across charts only on a complex manifold.
• $$J$$ completely specifies the complex structure.
• $J_p \pdv{}{z^\mu} = i\pdv{}{z^\mu},\quad J_p \pdv{\overline{z}^\mu} = -i \pdv{\overline{z}^\mu}.$
• $\color{red} J_p = i\dd{z^\mu} \otimes \pdv{}{z^\mu} - i \dd{\overline{z}^\mu} \otimes \pdv{\overline{z}^\mu} \Leftrightarrow \begin{pmatrix} i I_m & 0 \\ 0 & -i I_m \end{pmatrix}.$
• For $$\displaystyle Z = Z^\mu \pdv{}{z^\mu}$$, $J_p Z = iZ.$
• For $$\displaystyle Z = Z^\mu \pdv{}{\overline{z}^\mu}$$, $J_p Z = -iZ.$
• $$T_p M^{\mathbb{C}}$$ is separated into $T_p M^{\mathbb{C}} = T_p M^+ \oplus T_p M^{-}.$
• $\color{orange} T_p M^\pm = \set{Z\in T_p M^{\mathbb{C}} | J_p Z = \pm iZ}.$
• Projection operator $\mathcal{P}^\pm = \frac{1}{2}(I_{2m} \mp iJ_p).$
• $$Z$$ may be decomposed into $Z = Z^+ + Z^-.$
• $$Z \in T_p M^+$$ is called an holomorphic vector.
• Spanned by $$\qty{\partial/\partial z^\mu}$$.
• $$Z \in T_p M^-$$ is called an anti-holomorphic vector.
• Spanned by $$\qty{\partial/\partial\overline{z}^\mu}$$.
• Likewise, a vector field may be decomposed as $\mathcal{X}(M)^\mathbb{C} = \mathcal{X}(M)^+ \oplus \mathcal{X}(M)^-.$
• $$Z = Z^+ + Z^-$$ is real if and only if $Z^+ = \overline{Z^-}.$
• If $$X,Y\in \mathcal{X}(M)^{\pm}$$ then $[X,Y] \in \mathcal{X}(M)^{\pm}.$

#### Complex Differential Forms

##### Complexification of Real Differetial Forms
• Complex $$q$$-form $$\zeta = \omega + i\eta$$.
• Exterior product defined by $(\omega + i\eta)\wedge (\varphi + i\psi) = (\omega \wedge \varphi - \eta \wedge \psi) + i(\omega \wedge \psi + \eta \wedge \varphi).$
• Exterior derivative $\dd{\xi} = \dd{\omega} + i \dd{\eta}.$
• $$\dd{}{}$$ is a real operator, i.e. $\overline{\dd{\eta}} = \dd{\overline{\zeta}}.$
• $\omega \wedge \xi = (-1)^{qr} \xi \wedge \omega.$
• $\dd{\omega \wedge \xi} = \dd{\omega} \wedge \xi + (-1)^q \omega \wedge \dd{\xi}.$
##### Differential Forms on Complex Manifolds
• $$\Omega^{r,s}_p(M)^{\mathbb{C}}{}$$ is
• a subset of $$\Omega^{r+s}_p(M)^{\mathbb{C}}$$,
• where $\omega(V_1,\cdots,V_q) \neq 0$ only if $$r$$ of the $$V_i$$ are in $$T_p M^+$$ and $$s$$ of the $$V_i$$ are in $$T_p M^-$$.
• $$(r,s)$$ is the bidegree of $$\omega$$.
• A $$(r,s)$$-form may be written as $\omega = \frac{1}{r! s!} \omega_{\mu_1 \cdots \mu_r \nu_1\cdots \nu_s} = \dd{z^{\mu_1}} \wedge \cdots \wedge \dd{z^{\mu_r}} \wedge \dd{\overline{z}^{\nu_1}} \wedge \cdots \wedge \dd{\overline{z}^{\nu_s}}.$
• $r+s \le 2\dim_{\mathbb{C}}M.$
• Proposistion: Let $$\dim_{\mathbb{C}} M = m$$.
• If $$\omega \in \Omega^{q,r}(M)$$ then $\overline{\omega} \in \Omega^{r,q}(M).$
• If $$\omega \in \Omega^{q,r}(M)$$ and $$\xi\in\Omega^{q',r'}(M)$$, then $\omega \wedge \xi = \Omega^{q+q',r+r'}(M).$
• A complex $$q$$-form is uniquely written as $\omega = \sum_{r+s = q} \omega^{(r,s)}.$
• This gives rise to the decomposition $\Omega^q(M)^{\mathbb{C}} = \bigoplus_{r+s=q}\Omega^{r,s}(M).$
##### Dolbeault Operators
• $\dd = \partial + \overline{\partial}.$
• $\partial: \Omega^{r,s}(M) \rightarrow \Omega^{r+1,s}(M).$
• $\overline{\partial}: \Omega^{r,s}(M) \rightarrow \Omega^{r,s+1}(M).$
• $$\partial$$ and $$\overline{\partial}$$ are called Dolbeault operators.

Let $\omega = \omega_{\mu\bar{\nu}} \dd{z^\mu} \wedge \dd{\overline{z}^\nu}.$ Then $\partial \omega = \pdv{\omega_{\mu\bar{\nu}}}{z^\lambda} \dd{z^\lambda} \wedge \dd{z^\mu} \wedge \dd{\overline{z}^\nu},$ and $\overline{\partial} \omega = \pdv{\omega_{\mu\bar{\nu}}}{\overline{z}^\lambda} \dd{\overline{z}^\lambda} \wedge \dd{z^\mu} \wedge \dd{\overline{z}^\nu}.$

Theorem. Let $$\omega \in \Omega^q(M)^{\mathbb{C}}$$ and $$\xi\in \Omega^p(M)^{\mathbb{C}}$$. Then

• $\partial \partial = (\partial\overline{\partial} + \overline{\partial}\partial) \omega = \overline{\partial}\overline{\partial} \omega = 0.$

• $\partial\overline{\omega} = \overline{\overline{\partial}\omega},\quad \overline{\partial}\overline{\omega} = \overline{\partial\omega}.$

• $\partial(\omega \wedge \xi) = \partial \omega \wedge \xi + (-1)^q \omega\wedge \partial \xi.$

• $\overline{\partial} (\omega \wedge \xi) = \overline{\partial} \omega \wedge \xi + (-1)^q \omega \wedge \overline{\partial}\xi.$

• Holomorphic $$r$$-form: $$\omega\in \Omega^{r,0}(M)$$ that satisfies $\overline{\partial}\omega = 0.$

• A holomorphic $$0$$-form is a holomorphic function.
• Doubeault complex: the sequence of $$\mathbb{C}$$-linear maps $\Omega^{r,0}(M) \xrightarrow{\bar{\partial}} \Omega^{r,1}(M) \xrightarrow{\bar{\partial}}\cdots \xrightarrow{\bar{\partial}} \Omega^{r,m-1}(M) \xrightarrow{\bar{\partial}} \Omega^{r,m}(M)$ where $$m = \dim_{\mathbb{C}}M$$.

• The set of $$\overline{\partial}$$-closed $$(r,s)$$ forms is called the $$(r,s)$$-cocycle, denoted by $$Z^{r,s}_{\overline{\partial}}(M)$$.
• The set of $$\overline{\partial}$$-exact $$(r,s)$$ forms is called the $$(r,s)$$-coboundary, denoted by $$B^{r,s}_{\overline{\partial}}(M)$$.
• $$(r,s)$$-th $$\overline{\partial}$$-cohomology group $H^{r,s}_{\bar{\partial}}(M) = Z^{r,s}_{\bar{\partial}}(M) / B^{r,s}_{\bar{\partial}}(M).$

#### Hermitian Manifolds and Hermitian Differential Geometry

• $$g$$ may be extended by $g_p(X+iY,U+iV) = g_p(X,U) - g_p(Y,V) + i[g_p(X,V) + g_p(Y,U)].$
• $g_{\mu\nu}(p) = g_p(\partial/\partial z^\mu, \partial/\partial z^\nu).$
• $g_{\mu\bar{\nu}}(p) = g_p(\partial/\partial z^\mu, \partial/\partial \overline{z}^\nu).$
• $g_{\bar{\mu}\nu}(p) = g_p(\partial/\partial \overline{z}^\mu, \partial/\partial z^\nu).$
• $g_{\bar{\mu}\bar{\nu}}(p) = g_p(\partial/\partial \overline{z}^\mu, \partial/\partial \overline{z}^\nu).$
• Hermitian metric: for any $$X,Y\in T_p M$$, $g_p(J_p X, J_p Y) = g_p(X,Y).$
• Hermitian manifold: $$(M,g)$$ where $$g$$ is a Hermitian metric.
• $$J_p X$$ is orthogonal to $$X$$: $g_p(J_p X,X) = 0.$
• A complex manifold always admits a Hermitian metric.
• The Hermitian $$g$$ takes the form $g = g_{\mu\bar{\nu}} \dd{z^\mu} \otimes \dd{\overline{z}^\nu} + g_{\bar{\mu}\nu} \dd{\overline{z}^\mu} \otimes \dd{z^\nu}.$
• Inverse metric: $g_{\mu\bar{\lambda}} g^{\bar{\lambda}\nu} = \delta{_\mu}{^\nu},\quad g{^{\bar{\nu}\lambda}}g_{\lambda\bar{\mu}} = \delta{^{\bar{\nu}}}{_{\bar{\mu}}}.$
##### Kähler Form
• Kähler form: a two-form field defined by $\Omega_p(X,Y) = g_p(J_p X, Y),$ where $$X,Y\in T_p M$$.
• Invariant under $$J$$: $\Omega(JX, JY) = \Omega(X,Y).$
• $\Omega = -i g_{\mu\bar{\nu}} \dd{z^\mu} \wedge \dd{\overline{z}^\nu} = -J_{\mu\bar{\nu}} \dd{z^\mu} \wedge \dd{\overline{z}^\nu}.$
• $$\Omega$$ is a real form.
• A complex manifold is orientable.
##### Covariant Derivatives
• $\grad_\mu \pdv{}{z^\nu} = \Gamma{^\lambda}_{\mu\nu} \pdv{}{z^\lambda}.$
• $\grad_{\bar{\mu}} \pdv{\overline{z}^\nu} = \Gamma{^{\bar{\lambda}}}_{\bar{\mu}\bar{\nu}} \pdv{}{\overline{z}^\lambda} = \overline{\Gamma{^\lambda}_{\mu\nu}} \pdv{}{\overline{z}^\lambda}.$
• $\grad_\mu \pdv{}{\overline{z}^\nu} = \grad_{\bar{\mu}} \pdv{}{z^\nu} = 0.$
• $\grad_\mu \dd{z^\nu} = -\Gamma{^\nu}_{\mu\lambda} \dd{z^\lambda}.$
• $\grad_{\bar{\mu}} \dd{\overline{z}^\nu} = -\Gamma{^{\bar{\nu}}}_{\bar{\mu}\bar{\lambda}}\overline{z}^\lambda.$
• Covariant derivatives of vector: \begin{align*} \grad_\mu X^+ &= (\partial_\mu X^\lambda + X^\nu \Gamma{^\lambda}_{\mu\nu}) \pdv{}{z^\lambda}, \\ \grad_\mu X^- &= \partial_\mu X^{\bar{\lambda}} \pdv{}{\overline{z}^\lambda}, \\ \grad_{\bar{\mu}} X^+ &= \partial_{\bar{\mu}} X^\lambda \pdv{}{z^\lambda}, \\ \grad_{\bar{\mu}} X^- &= (\partial_{\bar{\mu}} X^{\bar{\lambda}} + X^{\bar{\nu}} \Gamma{^{\bar{\lambda}}}_{\bar{\mu}\bar{\nu}} \pdv{}{\overline{z}^\lambda}). \end{align*}{}
• Metric compatibility:
• $\Lambda{^\lambda}_{\kappa\mu} = g^{\bar{\nu}\lambda}\partial_\kappa g_{\mu\bar{\nu}},\quad \Gamma{^{\bar{\lambda}}}_{\bar{\kappa}\bar{\nu}} = g^{\bar{\lambda}\mu}\partial_{\bar{\kappa}} g_{\mu\bar{\nu}}.$
• Hermitian connection: a metric compatible connection for which $$\Gamma$$(mixed indices) vanish.
• The almost complex structure $$J$$ is convariantly constant with respect to the Hermitian connection. $\color{red} (\grad_\kappa J){_\nu}{^\mu} = (\grad_{\bar{\kappa}} J){_\nu}{^\mu} = (\grad_\kappa J){_{\bar{\nu}}}{^{\bar{\mu}}} = (\grad_{\bar{\kappa}} J){_{\bar{\nu}}}{^{\bar{\mu}}} = 0.$
##### Torsion and Curvature
• Nonzero components of $$T$$:
• $T{^\lambda}_{\mu\nu} = g^{\bar{\xi}\lambda}(\partial_\mu g_{\nu\bar{\xi}} - \partial_\nu g_{\mu\bar{\xi}}).$
• $T{^\lambda}_{\bar{\mu}\bar{\nu}} = g^{\bar{\lambda}\xi}(\partial_{\bar{\mu}}g_{\bar{\nu}\xi} - \partial_{\bar{\nu}} g_{\bar{\mu}\xi}).$
• Nonzero components of $$R$$:
• \begin{align*} R{^\kappa}_{\lambda\bar{\mu}\nu} &= -R{^\kappa}_{\lambda\nu\bar{\mu}} = \partial_{\bar{\mu}}(g^{\bar{\xi}\kappa}\partial_\nu g_{\lambda\bar{\xi}}),\\ R{^{\bar{\kappa}}}_{\bar{\lambda}\mu\bar{\nu}} &= -R{^{\bar{\kappa}}}_{\bar{\lambda}\bar{\nu}\mu} = \partial_\mu(g^{\bar{\kappa} \xi} \partial_{\bar{\nu}} g_{\xi\bar{\lambda}}). \end{align*}{}
• Ricci form: $\mathscr{R} = i\mathscr{R}_{\mu\bar{\nu}} \dd{z^\mu} \wedge \dd{\overline{z}^\nu} = -i\partial\overline{\partial}\log G.$
• $G = \det(g_{\mu\overline{\nu}}) = \sqrt{g}.$
• $\mathscr{R}_{\mu\bar{\nu}} = R{^\kappa}_{\kappa\mu\bar{\nu}} = -\partial_{\bar{\nu}}\partial_\mu \log G.$
• $$\mathscr{R}$$ is closed, i.e. $$\dd{\mathscr{R}} = 0$$.
• $$\mathscr{R}$$ is not exact.
• $$G$$ is not a scalar and $(\partial - \overline{\partial})G$ is not defined globally.
• $$\mathscr{R}$$ defines a non-trivial element $c_1(M) = [\mathscr{R}/2\pi] \in H^2(M;\mathbb{R}).$
• $$c_1(M)$$ is invariant under a smooth change of metric $g \rightarrow g+\delta g.$

#### Kähler Manifolds and Kähler Differential Geometry

##### Examples of Kähler Manifolds
• Kähler manifold:
• a Hermitian manifold $$(M,g)$$,
• whose Kähler form $$\Omega$$ is closed, i.e. $\dd{\Omega} = 0.$
• The metric $$g$$ is called the Kähler metric of $$M$$.
• A Hermitian manifold $$(M,g)$$ is a Kähler manifold if and only if the almost complex structure $$J$$ satisfies $\color{red}\grad_\mu J = 0,$ where $$\grad_\mu$$ is assocaited with $$g$$.
• The Riemann structure is compatible with the Hermitian structure in a Kähler manifold.
• Kähler potential: denoted by $$\mathcal{K}$$,
• Locally $g_{\mu\bar{\nu}} = \partial_\mu \partial_{\bar{\nu}} \mathcal{K},$
• $$\mathcal{K}\in \mathcal{F}(U_i)$$ for a chart $$U_i$$.
• $\Omega = i\partial \overline{\partial} \mathcal{K}.$
• Between different charts: $\mathcal{K}_j(w,\overline{w}) = \mathcal{K}_i(z,\overline{z}) + \phi_{ij}(z) + \psi_{ij}(\overline{z}),$
• $$\phi$$ is holomorphic and $$\psi$$ is anti-holomorphic in $$z$$.

Let $$M = \mathbb{C}^m$$. Let the metric be given by \begin{align*} \delta\qty(\pdv{}{x^\mu},\pdv{}{x^\nu}) &= \delta\qty(\pdv{}{y^\mu}, \pdv{}{y^\nu}) = \delta_{\mu\nu}, \\ \delta\qty(\pdv{}{x^\mu},\pdv{}{y^\nu}) &= 0. \end{align*}{} Since $J \pdv{}{x^\mu} = \pdv{}{y^\mu},\quad J\pdv{}{y^\mu} = -\pdv{}{x^\mu},$ we find that $$\delta$$ is Hermitian. In complex coordinates, we have \begin{align*} \delta\qty(\pdv{}{z^\mu}, \pdv{}{z^\nu}) &= \delta\qty(\pdv{}{\overline{z}^\mu}, \pdv{}{\overline{z}^\nu}) = 0, \\ \delta\qty(\pdv{}{z^\mu}, \pdv{}{\overline{z}^\nu}) &= \delta\qty(\pdv{}{\overline{z}^\mu}, \pdv{}{z^\nu}) = \frac{1}{2}\delta_{\mu\nu}. \end{align*}{} The Kähler form is given by $\Omega = \frac{i}{2} \sum_{\mu=1}^m \dd{z^\mu} \wedge \dd{\overline{z}^\mu}.$ The Kählter potantial is given by $\mathcal{K} = \frac{1}{2} \sum z^\mu \overline{z}^\mu.$ The Kähler manifold $$\mathbb{C}$$ is called the complex Euclid space.

Any orientable complex manifold $$M$$ with $$\dim_{\mathbb{C}} M = 1$$ is Kähler. One-dimensional compact orientable complex manifolds are Riemann surfaces.

$$\mathbb{C}P^m$$ is a Kähler manifold. We define the Ka¨hler potential on $$U_\alpha$$ by $\log \mathcal{K}_\alpha(p) = \log \sum_{\nu=1}^{m+1} \abs{\frac{z^\nu}{z^\alpha}}^2.$ The Kähler form may be defined by $\Omega = i\partial \overline{\partial} \log \mathcal{K}_\alpha.$ It can be shown that $g(X,Y) = \Omega(X,JY)$ is Hermitian. It's positive-definite since $g(X,X) = 2 \qty[\sum_\mu \abs{X^\mu}^2 \qty(\sum_\lambda \abs{\zeta^\lambda}^2 + 1) - \abs{\sum_\mu X^\mu \zeta^\mu}^2]\qty(\sum_\lambda \abs{\zeta^\lambda}^2 + 1)^{-2},$ where $$\zeta^\lambda$$ are the inhomogeneous coordinates. The metric is called the Fubini-Study metric.

• $$S^2$$ is the only sphere which admits a complex structure. $$S^2 \cong \mathbb{C}P^1$$ and therefore $$S^2$$ is a K¨hler manifold ($$S^6$$ unknown still).
• $$S^{2m+1}\times S^{2n+1}$$ always admits a complex structure. Only $$S^1 \otimes S^1$$ admits a Kähler metric.
• Any complex submanifold of a Kähler manifold is Kähler.
##### Kähler Geometry
• The Kähler metric is torsion free: \begin{align*} T{^\lambda}_{\mu\nu} &= g^{\bar{\xi}\lambda}(\partial_\mu g_{\nu\bar{\xi}} - \partial_\nu g_{\mu\bar{\xi}}) = 0, \\ T{^{\bar{\lambda}}}_{\bar{\mu}\bar{\nu}} &= g^{\bar{\lambda}\xi}(\partial_{\bar{\mu}} g_{\bar{\nu}\xi} - \partial_{\bar{\nu}}g_{\bar{\mu}\xi}) = 0. \end{align*}{}
• The Riemann tensor has an extra symmetry: $R{^\kappa}_{\lambda\mu\bar{\nu}} = R{^\kappa}_{\mu\lambda\bar{\nu}}.$
• Ricci form $\mathscr{R}_{\mu\bar{\nu}} = \mathrm{Ric}_{\mu\bar{\nu}}.$
• If $$\mathrm{Ric} = \mathscr{R} = 0$$, the Kähler metric is said to be Ricci flat.
• Let $$(M,g)$$ be a Kähler manifold. If $$M$$ admits a Ricci flat metric $$h$$, then its first Chern class must vanish.
• Calabi-Yau manifold: a compact Kähler manifold with vanishing first Chern class.
• Calabi conjecture (proved): If $$c_1(M) = 0$$, then the Kähler manifold $$M$$ admits a Ricci-flat metric.
##### The Holonomy Group of Kähler Manifolds
• The holonomy group is contained in $$\mathrm{U}(m) \subset \mathrm{O}(2m)$$.
• If $$g$$ is the Ricci-flat metric of an $$m$$-dimensional Calabi-Yau manifold $$M$$, the holonomy group is contained in $$\mathrm{SU}(m)$$.

#### Harmonic Forms and ∂̅-Cohomology Groups

• Inner product: $$\alpha,\beta \in \Omega^{r,s}(M)$$, $\color{orange}(\alpha,\beta) = \int_M \alpha \wedge\overline{*}\beta.$
• Hodge *: $\overline{*}\beta = \overline{*\beta} = *\overline{\beta}.$
• Adjoint operators: $\color{red} \partial^\dagger = -*\overline{\partial} *,\quad \overline{\partial}^\dagger = -*\partial *.$
• $\color{red} (\alpha,\partial\beta) = (\partial^\dagger \alpha,\beta),\quad (\alpha,\overline{\partial}\beta) = (\overline{\partial}^\dagger \alpha,\beta).$
• $(\partial^\dagger)^2 = (\overline{\partial}^\dagger)^2 = 0.$
##### Hodge Theory on Complex Manifolds
• $$(r,s)$$th $$\overline{\partial}$$-cohomology group: $H^{r,s}_{\overline{\partial}}(M) = Z^{r,s}_{\bar{\partial}}(M) / B^{r,s}_{\bar{\partial}}(M).$
• Laplacians: \begin{align*} \bigtriangleup_\partial &= (\partial + \partial^\dagger)^2 = \partial\partial^\dagger + \partial^\dagger \partial, \\ \bigtriangleup_{\bar{\partial}} &= (\overline{\partial} + \overline{\partial}^\dagger)^2 = \overline{\partial}\overline{\partial}^\dagger + \overline{\partial}^\dagger \overline{\partial}. \end{align*}{}
• Harmonic forms:
• $$\partial$$-harmonic: $\bigtriangleup_\partial \omega = 0.$
• Equivalently, $\partial\omega = \partial^\dagger \omega = 0.$
• $$\overline{\partial}$$-harmonic: $\bigtriangleup_{\bar{\partial}} \omega = 0.$
• Equivalently, $\overline{\partial}\omega = \overline{\partial}^\dagger \omega = 0.$
• $$\overline{\partial}$$-harmonic forms: $\mathrm{Harm}^{r,s}_{\bar{\partial}}(M) = \set{\omega\in \Omega^{r,s}(M) | \bigtriangleup_{\bar{\partial}}\omega=0}.$
• Hodge's theorem: $\Omega^{r,s}(M) = \overline{\partial} \Omega^{r,s-1}(M) \oplus \overline{\partial} \Omega^{r,s+1}(M) \oplus \mathrm{Harm}^{r,s}_{\bar{\partial}}(M).$
• Each $$(r,s)$$-form $$\omega$$ is uniquely expressed as $\omega = \overline{\partial}\alpha + \overline{\partial}^\dagger \beta + \gamma.$
• If $$M$$ is a Kähler manifold, then $\bigtriangleup = 2\bigtriangleup_{\partial} = 2\bigtriangleup_{\bar{\partial}}.$
• Any holomorphic form is harmonic with respect to the Kähler metric.
• Every harmonic form of bidegree $$(p,0)$$ is holomorphic.
##### The Hodge Numbers of Kähler Manifolds

Let $$M$$ be a Kähler manifold.

• Hodge number: $b^{r,s} = \dim H^{r,s}_{\bar{\partial}}(M).$
• Hodge diamond: $\begin{pmatrix} & & & b^{m,m} & & & \\ & & b^{m,m-1} & & b^{m-1,m} & & \\ b^{m,0} & b^{m-1,1} & & \cdots & & b^{1,m-1} & b^{0,m} \\ & & b^{1,0} & & b^{0,1} & & \\ & & & b^{0,0} & & & \end{pmatrix}.$
• Properties of hodge numbers:
• $b^{r,s} = b^{s,r}.$
• $b^{r,s} = b^{m-r,m-s}.$
• Number of independent Hodge numbers:
• $$m = \dim_{\mathbb{C}}M$$ even: $(\frac{1}{2}m+1)^2.$
• $$m = \dim_{\mathbb{C}}M$$ odd: $\frac{1}{4}(m+1)(m+2).$
• $b^p = \sum_{r+s=p} b^{r,s}.$
• $b^{2p-1} \text{ is even}.$
• $b^{2p}\ge 1.$

#### Almost Complex Manifolds

• Almost complex manifold: the pair $$(M,J)$$,
• $$J$$ is a tensor field of type $$(1,1)$$,
• $$J^2_p = -\operatorname{id}_{T_p M}$$.
• $$J$$ is called the almost complex structure.
• Integrable almost complex structure $$J$$:
• For any $$X,Y\in\mathcal{X}^+(M)$$, $[X,Y] \in \mathcal{X}^+(M).$
• Nijenhuis tensor field: $\color{orange} N(X,Y) = [X,Y] + J[JX,Y] + J[X,JY] - [JX,JY].$
• $N(X,Y) = X^\kappa Y^\nu [-J{_\lambda}{^\mu}(\partial_\nu J{_\kappa}{^\lambda}) + J{_\lambda}{^\mu}(\partial_\kappa J{_\nu}{^\lambda}) - J{_\kappa}{^\lambda}(\partial_\lambda) J{_\nu}{^\mu} + J{_\nu}{^\lambda} (\partial_\lambda J{_\kappa}{^\mu})] \pdv{}{x^\mu}.$
• An almost complex structure $$J$$ is integrable if and only if $$N(A,B) = 0$$ for any $$A,B\in \mathcal{X}(M)$$.

Newlander–Nirenberg Theorem. $\substack{\text{Integrable almost} \\ \text{complex structure}} \Leftrightarrow \substack{\text{Vanishing} \\ N(X,Y)} \Leftrightarrow \substack{\text{Complex} \\ \text{manifold}}.$

#### Orbifolds

##### Definition and Examples of Orbifolds
• Orbifold: $\Gamma = M/G.$
• $$M$$ a manifold.
• $$G$$ a discrete group acting on $$M$$.
• $$\Gamma$$ may not be a manifold.

$$\Gamma = \mathbb{C}/C_3$$ is a third of the complex plane. The holonomy group is $$C_3$$. $$\Gamma$$ is a manifold.

Let $M = T^2 = \mathbb{C}/\qty{z \sim z+m+ne^{i\pi/3}},$ and $$C_3$$ acts on $$T^2$$ as $z \mapsto e^{2\pi i/3 z}.$ Then $$\Gamma = T^2/C_3$$ is homeomorphic to $$S^2$$. The holonomy around each fixed point is $$C_3$$.

$$T = \mathbb{C}^3/L$$ is a three-dimensional complex torus where $$L$$ is a lattice in $$\mathbb{C}^3$$. $T = T_1 \times T_2 \times T_3.$ Let $$C_3$$ act on $$T$$ by $z_i \mapsto e^{2\pi i/3}z_i.$ There are $$27$$ fixed points in the orbifold, each of which is a conical singularity. The orbifold is not a manifold.

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Hubbard Model

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## Topological Quantum Chemistry

How do we discover topological materials from band representations?

#### Filling-Enforced Quantum Band Insulator

##### Atomic Insulator
• (TR symmetry) total spin per site is integer.
• Electron wavefunctions localized at Wyckoff positions.
• Minimum filling $$\nu^{\mathrm{AI}}_{\mathrm{min}}$$.
##### Band Insulator

FEQBI. Filling-Enforced Quantum Band Insulaor.

• Minimum filling $$\nu^{\mathrm{Band}}_{\mathrm{min}}$$.
• (Kramers degeneracy) $$\nu^{\mathrm{Band}}_{\mathrm{min}}$$ is even.
• If SOC is nonnegligible, a band insulator may not be an atomic insulator.
• Such nonatomic filling may not respect all crystal symmetries.
• Filling-enforced quantum band insulator: a band insulator realized at such nonatomic filling.
• Possibly realizable in SG No.199, 214, 220, 230.
• Physical manifestations of an FEQBI is still unknown.

#### Topological Insulators

TI. Topological Insulator.

##### Classification
• Strong TI: robust to perturbations that break all crystal symmetries.
• 2D:
• Chern insulator (no symmetry required).
• Quantum spin Hall insulator (time-reversal symmetry).
• 3D:
• $$\mathbb{Z}_2$$ TI with time reversal symmetry.

TCI. Topological Crystal Insulator.

• TCI:
• Weak TI: protected by translation symmetry.
• Mirror Chern insulator.

#### Basics of Group Representation

##### Group Representation

IR. Irreducible Representation.

• The number of non-equivalent IR of $$G$$ equals to the number of conjugacy classes in $$G$$.
• $\color{red} d_1^2 + \cdots + d_C^2 = \abs{G}.$
• IRs of abelian groups are always one-dimensional.
• Inner Product of Characters: $\color{orange}(\chi_1,\chi_2) = \frac{1}{\abs{G}} \sum_{g\in G} \chi_1^*(g) \chi_2(g).$
• Characters of IRs are orthonormal. $(\chi_i,\chi_j) = \delta_{ij}.$
• Expansion into IRs: \begin{align*} \color{red}T &\color{red}= m_1 R_1 \oplus \cdots \oplus m_c R_c, \\ \color{red}m_j &\color{red}= (\chi_T, \chi_{i}). \end{align*}
• Conjugacy classes are strictly distinguishable symmetry operations.

Conjugacy classes of $$D_3$$ includes $$E$$, $$2C_3$$, $$3C_2'$$, each of which has $$1$$, $$2$$ and $$3$$ elements.

##### Interlude: O(3)
• Character: $$P = \pm 1$$,
• $\chi_l(E) = 2l + 1.$
• $\chi_l(C_\theta) = \dfrac{\sin\qty(l+\frac{1}{2})\theta}{\sin \frac{\theta}{2}}.$
• $\chi_l(I) = (2l+1) P.$
• $\chi_l(\sigma) = \chi_l(C_\pi) P.$
• $\chi_l(S_\theta) = \chi_l(C_{\pi - \theta}) P.$
• For each IR of $$\mathrm{SO}(3)$$ there are two different IRs of $$\mathrm{O}(3)$$ denoted $$D_l^\pm$$.
• The superscript $$\pm$$ specify the behaviour under parity.
• $$P=+1$$: the representation is insensitive to parity, and therefore $$\rho(g) = \rho(g/\det g)$$ (I am just guessing).
• $$P=-1$$: the representation is sensitive to parity, and therefore $$\rho(g) = (\det g)\rho(g/\det g)$$ (again guessing).
• How does $$O(3)$$ acts on $$Y_l^m$$?
• $$Y_l^m$$ are sensitive to parity, with $$P = (-1)^l$$.

$$O(3)$$ acts on an electron on the p-orbit by $D_{1}^- \otimes D_{1/2}^{+} = D_{1/2}^- \oplus D_{3/2}^-.$ Note that on both sides we have a $$-1$$ parity.

When taking $$\mathrm{O}(3)$$ as a subgroup of $$\mathrm{SU}(2)$$, for $$j=l+1/2$$ we should have $\rho(C_{\theta}) \neq \rho(C_{\theta + 2\pi}).$ Therefore, each conjugacy class of $$\mathrm{O}_3$$ should split into $$C$$ and $$\overline{C}$$ ($$C$$ multiplied by $$C_{2\pi}$$ I guess).

• Double-valued representations have a bar on their notation. For example, $$\Gamma_1$$ is a single-valued representation and $$\overline{\Gamma}_7$$ is a double-valued one. Double conjugacy classes are also barred. For example, $$E$$ denote the identity operator while $\overline{E} = C_{2\pi} \cdot E.$

Every two-dimensional point group could be embedded into a three-dimensional point group. We may either map $$E$$ to $$E$$ or $$m$$ (reflection about the $$\mathbb{R}^2$$ plane). In the latter case we may obtain a double-valued representation.

##### Interlude: Little Group

Little Group $$=$$ Site Symmetry Group $$=$$ Isotropy Group $$=$$ Stabilizer.

little group: $$G_{\vb*{k}}$$ is the set of symmetries that leave $$\vb*{k}$$ invariant. $\color{warning} G_{\vb*{k}} = \Set{g\in G|g\vb*{k} = \vb*{k} + \vb*{K}},$ where $$\vb*{K}$$ is a reciprocal lattice vector.

• The Bloch waves $$\qty{\psi_{\vb*{k}}^a}$$ transform among themselves under $$G_{\vb*{k}}$$.
• This defines the small representation $$T_{\vb*{k}}$$ of the little group $$G_{\vb*{k}}$$.

$$G_{\vb*{k}}$$ is infinite.

Little cogroup: $$\tilde{G}_{\vb*{k}}$$ defined by $\tilde{G}_{\vb*{k}} = G_{\vb*{k}} / T$ where $$T\in G$$ is the subgroup of lattice translations. The little cogroup is isomorphic to a point group.

Site symmetry groups: $$G_{\vb*{q}}$$ defined by $G_{\vb*{q}} = \Set{g\in G|g\vb*{q} = \vb*{q}},$ where $$g$$ may contain a nonzero translation.

$$G_{\vb*{q}}$$ is finite.

#### Crystal Symmetry

##### Symmetry Operations

Roto-Reflections: Rotation combined with a reflection about a plane perpendicular to the axis.

• $$C_n$$: rotations.
• $$\chi_V(C_n) = 1 + 2\cos\theta_n$$.
• $$\chi_R(C_n) = 1 + 2\cos\theta_n$$.
• $$\sigma$$: reflection planes.
• $$\chi_V(\sigma) = 1$$.
• $$\chi_R(\sigma) = -1$$.
• $$I$$: inversion.
• $$\chi_V(I) = -3$$.
• $$\chi_R(I) = 3$$.
• $$S_n$$: roto-reflections.
• $$\chi_V(S_n) = -1 + 2\cos\theta_n$$.
• $$\chi_R(S_n) = 1 - 2\cos\theta_n$$.

Trace representation trace: $\chi_M(g) = n_{\mathrm{inv}}(g)\chi_V(g),$ where $$n_{\mathrm{inv}}(g)$$ is the number of atoms fixed under this operation.

By $\qty{R,\vb*{l}}$ we denote a operation $\vb*{r} \rightarrow R\vb*{r} + \vb*{l}.$

##### Mulliken Symbols
• A: singly degenerate.

• Only one orbital has that particular symmetry and energy.
• Symmetric with respect to primary rotational axis.
• B: singly degenerate.

• Only one orbital has that particular symmetry and energy.
• Antisymmetric with respect to primary rotational axis.
• E: doubly degenerate.

• Two orbitals has the same symmetry and energy
• T: triply degenerate.

• Three orbitals has the same symmetry and energy.
• g: symmetric with respect to inversion center.

• u: antisymmetric with respect to inversion center.

• 1: symmetric with respect to perpendicular $$C_2$$ axis.

• 2: antisymmetric with respect to perpendicular $$C_2$$ axis.

• ': designates symmetric with respect to horizontal reflection plane.

• '': designates antisymmetric with respect to horizontal reflection plane. In particular, we have dimensions \begin{align*} \color{red}\dim A &\color{red}= \dim B = 1, \\ \color{red}\dim E &\color{red}= 2, \\ \color{red}\dim T &\color{red}= 3. \end{align*}

• $$\sigma_h$$: a plane perpendicular to the axis; horizontal.

• $$\sigma_v$$: a vertical mirror containing the main axis.

• $$\sigma_d$$: if such plane bisects the angle between two $$C_2$$ axes.
##### Wyckoff Positions

For Wyckoff positions of a specific SG, look out the Bilbao Crystallographic Server.

A Wyckoff position is a set of points whose site symmetry groups are conjugate to each other. A Wyckoff position may contain multiple points.

Wyckoff positions are labelled $$n\alpha$$, where $$n$$ is the multiplicity of the position and $$\alpha$$ is a letter that orders Wyckoff positions in a particular space group by ascending $$n$$.

Maximal Wyckoff positions: those whose site-symmetry groups are not a proper subgroup of any other site-symmetry group.

$$\vb*{k}^\star$$ denote all points $$\set{g\vb*{k}| g\in G}$$ in the BZ.

IR of the space groups are labelled by $$\vb*{k}^\star$$ and are induced from IRs of the little groups.

#### Group Representation and Crystal Symmetry

##### Representation of Point Group

Point group:

• $$M$$ for mechanical representation:
• Write down the position vectors of all atoms: $u = \begin{pmatrix} \vb*{u}_1 & \cdots & \vb*{u}_n \end{pmatrix}^T.$
• For each symmetry operation, $$M$$ could be represented by a matrix that consists of $$3\times 3$$-blocks.
• Only diagonal terms (i.e. atom fixed under this operation) contribute to the trace.
• $$V$$ for vector representation: how vectors transform under the elements of the symmetric group.
• $$R$$ for rotational representation: how pseudovectors transform under the lements of the symmetry group.
• $$\mathrm{Vib}$$ for vibrational representation.

Point group of $$\ce{CH4}{}$$:

• Vector representation:
• The four vertices mix under symmetry operations, therefore triply degenerate.

\begin{align*} M &= A_1 \oplus E \oplus 3T_2 \oplus T_1, \\ V &= T_2, \\ R &= T_1, \\ \mathrm{Vib} &= M-V-R = A_1 \oplus E \oplus 2T_2. \end{align*} Since $$\dim \mathrm{Vib} = 9$$ we have $$9$$ vibration modes, $$1$$ non-degenerate, $$1$$ doubly and $$2$$ triply degenerate.

Point group of $$\ce{CH_3D}{}$$:

$$C_{3v}{}$$ $$E$$ $$2C_3$$ $$3C'_2$$
$$A_1$$ $$1$$ $$1$$ $$1$$
$$A_2$$ $$1$$ $$1$$ $$-1$$
$$E$$ $$2$$ $$-1$$ $$0$$
• Vector representation:
• The lower three vertices mix under symmetry operations, therefore doubly degenrate; plus
• The upper vertex in invariant under any symmetry operations.

\begin{align*} M &= 4A_1 \oplus A_2 \oplus 5E, \\ V &= A_1 \oplus E, \\ R &= A_2 \oplus E, \\ \mathrm{Vib} &= M - V - R = 3A_1 \oplus 3E. \end{align*} Since $$\dim \mathrm{Vib} = 9$$ we have $$9$$ vibration modes, $$3$$ non-degenerate and $$3$$ doubly degenerate.

##### Molecular Vibrations
• For $$T_d$$ symmetry, $\mathrm{Vib} = A_1(q_1) \oplus E(q_2,q_3) \oplus T_2(q_4,q_5,q_6) \oplus T_2(q_7,q_8,q_9).$
• The potential energy (under the normalization$$M = \mathbb{1}$$) is given by \begin{align*} U_{ij}q_iq_j &= aq_1 + b(q_2^2+q_3^3) + c(q_4^2+q_5^2+q_6^2) \\ &\phantom{{}={}} d(q_7^2 + q_8^2 + q_9^2) + 2e(q_4 q_7 + q_5 q_8 + q_6 q_9). \end{align*}{}
• After permuting $$\qty{q_4,\cdots,q_6}$$, we find $\begin{pmatrix} a & & & & & & & & \\ & b & & & & & & & \\ & & b & & & & & & \\ & & & c & e & & & & \\ & & & e & d & & & & \\ & & & & & c & e & & \\ & & & & & e & d & & \\ & & & & & & & c & e \\ & & & & & & & e & d \end{pmatrix}.$
• We find $$4$$ sequences, $$1$$ non-degenerate, $$2$$ doubly and $$2$$ triply degenerate.
• In general, if $\color{red} \mathrm{Vib} = \cdots \oplus m_i R_i \oplus \cdots,$ then there are $$m$$ different frequencies, each $$\dim R_i$$-degenerated.
##### Degeneration Splitting

From character tables

$$C_{3v}{}$$ $$E$$ $$2C_3$$ $$3C'_2$$
$$A_1$$ $$1$$ $$1$$ $$1$$
$$A_2$$ $$1$$ $$1$$ $$-1$$
$$E$$ $$2$$ $$-1$$ $$0$$
$$T_d$$ $$E$$ $$8C_3$$ $$3C'_2$$ $$6S_4$$ $$6\sigma_d$$
$$A_1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$A_2$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$E$$ $$2$$ $$-1$$ $$2$$ $$0$$ $$0$$
$$T_2$$ $$3$$ $$0$$ $$-1$$ $$1$$ $$-1$$
$$T_1$$ $$3$$ $$0$$ $$-1$$ $$-1$$ $$1$$

we find the subduction table

 $$T_d$$ $$A_1$$ $$A_2$$ $$E$$ $$T_2$$ $$T_1$$ $$C_{3v}{}$$ $$A_1$$ $$A_2$$ $$E$$ $$A_1\oplus E$$ $$A_2\oplus E$$

Triply degenerate level of a $$T_d$$ system will split into two levels, one if which double degenerate.

In the central field approximation $H\ket{nlm} = E_{nl}\ket{nlm},$ each level has a degeneracy $$2(2l+1)$$. In crystal, the symmetry is reduced from $$\mathrm{O}(3)$$ to the local point group.

Weak spin-orbit: from the character table $$\mathrm{O}(3)$$ to $$C_{3v}{}$$

$$C_{3v}{}$$ $$E$$ $$2C_3$$ $$3C'_2$$
$$A_1$$ $$1$$ $$1$$ $$1$$
$$A_2$$ $$1$$ $$1$$ $$-1$$
$$E$$ $$2$$ $$-1$$ $$0$$
$$O(3)$$ $$E$$ $$2C_3$$ $$3C'_2$$
$$D_0^+$$ $$1$$ $$1$$ $$1$$
$$D_1^-$$ $$3$$ $$0$$ $$1$$
$$D_2^+$$ $$5$$ $$-1$$ $$1$$
$$D_3^-$$ $$7$$ $$1$$ $$1$$

we find \begin{align*} s &\rightarrow a_1, \\ p &\rightarrow a_1 \oplus e, \\ d &\rightarrow a_1 \oplus 2e, \\ f &\rightarrow 2a_2 \oplus a_2 \oplus 2e. \end{align*}

Strong spin-orbit: from the character table $$\mathrm{SU}(2)$$ to $$C_{3v}{}$$

$$C_{3v}{}$$ $$E$$ $$\overline{E}{}$$ $$2C_3$$ $$2\overline{C}_3$$ $$3C'_2$$ $$3\overline{C}'_2$$
$$A_1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$A_2$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$^1\overline{E}{}$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$i$$ $$-i$$
$$^2\overline{E}{}$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-i$$ $$i$$
$$E$$ $$2$$ $$2$$ $$-1$$ $$-1$$ $$0$$ $$0$$
$$\overline{E}_1$$ $$2$$ $$-2$$ $$1$$ $$-1$$ $$0$$ $$0$$
$$O(3)$$ $$E$$ $$\overline{E}{}$$ $$2C_3$$ $$2\overline{C}_3$$ $$3C'_2$$ $$3\overline{C}'_2$$
$$D_{1/2}^-$$ $$2$$ $$-2$$ $$1$$ $$-1$$ $$0$$ $$0$$
$$D_{3/2}^-$$ $$4$$ $$-4$$ $$-1$$ $$1$$ $$0$$ $$0$$

we find \begin{align*} D^{-}_{1/2} &= \overline{E}_1, \\ D^{-}_{3/2} &= \overline{E}_1 \oplus {}^1\overline{E} \oplus {}^2\overline{E}. \end{align*}

##### Example: Group p2

Group p2:

• Point group: $$C_2$$.
• The group p2 contains coutains four centers of order 2 (180°), but no reflections or glide reflections.

The rotation centers in a p2 crystal are also Wyckoff positions:

• : 1a, symmetry group $$\qty{C_{2z}, \vb*{0}}$$, where $$C_{2z}$$ denote a $$C_2$$ rotation around $$1a$$.
• : 1b, symmetry group $$\qty{C_{2z}, \hat{\vb*{x}}}$$,
• : 1c, symmetry group $$\qty{C_{2z}, \hat{\vb*{y}}}$$,
• : 1d, symmetry group $$\qty{C_{2z}, \hat{\vb*{x}} + \hat{\vb*{y}}}$$,
• General position: 2e, which consists of two points $$(x,y)$$ and $$(-x,-y)$$. Symmetry group trivial.

Note that the symmetry group at 1a, 1b, 1c and 1d are isomorphic but not conjugate to each other.

1a, 1b, 1c and 1d are maximal Wyckoff positions while 2e is not.

Little groups of points of high symmetry in the BZ ($$\Gamma,X,Y,M$$) equal to the full group $$G$$. The little groups thereof are $$C_2$$.

Since there are two IRs to $$C_2$$, the band representations are given by $\rho_{G,1\alpha,A/B}(g) = \begin{cases} e^{-i\vb*{k}\cdot \vb*{t}}, & \text{if } g = \qty{E|\vb*{t}}, \\ \pm e^{i\vb*{k}\cdot (\vb*{t} - 2\vb*{q}_\alpha)}, & \text{if } g = \qty{C_2|\vb*{t}}. \end{cases}$ Different representations are distinguished by their $$\qty{C_2|\vb*{0}}$$ eigenvalues.

##### Band Representation
• $$G$$ denotes the space group.
• $$G_{\vb*{q}}$$ is the site-symmetry group of some site $$\vb*{q} = \vb*{q}_1$$.
• $$\qty{q_{\alpha}}$$ are the sites in the Wyckoff position of $$\vb*{q}$$ in the primitive cell, where $$\alpha = 1,2,\cdots,n$$.
• $$g_\alpha\in G$$ such that $$\vb*{q}_\alpha = g_\alpha \vb*{q}$$.
• $$G$$ admits the coset decomposition $G=\bigcup_{\alpha=1}^n g_\alpha(G_{\vb*{q}} \ltimes T).$
• A representation of $$G_{\vb*{q}}$$ induces a representation of $$G$$ by $[\rho_G(g)]_{(i,\alpha,\vb*{t}), (j,\beta,\vb*{t}')} = \qty[\tilde{\rho} \qty(g^{-1}_\alpha \qty{E|\vb*{t}} g \qty{E|\vb*{t}'}^{-1} g_\beta)]_{ij}.$
• This representation is denoted by $\rho_{\vb*{q}} \uparrow G.$
• Band representation: a direction sum of representations, each induced from a representation of the site-symmetry group of a Wyckoff position. $\color{orange} \bigoplus_{\vb*{q}} (\rho_{\vb*{q}} \uparrow G).$
• $$\vb*{q}$$ runs through representative site in all Wyckoff positions.
##### Wannier Basis
• A Wannier function $$W_{i\alpha}(\vb*{r})$$ carries three indices:
• $$i = 1,\cdots,\dim(\rho)$$ spans the dimension of the representation. I guess this denote the orbital.
• $$\alpha$$ labels sites of a Wyckoff position (I guess) in the primitive cell.
• $$\vb*{t}$$ is a lattice vector.
• Wannier functions of at the same site of different orbitals are related by $gW_{i1}(\vb*{r}) = [\rho(g)]_{ji}W_{j1}(\vb*{r}).$
• $$g\in G_{\vb*{q}}$$.
• Wannier functions provide a representation of $$G_{\vb*{q}}$$.
• Wannier functions of the same orbital $$i$$ at different $$\vb*{q}_\alpha$$ are defined by $W_{i\alpha} = g_\alpha W_{i1}(\vb*{r}) = W_{i1}(g_\alpha^{-1}\vb*{r}).$
• Wanier functions are extended outside the primitive cell by $\qty{E|\vb*{t}} W_{i\alpha}(\vb*{r}) = W_{i\alpha}(\vb*{r} - \vb*{t}).$
• $$W_{i\alpha}(\vb*{r} - \vb*{t})$$ forms a set of $\dim(\rho) \times n \times \mathcal{N}$ functions, where $$\mathcal{N}$$ denotes the number of unit cells.
• Now the Wannier functions provide a representation of $$G$$.
• With the decomposition $G = \bigcup_{\alpha = 1}^n g_\alpha(G_{\vb*{q}}\ltimes T)$ we write for a general $$g\in G$$ that $gg_\alpha = \qty{E | \vb*{t}_{\beta\alpha}} g_\beta g_{\vb*{q}}.$
• $$g_\beta$$ denote the representative element in the coset decomposition.
• $$\vb*{t}_{\beta\alpha} = g\vb*{q}_\alpha - \vb*{q}_\beta$$.
• $$g_{\vb*{q}}\in G_{\vb*{q}}$$.
• Wannier functions provide a representation by $\color{orange} \rho_G(g) W_{i\alpha}(\vb*{r} - \vb*{t}) = \sum_{j=1}^{\dim(\rho)} [\rho(g)]_{ji}W_{j\beta}(\vb*{r} - R\vb*{t} - \vb*{t}_{\beta\alpha}).$
• $$\rho$$ is a representation of $$G_{\vb*{q}}$$.
• I guess $$R$$ is the $$\mathrm{O}(3)$$ part of $$g$$.
• Fourier transformed Wannier functions: $a_{i\alpha}(\vb*{k},\vb*{r}) = \sum_{\vb*{t}} e^{i\vb*{k}\cdot \vb*{t}} W_{i\alpha}(\vb*{r} - \vb*{t}).$
• Isn't this just a Bloch wave generated by orbital $$i$$ of $$\alpha$$ atoms?
• Momentum space Wannier functions provide a representation by $\color{orange} \rho_G(g) a_{i\alpha}(\vb*{k},\vb*{r}) = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \sum_{j=1}^{\dim(\rho)} [\rho(g)]_{ji} a_{j\beta}(R\vb*{k}, \vb*{r}).$
• Represented by a $$\mathcal{N}\times \mathcal{N}$$ matrix of blocks $$(\vb*{k},\vb*{k}')$$, each $$(i\alpha,j\beta)$$ of size $n\dim(\rho)\times n\dim(\rho).$
• Most of the blocks are zero, unless $$\vb*{k}' = R\vb*{k}$$.
• Such nonzero blocks are denoted by $$\rho^{\vb*{k}}_G(g)$$.
• The above matrix $$\rho_G(g)$$ is a block diagonal matrix in $$\vb*{k}$$ if and only if $g\in G_{\vb*{k}}.$
• Band representation: a function takes $$\vb*{k}$$ to a representation $\rho_G \downarrow G_{\vb*{k}}.$
• This subduced representation is reducible (I guess).
• Under this definition, $$\rho_G(g)$$ does not make sense for every $$g$$, inconsistent with parts of the text remaining.
• Two band representations $$\rho_G$$ and $$\sigma_G$$ are equivalent if and only if there exists a unitary matrix valued tunction $$S(\vb*{k}, 1, g)$$ that
• is smooth in $$\vb*{k}$$,
• is continuous in $$t$$, and
• for all $$g\in G$$, $S(\vb*{k},0,g) = \rho^{\vb*{k}}_G(g),\quad S(\vb*{k},1,g) = \sigma_G^{\vb*{k}}(g).$
• In other words, equivalent representations are homotopy when regarded as functions of $$\vb*{k}$$.

EBR. Elementary Band Representation.

• Elementary band representation: a band representation that is not equivalent to a direct sum of two or more other band representations.
• Otherwise the representation is composite.
• EBRs are not IRs of the space group.
• There are a finite number of EBRs, indexed by IRs of maximal Wyckoff positions.
##### Electronic Bands

RL: Reciprocal Lattice.

• $$\vb*{k}_R$$ denote a reciprocal vector. $$\mathcal{T}$$ denotes the translation group.
• Representations of the translation group:
• All IRs of the translation group are one-dimensional.
• $\tau_{\vb*{k}}(\vb*{t}) = e^{i\vb*{k}\cdot \vb*{t}}.$
• $\vb*{k} + \vb*{k}_R \sim \vb*{k}.$
• Tight binding approximation: $\psi_{\vb*{k}}^a(\vb*{r}) = \sum_{\vb*{t}\in \mathcal{T}} \phi^a(\vb*{r} - \vb*{r}_a - \vb*{t}) e^{i\vb*{k}\cdot \vb*{t}}.$
• Band structure obtained by diagonalizing $\bra{\psi_{\vb*{k}}^a} H\ket{\psi_{\vb*{k}}^b} = H^{ab}(\vb*{k})\delta_{\vb*{k},\vb*{k}'}.$
• There are $$N_o$$ orbitals in total.

Character of the little group of $$p_z$$-orbital: $\color{red} \chi_T(g,\vb*{k}) = \sum_{i=1}^A \varphi_i(g,\vb*{k}) \theta_i(g) \chi_\tau(g).$

• $$A$$: number of atoms in the primitive cell.
• $$\theta_i(g)$$: $$1$$ if atom $$i$$ is invariant under $$g$$, zero otherwise.
• $$\varphi_i(g,\vb*{k})$$: additional phase.
• $$\chi_\tau(g)$$: the character of the IR of the orbitals.

The atoms in graphene all belong to the same Wyckoff position. There are two atoms per unit cell in graphene, $$8$$ orbitals in total: $\qty{\phi^1_s, \phi^1_{p_x}, \phi^1_{p_y}, \phi^1_{p_z},\phi^2_s, \phi^2_{p_x}, \phi^2_{p_y}, \phi^2_{p_z}}.$ $$(s,p_x,p_y)$$ are even while $$p_z$$ is odd under $$\sigma_h$$. We discard $$\sigma_h$$ and work with $$C_{6v}$$ only: $C_{6v} = \qty{E, C_2, C_3^\pm, C_6^\pm, \sigma_{di}, \sigma_{vi}}.$ The site symmetry group for each kind of atom is $C_{3v} = \qty{E, C_3^\pm, \sigma_{di}}.$ We have $D_1^- = A_1(p_z) \oplus E(p_x,p_y)$ and therefore $$p_z \rightarrow a_1$$.

We introduce two Bloch waves here: $\psi_{\vb*{k}}^a(\vb*{r}) = \sum_{\vb*{t}\in \mathcal{T}} \phi_{p_z}(\vb*{r} - \vb*{r}_a - \vb*{t}) e^{i\vb*{k}\cdot \vb*{t}}$ where $$a = 1,2$$ denotes the two kinds of atoms. $$\phi^1_{\vb*{k}}$$ and $$\psi^2_{\vb*{k}}$$ defines the 2-dimensional representation $$T_{\vb*{k}}$$ of the little group. Now we obtain the characters of little groups at points of high symmetry.

• $T_\Gamma = A_1 \oplus B_2 = \Gamma_1 \oplus \Gamma_3.$
• $T_K = K_3.$
• $T_M = M_1 \oplus M_2.$
• If $$k$$ is on a symmetry line, then $$T_{\vb*{k}}$$ is a subgroup of the little groups for the endpoints.

We could associate the names of IRs with bands, since $$G$$ acts on them.

• At each $$\vb*{k}$$ we obtain a representation of the little group. If no degeneracy occurs, we label the bands by the names of the IRs.
• If degeneracy occurs, then there must be a IR of more than one-dimension. We label the intersection of the bands by this IR.
##### Topological Systems

Let $$\vb*{k}_1$$ be a point near a high-symmetry point $$\vb*{k}_0$$. We have $G_{\vb*{k}_1} \subset G_{\vb*{k}_0}$ and the compatibility relations $\color{orange} \rho_1 = \rho_0 \downarrow G_{\vb*{k}_1}.$

• Quasiband representation: any little group representations satisfying these compatibility relations.
• Band representations are also quasiband representations, but not vice versa.
• Quasiband representation may lack exponentially localized Wannier functions. Such quasiband representations are topological bands.
• Symmetry indicated topological bands: topological bands that can be distinguished by their little group representations.
• Symmetry indicated topological bands $$\Leftrightarrow$$ nonnegative integer vectors $$\vb*{v}$$ that are not expressible as nonnegative integer sums of EBR vectors $$\vb*{e}_a$$.
• Stable symmetry-indicated topological bands: $$\vb*{v}$$ cannot be expressed as any integer (positive or not) linear combination of EBR vectors.
• Fragile symmetry-indicated topological bands: $$\vb*{v}$$ can only be expressed as a linear combination of EBR vectors with at least one negative coefficient.
• Let $$\rho_{i[\vb*{k}]}$$ denote the $$i$$th IR of the little group at $$\vb*{k}$$.
• $$[\vb*{k}]$$ denotes the symmetry-equivalent class of $$\vb*{k}$$.
• We form a vector space $$V_G$$ spanned by $\vb*{v} = \bigoplus_{i,[\vb*{k}]} n_{i[\vb*{k}]} \rho_{i[\vb*{k}]}.$
• Every quasiband representation corresponds to a $$\vb*{v}$$ whose $$n_{i[\vb*{k}]}$$ satisfy the compatibility relations.
• Each EBR $$\rho_{\vb*{k}}^a$$ corresponds to a vector $$\vb*{e}_a$$ in that space.
• Each atomic-limit band structure can be identified with a vector $\vb*{a} = \bigoplus_{a} n_a\vb*{e}_a.$
• We organize $$\vb*{e}_a$$'s into a matrix $$\vb*{A}$$. The problem of expanding a quasiband representation into EBR is formulated as $\vb*{v} = \vb*{A} \vb*{n}.$
• We demand in the above expressions that $$n$$'s be non-negative.

After a linear transformation, $$\vb*{v} = \vb*{A}\vb*{n}$$ could be reformulated into $\vb*{v}' = \vb*{D} \vb*{n}',$ where $$\vb*{D}$$ is a diagonal matrix of size $$n\times m$$.

• Trivial bands are the integer solutions.
• The space $$\mathcal{V}_G$$ of symmetry indicated topological bands is isomorphic to $\color{darkcyan} \mathcal{V}_G \approx \bigoplus_i \mathbb{Z}_{d_i}.$
• $$\mathcal{V}_G$$ is the symmetry indicator group for the space group $$G$$.

The $$\vb*{k}$$ points of p2 fall into five categories:

• $$C_2$$, which admits two irreducible representations, denoted $$A$$ and $$B$$: $$\Gamma$$, $$X$$, $$Y$$ and $$M$$.
• Trivial: GP. Therefore, $$V_G$$ has nine elements in its basis: $\Gamma_\pm, X_\pm, Y_\pm, M_\pm, GP.$ Compatibility relations: since lines connecting different $$\vb*{k}$$'s are trivial, the only compatibility relation is that the dimension of the representation at each $$\vb*{k}$$ be the same. $n_{\Gamma+} + n_{\Gamma-} = n_{X+} + n_{X-} = n_{Y+} + n_{Y-} = n_{M+} + n_{M-} = n_{GP}.$
• Quasiband representation: $\vb*{v} = (n_{\Gamma+}, N - n_{\Gamma+}, n_{X+}, N-n_{X+}, n_{Y+}, N-n_{Y+}, n_{M+}, N-n_{M+}, N).$
• EBR: \begin{align*} \vb*{e}_{1aA} &= (1,0,1,0,1,0,1,0,1), \\ \vb*{e}_{1aB} &= (0,1,0,1,0,1,0,1,1), \\ \vb*{e}_{1bA} &= (1,0,0,1,1,0,0,1,1), \\ \vb*{e}_{1bB} &= (0,1,1,0,0,1,1,0,1), \\ \vb*{e}_{1cA} &= (1,0,1,0,0,1,0,1,1), \\ \vb*{e}_{1cB} &= (0,1,0,1,1,0,1,0,1), \\ \vb*{e}_{1dA} &= (1,0,0,1,0,1,1,0,1), \\ \vb*{e}_{1dB} &= (0,1,1,0,1,0,0,1,1). \end{align*} We have $D = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}.$
• $$0$$ of multiplicity $$3$$: we have only $$5$$ degrees of freedom.
• $$1$$ of multiplicity $$4$$: four of the dimension are trivial.
• $$2$$ of multiplicity $$1$$: topologically nontrivial part, $$\mathbb{Z}_2$$ symmetry.
• Indicator: $\nu = n_{\Gamma+} + n_{M+} - n_{X+} - n_{Y+} \operatorname{mod} 2 \in \mathbb{Z}_2.$
##### Example: Group P1̅1'

Symmetries:

• Three linearly independent lattice translations.
• Inversion.
• Time reversal.

TRIM. Time-Reversal Invariant Momenta.

Wyckoff positions:

• 1a to 1h, corresponding to the eight inversion centers in the unit cell, $\frac{1}{2}(n,m,l),$ where each of $$n,m,l$$ takes value $$0$$ or $$1$$. They are also called time-reversal invariant momenta.

EBRs: 16 in total.

• Induced from a Kramers pair of orbitals with either $$+1$$ or $$-1$$ inversion eigenvalue at one of these eight Wyckoff positions.

HOTI. High Order Topological Insulator.

The Indicator group of P1̅1' is $$\mathbb{Z}_4$$, and $\nu_4 = \sum_{\mathrm{TRIMs} \vb*{k}_i} n_{\vb*{k}_i -} \operatorname{mod} 4,$ where $$n_{\vb*{k}_i -}$$ the multiplicity of IRs at the TRIM $$\vb*{k}_i$$ with negative inversion eigenvalue.

##### Disconnected Elementary Band Representations
• Suppose $$\rho_{\vb*{k}}$$ is disconnected.
• $$\rho_{\vb*{k}}$$ could be written as a sum of quasiband representations $\rho_{\vb*{k}} = \rho_{1\vb*{k}} \oplus \rho_{2\vb*{k}}.$
• $$\rho_1$$ and $$\rho_2$$ are quasiband representations but cannot be both band representations.
• Topologically nontrivial.

In the Kane-Mele model of the p6mm group, the four-bane EPR induced from $$p_z$$ orbitals at honeycomb sites (2b Wyckoff position) may split into two disconnected components (in the topological phase), each of which has a nontrivial $$\mathbb{Z}_2$$ index.

#### Band Representation and Its Applications

##### Band Representation
• What do we need to construct a band representation? We need to pick a site $$\vb*{q}$$, which belongs to a Wyckoff position of multiplicity $$n$$, and an $$d_q$$-dimensional representation $$\rho$$ of $$G_{\vb*{q}}$$.
• It's the Chemists' habit to let the representation act on Wannier functions, i.e. $g_{\vb*{q}} W_{i1}(\vb*{r}) = [\rho(g_{\vb*{q}})]_{ji} W_{j1}(\vb*{r}),\quad \text{for}\quad g\in G_{\vb*{q}}.$
• Now we have the linear space where this representation works on: $V = \set{W_{i1}(\vb*{r}) | i = 1,\cdots,d_q}.$
• I think (just guessing) that the Wannier functions here are just those simple basis functions like $$x$$, $$y$$, and $$z$$ for $$p$$-orbitals and $$xy$$ for $$d$$-orbitals, etc, not the calculated Wannier functions obtained after we know the exact wavefunctions.
• Now we extend the representation to the whole point group by defining the action of representative elements as $W_{i\alpha}(\vb*{r}) = g_\alpha W_{i1}(\vb*{r}),$ where $\vb*{q}_\alpha = g_\alpha \vb*{q}_1.$
• We extend the representation further to the whole space group by $\qty{E|\vb*{t}} W_{i\alpha}(\vb*{r}) = W_{i\alpha}(\vb*{r} - \vb*{t}).$
• Basis of our huge representation: $\qty{W_{i\alpha}(\vb*{r} - \vb*{t})} = \bigcup_{\alpha,\vb*{t}} \qty{g_\alpha | \vb*{t}} V,$ i.e. Wannier function of orbital $$\alpha$$ at cell $$\vb*{t}$$ and the site labelled by $$i$$.
• Dimension: $n\times d_q \times N_1 N_2 N_3.$
• This is exactly what we had for induced representations. Let figure it out.
• Which space does $$g$$ move $$W$$ to? i.e. $g \cdot \qty{g_\alpha | \vb*{t}_\alpha} W_{i1}(\vb*{r}) = \qty{g_{?}|\vb*{t}_?} g_{\vb*{q}} W_{i1}(\vb*{r}).$
• The representation is determined once the $$?$$ is determined.
• With the Wannier functions we recover the Bloch functions (not necessarily the real Bloch functions: just toy Bloch functions), and let $$g\in G$$ acts on it: $a_{i\alpha}(\vb*{k}, \vb*{r}) = \sum_{\vb*{t}} e^{i\vb*{k}\cdot \vb*{t}} W_{i\alpha}(\vb*{r} - \vb*{t}).$
• This amounts to a change of basis in out huge linear space $\bigcup_{\alpha,\vb*{t}} \qty{g_\alpha|\vb*{t}} V.$
• Why do we need this when our huge set of Wannier functions already serve as a good representation? Because our huge representation is obviously reducible. By doing the Fourier transform, our basis functions are made into eigenfunctions of translations, i.e. pre-diagonalization $\qty{E|\vb*{t}} a_{i\alpha}(\vb*{k}, \vb*{r}) = e^{-i\vb*{k}\cdot \vb*{t}} a_{i\alpha}(\vb*{k}, \vb*{r}).$
• Band representation: the band representation $$\rho_G$$ induced from the $$d_q$$-dimensional representation $$\rho$$ of the site-symmetry group $$G_{\vb*{q}}$$ of a particular point $$\vb*{q}$$ whose orbit contains the sites $$\qty{\vb*{q}_\alpha = g_\alpha \vb*{q}}$$ in the unit cell, is defined by the action $\color{orange} [\rho_G(g)a]_{i\alpha}(\vb*{k},\vb*{r}) = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \sum_{i'=1}^{d_{\vb*{q}}} \rho_{i'i}(g_\beta^{-1} \qty{E\vert - \vb*{t}_{\beta\alpha}}gg_\alpha) a_{i'\beta}(R\vb*{k},\vb*{r}) = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \sum_{i'=1}^{d_{\vb*{q}}} \rho_{i'i}(g_{\vb*{q}}) a_{i'\beta}(R\vb*{k},\vb*{r}),$
• $g = \qty{R|\vb*{v}}.$
• $gg_\alpha = \qty{E|\vb*{t}_{\beta\alpha}} g_\beta g_{\vb*{q}},$
• $\vb*{t}_{\beta\alpha} = h\vb*{q}_\alpha - \vb*{q}_\beta.$
• Block form: $$\rho$$ connects $\vb*{k'} = R\vb*{k}$ with $$\vb*{k}$$.
• $\rho^{\vb*{k}}_G(g)_{j\beta,i\alpha} = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \rho_{ji}\qty(g_\beta^{-1}\qty{E|-\vb*{t}_{\beta\alpha}}gg_\alpha).$
• At each $$\vb*{k}$$ we have a $$n\times d_q$$-dimensional representation of the space group.
• Equivalent band representations: two band representations $$\rho_G$$ and $$\sigma_G$$ are equivalent if and only if there exists a unitary matrix-valued function $$S(\vb*{k}, t, g)$$ that is smooth in $$\vb*{k}$$ and continuous in $$t$$ such that
• $$S(\vb*{k},t,g)$$ defines a band representation by $S(\vb*{k},t,g)_{j\beta, i\alpha} = \tau^{\vb*{k}}_G(g)_{j\beta,i\alpha}.$
• $S(\vb*{k},0,g) = \rho^{\vb*{k}}_G(g).$
• $S(\vb*{k},1,g) = \sigma^{\vb*{k}}_G(g).$
• If two band representations are equivalent, then their subduced representation at little groups $$G_{\vb*{k}}$$ (small representations) are equivalent as well.
• Equivalence of small representations could be verified by characters.
• This is only a necessary condition.
##### Small Representations
• The little group of $$\vb*{k}$$: $G_{\vb*{k}} = \set{g=\qty{R|\vb*{v}} | R\vb*{k} = \vb*{k} + \vb*{K}, g\in G}.$
• $$\qty{R|\vb*{v}}$$ and $$\qty{R|\vb*{v}'}$$ differs only by a factor of $$e^{-i\vb*{k}\cdot \vb*{t}}$$.
• Small representation: $\rho_G \downarrow G_{\vb*{k}}$ restricted to those elements in $$G_{\vb*{k}}$$, and retain only the block $\rho^{\vb*{k}}_G(g)_{j\beta,i\alpha} = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \rho_{ji}\qty(g_\beta^{-1}\qty{E|-\vb*{t}_{\beta\alpha}}gg_\alpha).$
• Dimension: $n\times d_{q}.$
• Character: $\chi^{\vb*{k}}_G(g) = \sum_\alpha e^{-i(R\vb*{k})\cdot \vb*{t}_{\alpha\alpha}}\tilde{\chi}\qty[\rho(g_\alpha^{-1}\qty{E|-\vb*{t}_{\alpha\alpha}}gg_\alpha)],$ where $\tilde{\chi}[\rho(g)] = \begin{cases} \chi[\rho(g)], & \text{if } g\in G_{\vb*{q}}, \\ 0, & \text{otherwise}. \end{cases}$ For the trace to be nonzero, we demand $g_\alpha^{-1}\qty{E|\vb*{t}_{\alpha\alpha}} g g_\alpha = g_{\vb*{q}}$ for some carefully picked $$g_\alpha$$ and $$\vb*{t}_{\alpha\alpha}$$.
• Since $\rho^{\vb*{k}}_G = (\rho \uparrow G) \downarrow G_{\vb*{k}}$ is a representation of $$G_{\vb*{k}}$$, it may be decomposed into a direct sum of IRs of $$G_{\vb*{k}}$$ as $(\rho \uparrow G) \downarrow G_{\vb*{k}} = \bigoplus_i m_i^{\vb*{k}} \sigma_i^{\vb*{k}}.$ The coefficients may be obtained directly from the characters.
##### Qusaiband Representation

QBR. Quasiband Representation.

TQBR. Topological Quasiband Representation.

• Quasiband representation: any solution to the compatibility relations.
• Topological quasiband representation: a QBR that is not a (composite or elementary) band representation.
##### Elementary Band Representation
• Composite band representation: a band representation that is equivalent to a direct sum of other band representations.
• Elementary band representation: a band representation that is not composite.
• All elementary band representations could be induced from IRs of the maximal site symmetry groups.
• A band representation $$\rho_G$$ is elementary if and only if it can be induced from an IR $$\rho$$ of a maximal site-symmetry group $$G_{\vb*{q}}$$, with only a few exceptions.
##### Classification of Elementary representations
• (Fully filled) Connected EBR ⬌ Exponentially localized Wannier orbitals that respect to the symmetries of crystals ⬌ Topologically trivial.
• Fractionally filled EBR ⬌ Protected (semi-)metals.
• Disconnected EBR ⬌ Topological bands ⬌ At least one group of bands is a weak, strong, or crystalline topological insulator.
• Isolated bands not equivalent to a band representation ⬌ strong, weak, or crystalline topological insulator.
##### Topological Bands
• Bands in the atomic limit: bands that can be induced from localized Wannier functions consistent with the crystalline symmetry of the SG.
• Topological bands: bands not in the atomic limit.
##### Hidden Obstructed Atomic Limits

OAL. Obstructed Atomic Limits.

• Bands in the obstructed atomic limit: a set of bands that possess symmetric, localized Wannier functions that reside on a Wyckoff position distinct from the Wyckoff position distinct from the Wyckoff position of the underlying atoms and cannot be smoothly defoemed to the ionic position.

IR-equivalent bands: two groups of bands with the same IRs at all high-symmetry points in the BZ.

It is possible that two such bands transforms identically under all symmetries at each point in the BZ but differ by a topologically nontrival global gauge transformation. Such band representations are called obstructed atomic limits.

In F222 each EBR induced from the 4a position is IR-equivalent to that induced from the 4b position. The basis functions are related by $\ket{w_b} = e^{i(k_2+k_3-k_1)/2} \ket{w_a}.$ If $$H(t)$$ has occupied band 4a at $$t=0$$ and 4b at $$t=1$$, then a gap-closing phase transition must occur at some intermediate $$t$$.

#### Examples

##### Position 1a in p6mm
$$C_{6v}{}$$ $$E$$ $$C_{3z}{}$$ $$C_{2z}{}$$ $$C_{6z}{}$$ $$m$$ $$C_{6z}m$$ $$\overline{E}{}$$
$$\Gamma_1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\Gamma_2$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
$$\Gamma_3$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$
$$\Gamma_4$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\Gamma_5$$ $$2$$ $$-1$$ $$2$$ $$-1$$ $$0$$ $$0$$ $$2$$
$$\Gamma_6$$ $$2$$ $$-1$$ $$-2$$ $$1$$ $$0$$ $$0$$ $$2$$
$$\overline{\Gamma}_7$$ $$2$$ $$-2$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-2$$
$$\overline{\Gamma}_8$$ $$2$$ $$1$$ $$0$$ $$-\sqrt{3}{}$$ $$0$$ $$0$$ $$-2$$
$$\overline{\Gamma}_9$$ $$2$$ $$1$$ $$0$$ $$\sqrt{3}{}$$ $$0$$ $$0$$ $$-2$$
• Little group: $$C_{6v}{}$$.
• Sites: $\qty{q_\alpha} = \qty{q_1}.$
• Band representation: $[\rho_G(\qty{R|\vb*{t}})a]_i(\vb*{k}, \vb*{r}) = e^{-i(R\vb*{k})\cdot \vb*{t}} \sum_{i'} [\Gamma(R)]_{i' i}a_{i'}(R\vb*{k}, \vb*{r}).$
##### Position 2b in p6mm
$$C_{3v}{}$$ $$E$$ $$C_{3}{}$$ $$m$$ $$\overline{E}{}$$
$$\overline{\Gamma}_4$$ $$1$$ $$-1$$ $$-i$$ $$-1$$
$$\overline{\Gamma}_5$$ $$1$$ $$-1$$ $$i$$ $$-1$$
$$\overline{\Gamma}_6$$ $$2$$ $$1$$ $$0$$ $$-2$$
$$C_{2v}{}$$ $$E$$ $$C_{2}{}$$ $$m$$ $$C_{2}m$$ $$\overline{E}{}$$
$$\overline{\Gamma}_5$$ $$2$$ $$0$$ $$0$$ $$0$$ $$-2$$
$$C_{s}{}$$ $$E$$ $$m$$ $$\overline{E}{}$$
$$\overline{\Gamma}_3$$ $$1$$ $$-i$$ $$-1$$
$$\overline{\Gamma}_4$$ $$1$$ $$i$$ $$-1$$
• Little group: $$C_{3v}{}$$.
• Sites: $\qty{q_{\alpha}} = \qty{C_2 q_1, q_1}.$
• $g_1 = \qty{E | \vb*{0}}.$
• $g_2 = \qty{C_2, \vb*{0}}.$
• We pick a specific IR of the little group: $$\overline{\Gamma}_6$$, which is the spin-1/2 representation and describes the behavior of the $$p_z$$ orbital.
• In particular, the orbital index $$i$$ takes two values, and the site index $$\alpha$$ takes two values, giving rise to a four-dimensional representation at each site $$\vb*{k}$$.
• We are now able to compute for each $$\vb*{k}$$ the matrix $\rho^{\vb*{k}}_G(g)_{j\beta,i\alpha}$ and restrict $$g$$ to $$g\in G_{\vb*{k}}$$. This furnishes a representation of $$G_{\vb*{k}}$$, which might be reducible. \begin{align*} \rho^{\Gamma} &= \overline{\Gamma}_8 \oplus \overline{\Gamma}_9, \\ \rho^K &= \overline{K}_4 \oplus \overline{K}_5 \oplus \overline{K}_6, \\ \rho^M &= \overline{M}_5 \oplus \overline{M}_5. \end{align*}{}
• Suppose that we are now given a group of bands that
• transforms at $$\Gamma$$ according to $\overline{\Gamma}_8 \oplus \overline{\Gamma}_9;$
• transforms at $$K$$ according to $\overline{K}_4 \oplus \overline{K}_5 \oplus \overline{K}_6;$
• transforms at $$M$$ according to $\overline{M}_5 \oplus \overline{M}_5.$ Then how are they possibly connected on the line joining these points? The little group of each point on the line are isomorphic to $$C_s$$, all generated by a mirror combined with translations. From the character tables we obtain the compatibility relations
• from $$\Gamma$$: \begin{align*} \overline{\Gamma}_8 &\rightarrow \overline{\Gamma}_3 + \overline{\Gamma}_4, \\ \overline{\Gamma}_9 &\rightarrow \overline{\Gamma}_3 + \overline{\Gamma}_4; \end{align*}
• from $$K$$: \begin{align*} \overline{K}_4 & \rightarrow \overline{\Gamma}_3, \\ \overline{K}_5 & \rightarrow \overline{\Gamma}_4, \\ \overline{K}_6 & \rightarrow \overline{\Gamma}_3 + \overline{\Gamma}_4; \\ \end{align*}
• from $$M$$: \begin{align*} \overline{M}_5 &\rightarrow \overline{\Gamma}_3 + \overline{\Gamma}_4. \end{align*}
• The compatibility relations per se allow different connections between.
• In fig (a) the lower band is equivalent to the $$\overline{\Gamma}_9$$ representation induced from the 1a position. However, the upper band is not equivalent to any band representation and is therefore topological.
• The Wannier function of the upper band is given by (see Wannier representation of Z₂ topological insulators) $\ket{p_z \uparrow + \downarrow}$ on the $$A$$ sites, and $\ket{p_z \uparrow - \downarrow}$ on the $$B$$ sites.
• The Wannier functions does not respect the symmetries (for example, $$C_{3v}{}$$) of the 2b positions.
• In fig (b), the bands are connected and therefore nontopological.
• Only with a given crystal with its bands calculated could we know which connection scheme do the bands adopt and thereby find whether the band is topological or not.

#### Miscellaneous

Little group and little cogroup?

What do we mean when we say a time-reversal symmetric representation?

• I don't know. I am making this up. Under time-reversal we have $\psi_{\vb*{k},\sigma} \rightarrow \psi_{-\vb*{k},-\sigma}.$ I guess that, for $$\rho$$ being a time-reversal symmetric IR of a site symmetry, $$\rho(T)$$ does not connect two different degenerate spaces.
• What the heck is that?

Band representation and quasiband representation?

• Both satisfy the compatibility relations. However, band representations allows Wannier functions that exponentially decay while quasiband representations may or may not.

What do we mean when we say %#@&^! is protected by $&*<# symmetry? • As long as the symmetry exists, the property is there disregarding perturbations. Expand ## Neutral Network Help me! My project is due tomorrow and I don't have access to Wikipedia! More than one billion people have no access to Wikipedia, Google, etc. Don't wanna be one of them? See if this post could help you. • I need Google immediately! Tell me how to setup my computer/phone! • First you need a V2ray client installed on your device. See Setup My Client for instructions. • Then you should import a server into your V2ray client. You may either • Start your V2ray client, and see if you have access to Google. • Ze, your servers are too slow! I wanna set up my own one! • My apt/npm/gradle build hangs. What should I do? • Switch to a mirror when there is one. • Otherwise, setup a proxy. I should provide more detail, maybe in another post. ### Client #### Configuration See Setup My Client first if you don't have a V2ray client installed yet. #### Setup My Client ##### Mac I'm currently using V2rayU as my client on Mac. The design of the UI is quite intuitive and you may be able to figure out the usage on your own with little effort. Nevertheless, the following tips may save your time. • Don't forget to click Turn v2ray-core On in the menu. • Set the Proxy Mode to Global Mode (or Pac mode). • Tick a working server in the menu. • The time on your computer is set correctly. ##### Windows I'm currently using v2rayN as my client on Windows. Generally, you may be able to figure out its usage after a short time. If it doesn't work, you may check that • Proxy is turned on, • Proxy mode is set to Global or Pac, • You've ticked a working server, and • The time on your computer is set correctly. ##### Linux I'm currently using v2ray-linux-64 as my client on Ubuntu. I have not yet tested it on other distributions. • GitHub page • In case the link breaks down: download here. • You should put the config.json from the Configuration section into the directory where the executable v2ray lies. You should be a far better Linux user than I am. So, I don't have much advice here. Just • Don't forget to properly setup the proxy of your computer, and • Make sure that the time on your computer is set correctly. Also check out v2ray-core as well as other clients. ##### iOS I'm currently using Shadowrocket as my client on iOS. • This software is priced$2.99, and is not available in Mainland China App Store.
• There is a FAKE software named Shadowrocket in Mainland China App Store. DON'T BE FOOLED.
• To acquire the software, you may
• Download it with your own Apple ID if it is NOT a Mainland China Apple ID, or
• ID: rxz530@163.com
• Password: SRGB#0ff
• Security Questions:
• Meal: wp123
• Book: wp345
• Album: wp456
• Birthdate: 1988 01 15
• You may import a server via the QR Code, vmess link, or JSON from the Configuration section.
##### Android

I'm currently using v2rayNG as my client on Android.

##### Still Doesn't Work?

Don't hesitate to contact me!

### Server

For those who wanna setup their own servers only. If you are already satisfied with whatever you get from the previous section, you don't have to go ahead.

For a V2ray server to work, you need

• A domain name.
• A VPS.
• A V2ray server properly configured on the VPS.

Theoretically, you don't need a domain name for the server to work. However, your server would be unlikely to survive the powerful censorship in that case.

The procedure could take around an hour. Note that if you are getting your domain name from Aliyun, it may take additional procedures and time for Aliyun to get your domain name ready. I would therefore advise you to acquire a domain name before you proceed.

#### VPS

Make sure that the operating system of your VPS is CentOS 7. Exactly 7.

I setup my VPS

These are about the fastest I could get. You may have your own choice and don't forget to leave a comment below if you find a faster one.

Typically, it would take from 10 minutes to an hour for the VPS to be ready. Follow the instructions from your server provider so that you could ssh to your server. I couldn't give a detailed procedure here since it varies from one provider to another.

#### Domain Name

1. Acquire a domain name from, for example, Aliyun.
2. Bind the domain name to the server you've just obtained (A Record).
3. ping your domain name and make sure it is resolved to the IP address of your VPS.

#### V2ray Server

The following procedure may not be the simplest one. I will revise it in the future.

1. Make sure the time on your VPS is properly set.
• Calibrate the time with, for example, ntpdate ntp.ubuntu.com.
2. Install V2ray on your server:
1. git clone https://github.com/v2fly/fhs-install-v2ray.git
2. cd fhs-install-v2ray
3. chmod 777 install-release.sh if necessary,
4. ./install-release.sh
5. systemctl enable v2ray
6. You are gonna have a config.json in /usr/local/etc/v2ray. Confirm that with ls.
• Replace config.json with this config.json.
• wget http://zechen.site/scribble-files/config.json
• Create the directory if you don't have it.
• wget http://zechen.site/scribble-files/v2ray_ws_tls.sh
• Before 2021, this script itself could do the whole procedure. However, indescribable things happened and we have to revise our procedure.
4. chmod 777 v2ray_ws_tls.sh if necessary.
5. ./v2ray_ws_tls.sh
• It typically takes 10 minutes or so to complete the installation.
• You will be asked to type the domain name to your server. Make sure that your domain name is ready before you run this script.
• Finally you are gonna get something on your screen like
===========配置参数============
地址：abcdef.xyz
端口：443
uuid：33c917a5-c9d6-453c-9d02-a464d2474030
额外id：64
加密方式：aes-128-gcm
传输协议：ws
别名：myws
路径：0368
底层传输：tls

• Don't clear your screen. Save the above output to somewhere.
6. Edit your /usr/local/etc/v2ray/config.json:
• Change
"id": "1a7330b8-878c-4a47-8dfe-1c9066ae938b",

to
"id": "whatever you see from the above output, e.g. 33c917a5-c9d6-453c-9d02-a464d2474030",

Don't forget the comma!
• Change
"path": "/8f2f"

to
"path": "whatever you see from the above output, e.g. 0368"

• Save the changes.
7. systemctl start v2ray

Don't hesitate to contact me if there is any problem!

Expand
 Quantum Field Theory (III) Feynman Diagrams

## Quantum Field Theory (IV)

Renormalization

やはりディラックの量子電磁力学はまちがっている。

#### Critical Exponents

 Quantum Field Theory (V) Applications of Feynman Diagrams

Expand
 Index Index

## Quantum Field Theory (I)

Mathematical Prerequisites

#### Grassman Number

• Properties:
• Anticommutative: $\theta\eta = -\eta\theta.$
• Nilpotent: $\theta^2 = 0.$
• Integration: $\int \dd{\theta} f(\theta) = \int \dd{\theta} \qty(A+B\theta) = B.$
• Rule of proximity: $\int \dd{\theta} \int \dd{\eta} \eta\theta = 1.$
• Rule of proximity for derivatives: $\dv{}{\eta} \theta\eta = -\dv{}{\eta} \eta\theta = -\theta.$
• Complex conjugation: $(\theta\eta)^* = \eta^* \theta^* = -\theta^* \eta^*.$
• Integration of complex Grassman numbers: $\theta = \frac{\theta_1 + i\theta_2}{\sqrt{2}},\quad \theta^* = \frac{\theta_1 - i\theta_2}{\sqrt{2}}$ treated as independent.

Gaussian integral $\qty(\prod_i \int \dd{\theta^*_i} \dd{\theta}_i) e^{-\theta^*_i B_{ij} \theta_j} = \qty(\prod_i \int \dd{\theta^*_i} \dd{\theta_i}) e^{-\sum_i \theta^*_i b_i \theta_i} = \det B.$ $\qty(\prod_i \int \dd{\theta^*_i} \dd{\theta_i}) \theta_k \theta^*_l e^{-\theta^*_i B_{ij} \theta_j} = (\det B) (B^{-1})_{kl}.$

#### Gamma Matrices

##### Properties
• $$\gamma_a$$ satisfies the commutation relations $\color{orange}\qty{\gamma_a,\gamma_b} = 2\delta_{ab} \mathbb{1}.$
• A series of product of Gamma matrices may be simplified using the anti-commutation relation. These products form the group $$\Gamma_N$$.
• $$\gamma_a$$'s are Hermitian and Unitary.
• $\gamma^{(N)}_\chi = \gamma_1 \gamma_2 \cdots \gamma_N,\quad \qty(\gamma^{(N)}_\chi)^2 = (-1)^{N(N-1)/2}\mathbb{1}.$
• For odd $$N$$, the representation may be given by a simple modification of the case of even $$N$$.
• For $$N=2l$$, the order $$\abs{\Gamma_N} = 2^{2l+1}$$.
• The dimension is given by $$d^{(2l)} = 2^l$$.
• We may define $\gamma^{2l}_f = (-1)^l \gamma_1 \cdots \gamma_{2l}$ such that $\qty(\gamma^{(2l)}_f)^2 = \mathbb{1}.$
• Let $$\Gamma'_{2l} = \Gamma_{2l}/(x\sim -x)$$. Then elements of $$\Gamma'$$ are linearly independent.
• There are $$2^{2l}+1$$ conjugacy classes. $$+\mathbb{1}$$, $$-\mathbb{1}$$, as well as $$2^{2l}-1$$ other $$\qty{\pm S}$$. Therefore, there are $$2^{2l}+1$$ inequivalent irreducible representations. $$2^{2l}$$ of which are one-dimensional unfaithful representations.
• We are left with a unique faithful representation of $$\Gamma_{2l}$$, of dimension $$2^l$$.

Theorem. If $$N=2l$$, then representations of dimension $$2^l$$ are equivalent.

##### Four-Dimensional Case

We define $\color{orange}\gamma^5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3,$ which satisfies the same conditions as $$\gamma_0$$ to $$\gamma_4$$:

• $(\gamma^5)^\dagger = \gamma^5;$
• $(\gamma^5)^2 = \mathbb{1}.$
• $\qty{\gamma^5,\gamma^\mu} = 0.$
##### Weyl (Chiral) Representation, Pauli Matrices, etc.

We define

• $\sigma^\mu = (\mathbb{1},\vb*{\sigma}),$
• $\overline{\sigma}^\mu = (\mathbb{1}, -\vb*{\sigma}).$

The Weyl or Chiral representation is given by $\gamma^0 = \begin{pmatrix} 0 & \mathbb{1} \\ \mathbb{1} & 0 \end{pmatrix},\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix}.$

Or, $\gamma^\mu = \begin{pmatrix} 0 & \sigma^\mu \\ \overline{\sigma}^\mu & 0 \end{pmatrix}.$

Under this representation, $\gamma_5 = i\gamma^0\gamma^1\gamma^2\gamma^3 = \begin{pmatrix} -\mathbb{1} & 0 \\ 0 & \mathbb{1} \end{pmatrix}.$

Useful identities of $$\sigma$$:

• $\color{darkcyan} \sigma^2 \vb*{\sigma}^* = -\vb*{\sigma}\sigma^2.$
• $(p\cdot \sigma)(p\cdot \overline{\sigma}) = p^2 = m^2.$

Useful identities of $$\gamma$$:

• $\color{darkcyan} \Lambda_{1/2}^{-1} \gamma^\mu \Lambda_{1/2} = \Lambda{^\mu}{_\nu} \gamma^\nu.$
##### Dicrete Transformation
• Charge conjugation $$C$$ is a unitary matrix such that $C^\dagger \gamma_a C = -\gamma_a^T.$
• We have $C^T = \lambda C,$ where $\lambda = (-1)^{l(l+1)/2}.$
• Parity $$P$$ is a unitary matrix such that $P^\dagger \gamma_a P = \gamma_a^T.$
• We have $P^T = \tau P,$ where $\tau = (-1)^{l(l+1)/2}.$
• $$C$$ and $$P$$ are related by $P = \gamma_f C.$
• For $$N=4$$ we have $C^T = -C,\quad P^T = -P.$

#### Lorentz Invariance

##### How do Tuples Transform?

Vectors are transformed in the familiar way: $\partial_\mu \phi(x) \rightarrow (\Lambda^{-1}){^\nu}{_\mu} (\partial_\nu \phi) (\Lambda^{-1} x).$

On the LHS we have quantities in the old frame while on the RHS in the new frame.

For a general $$n$$-component multiplet, the transformation law is given by an $$n$$-dimensional representation of the Lorentz group: $\color{red} \varphi_a(x) \rightarrow U^{-1}(\Lambda) \varphi_a(x) U(\Lambda) = L{_a}{^b}(\Lambda) \varphi_b(\Lambda^{-1}x).$ This is the most general transformation rule! You should fucking remember it! As well as its infinitesimal form! $\color{red} \varphi_a(x) \rightarrow (\delta{_a}{^b} + \frac{i}{2} \delta \omega_{\mu\nu} (S^{\mu\nu}){_a}{^b}) \varphi_b.$ Don't give a shit to $$S^{\mu\nu}$$ now. We don't know it until we obtain a specific representation of the Lorentz group.

In infinitesimal form,

• $\Lambda{^\mu}{_\nu} = \delta{^\mu}{_\nu} + \delta\omega{^\mu}{_\nu},$
• $U(1+\delta\omega) = \mathbb{1} + \frac{i}{2}\delta\omega_{\mu\nu} M^{\mu\nu},$
• $L{_a}{^b}(1+\delta\omega) = \delta{_a}{^b} + \frac{i}{2} \delta\omega_{\mu\nu} (S^{\mu\nu}){_a}{^b},$
• $[\varphi_a(x), M^{\mu\nu}] = \mathcal{L}^{\mu\nu}\varphi_a(x) + (S^{\mu\nu}){_a}{^b} \varphi_b(x),$ where $\color{blue} \mathcal{L}^{\mu\nu} = \frac{1}{i}(x^\mu \partial^\nu - x^\nu\partial^\mu).$
##### Representation of the Lorentz Group
• There are six generators to the Lorentz group.
• The six generators are organized into a $$4\times 4$$ anti-symmetric matrix $$M^{\mu\nu}$$.
• They satisfy the commutation relation $[M^{\mu\nu}, M^{\rho\sigma}] = i(g^{\nu\rho} M^{\mu\sigma} - (\mu\leftrightarrow \nu)) - (\rho \leftrightarrow \sigma).$
• $$M^{\mu\nu}$$, $$\mathcal{L}^{\mu\nu}$$ and $$(S^{\mu\nu}){_a}{^b}$$ should satisfy the same commutation relation.
• The six generators may be divided into
• angular momentum: $\color{blue} M^{12} = J_3,\ M^{31} = J_2,\ M^{23} = J_1,$
• boosts: $\color{blue} K_i = M^{i0}.$
• We have the commutation relations between them: \begin{align*} [J_i,J_j] &= +i\epsilon_{ijk} J_k, \\ [J_i,K_j] &= +i\epsilon_{ijk} K_k, \\ [K_i,K_j] &= -i\epsilon_{ijk} J_k. \end{align*}
• A new set of operators \begin{align*} N_i &= \frac{1}{2} (J_i - iK_i), \\ N_i^\dagger &= \frac{1}{2} (J_i + iK_i) \end{align*} yields \begin{align*} [N_i,N_j] &= i\epsilon_{ijk} N_k, \\ [N^\dagger_i, N^\dagger_j] &= i\epsilon_{ijk} N^\dagger_k, \\ [N_i, N^\dagger_j] &= 0. \end{align*}
• The representations of the Lorentz group may be specified by two half-integers.
• $$(0,0)$$: scalar or singlet representation;
• $$(1/2, 0)$$: left-handed spinor representation;
• $$(0, 1/2)$$: right-handed spinor representation;
• $$(1/2,1/2)$$: vector representation.
##### What the fuck are Spinors?

Go back and see the transformation rule (marked in red). Left-handed / right-handed spinors transforms according to the $$S^{\mu\nu}_{\mathrm{L}}$$ / $$S^{\mu\nu}_{\mathrm{R}}$$ specified below $\color{red} \psi_a(x) \rightarrow (\delta{_a}{^b} + \frac{i}{2} \delta \omega_{\mu\nu} (S^{\mu\nu}){_a}{^b}) \psi_b.$ where \begin{align*} \color{blue}S^{0i}_{\mathrm{L}} &\color{blue}= -\frac{i}{2}\sigma^i, \\ \color{blue}S^{0i}_{\mathrm{R}} &\color{blue}= +\frac{i}{2}\sigma^i, \\ \color{blue}S^{ij}_{\mathrm{L}/\mathrm{R}} &\color{blue}= \frac{1}{2}\epsilon^{ijk} \sigma^k.\\ \end{align*} Or, written explicitly, as \begin{align*} \color{red}\psi_{\mathrm{L}} &\color{red}\rightarrow (1 - \frac{i}{2}\vb*{\theta}\cdot \vb*{\sigma} - \frac{1}{2}\vb*{\beta}\cdot \vb*{\sigma})\psi_{\mathrm{L}}, \\ \color{red}\psi_{\mathrm{R}} &\color{red}\rightarrow (1 - \frac{i}{2}\vb*{\theta}\cdot \vb*{\sigma} + \frac{1}{2}\vb*{\beta}\cdot \vb*{\sigma})\psi_{\mathrm{R}}. \end{align*}

##### And Dirac Spinors?

Dirac spinors has four components. It transforms according to $\color{red} \psi_a(x) \rightarrow (\delta{_a}{^b} + \frac{i}{2} \delta \omega_{\mu\nu} (S^{\mu\nu}){_a}{^b}) \psi_b.$ where $\color{blue} S^{\mu\nu} = \frac{i}{4}[\gamma^\mu,\gamma^\nu] = \begin{pmatrix} S^{\mu\nu}_{\mathrm{L}} & 0 \\ 0 & S^{\mu\nu}_{\mathrm{R}} \end{pmatrix}.$

The upper two components transforms like a left-handed spinor while the lower two like a right-handed. $\color{orange} \psi = \begin{pmatrix} \psi_{\mathrm{L}} \\ \psi_{\mathrm{R}} \end{pmatrix}.$

In Peskin the exponent of the infinitesimal transform is denoted by $$\Lambda_{1/2}$$: $\delta{_a}{^b} + \frac{i}{2} \delta \omega_{\mu\nu} (S^{\mu\nu}){_a}{^b} \xrightarrow{\exp} \Lambda_{1/2} = \exp\qty(-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}).$

We define $\color{orange} \overline{\psi} = \psi^\dagger \gamma^0.$

The transformation laws are given by

• $\color{red} \psi(x) \rightarrow \Lambda_{1/2} \psi(\Lambda^{-1} x).$
• $\color{red} \overline{\psi}(x) \rightarrow \overline{\psi}(\Lambda^{-1}x) \Lambda^{-1}_{1/2}.$
• $$\overline{\psi}\psi$$ is a Lorentz scalar.
• $$\overline{\psi}\gamma^\mu \psi$$ is a Lorentz vector.
##### Dirac Bilinears
Definition Count
$$1$$ 1
$$\gamma^\mu$$ 4
$$\gamma^{\mu\nu} = \gamma^{[\mu}\gamma^{\nu]} = -i\sigma^{\mu\nu}{}$$ 6
$$\gamma^{\mu\nu\rho} = \gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]}{}$$ 4
$$\gamma^{\mu\nu\rho\sigma} = \gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\rho^{\sigma]}{}$$ 1

$$\overline{\psi} \gamma^{\mu\nu} \psi$$ transforms like a tensor of rank 2: $\overline{\psi} \gamma^{\mu\nu} \psi \rightarrow \Lambda{^\mu}{_\alpha} \Lambda{^\nu}{_\beta} \overline{\psi} \gamma^{\alpha\beta} \psi.$

Definition Transformation Count
$$1$$ $$\overline{\psi}\psi$$: scalar 1
$$\gamma^\mu$$ $$\overline{\psi}\gamma^\mu\psi$$: vector 4
$$\displaystyle \sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu]$$ $$\overline{\psi}\sigma^{\mu\nu}\psi$$: tensor 6
$$\gamma^\mu \gamma^5$$ $$\overline{\psi}\gamma^\mu\gamma^5\psi$$: pseudo-vector 4
$$\gamma^5$$ $$\overline{\psi}\gamma^5\psi$$: pseudo-scalar 1
 Quantum Field Theory (II) Quantization of Fields

Expand

## General Relativity (I)

Differential Geometry

#### Miscellaneous

##### Penrose Graphical Notation

See https://en.wikipedia.org/wiki/Penrose_graphical_notation. Not particularly useful in my homework.

Expand

## General Relativity (IV)

Perturbation Theory

#### Linearized Gravity

##### Metric and Lagrangian
• Metric \begin{align*} g_{\mu\nu} &= \eta_{\mu\nu} + h_{\mu\nu}, \\ g^{\mu\nu} &= \eta^{\mu\nu} - h^{\mu\nu}. \end{align*}{}
• Raising and lowering using $$\eta$$.
• Riemann tensor $R_{\mu\nu\rho\sigma} = \frac{1}{2}\qty(\partial_\rho \partial_\nu h_{\mu\sigma} + \partial_\sigma \partial_\mu h_{\nu\rho} - \partial_\sigma \partial_\nu h_{\mu\rho} - \partial_\rho \partial_\mu h_{\nu\sigma}).$
• Ricci tensor $R_{\mu\nu} = \frac{1}{2}\qty(\partial_\sigma \partial_\nu h{^\sigma}{_\mu} + \partial_\sigma \partial_\mu h{^\sigma}{_\nu} - \partial_\mu \partial_\nu h - \partial^2 h_{\mu\nu})$
• Ricci scalar $R = \partial_{\mu} \partial_{\nu} h^{\mu\nu} - \partial^2 h.$
• Einstein tensor \begin{align*} G_{\mu\nu} &= R_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu} R \\ &= \frac{1}{2}(\partial_\sigma \partial_\nu h{^\sigma}{_\mu} + \partial_\sigma \partial_\mu h{^\sigma}{_\nu} - \partial_\mu \partial_\nu h - \partial^2 h_{\mu\nu} - \eta_{\mu\nu}\partial_\rho \partial_\lambda h^{\rho\lambda} + \eta_{\mu\nu} \partial^2 h). \end{align*}{}
• Lagrangian $\mathcal{L} = \frac{1}{2}\qty[(\partial_\mu h^{\mu\nu})(\partial_\nu h) - (\partial_\mu h^{\rho\sigma})(\partial_\rho h{^\mu}{_\sigma}) + \frac{1}{2}(\partial_\mu h^{\rho\sigma})(\partial_\nu h_{\rho\sigma}) - \frac{1}{2}\eta^{\mu\nu}(\partial_\mu h)(\partial_\nu h)].$
• Gauge transformation: $h^{(\epsilon)}_{\mu\nu} = h_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu.$
• LHS the pullback under a diffeomorphism generated by $$\xi^\mu$$.
• Riemann tensor left intact: $\delta R_{\mu\nu\rho\sigma} = 0.$
##### Decomposition
• Metric \begin{align*} h_{00} &= -2\Phi, \\ h_{0i} &= w_i, \\ h_{ij} &= 2s_{ij} - 2\Psi \delta_{ij}. \end{align*}{}
• $\Psi = -\frac{1}{6}\delta^{ij} h_{ij}.$
• $s_{ij} = \frac{1}{2}\qty(h_{ij} - \frac{1}{3}\delta^{kl}h_{kl}\delta_{ij}).$
• $$s_{ij}$$ is traceless.
• Christoffel symbols: \begin{align*} \Gamma{^0}_{00} &= \partial_0 \Phi, \\ \Gamma{^i}_{00} &= \partial_i \Phi + \partial_0 w_i, \\ \Gamma{^0}_{j0} &= \partial_j \Phi, \\ \Gamma{^i}_{j0} &= \partial_{[j}w_{i]} + \frac{1}{2} \partial_0 h_{ij}, \\ \Gamma{^0}_{jk} &= -\partial_{(i}w_{k)} + \frac{1}{2}\partial_0 h_{jk}, \\ \Gamma{^i}_{jk} &= \partial_{(j} h_{k)i} - \frac{1}{2} \partial_i h_{jk}. \end{align*}
• Geodesic equation: \begin{align*} \dv{E}{t} &= -E \qty[\partial_0 \Phi + 2(\partial_k \Phi) v^k - \qty(\partial_{(j} w_{k)} - \frac{1}{2}\partial_0 h_{jk})v^j v^k], \\ \dv{p^i}{t} &= -E\qty[\partial_i \Phi + \partial_0 w_i + 2(\partial_{[i} w_{j]} + \partial_0 h_{ij})v^j + \qty( \partial_{(j}h_{k)i} - \frac{1}{2}\partial_i h_{jk} )v^j v^k]. \end{align*}{}
• Riemann tensor: \begin{align*} R_{0j0l} &= \partial_j \partial_l \Phi + \partial_0 \partial_{(j}w_{l)} - \frac{1}{2}\partial_0 \partial_0 h_{jl}, \\ R_{0jkl} &= \partial_j \partial_{[k} w_{l]} - \partial_0 \partial_{[k} h_{l]j}, \\ R_{ijkl} &= \partial_j \partial_{[k} h_{l]i} - \partial_i \partial_{[k} h_{l]j}. \end{align*}{}
• Ricci tensor: \begin{align*} R_{00} &= \grad^2 \Phi + \partial_0 \partial_k w^k + 3\partial_0^2 \Phi, \\ R_{0j} &= -\frac{1}{2} \grad^2 w_j + \frac{1}{2} \partial_j \partial_k w^k + 2\partial_0 \partial_j \Psi + \partial_0 \partial_k s{_j}{^k}, \\ R_{ij} &= -\partial_i \partial_j (\Phi - \Psi) - \partial_0 \partial_{(i} w_{j)} + \partial^2 \Psi \delta_{ij} - \partial^2 s_{ij} + 2\partial_k \partial_{(i} s_{j)}{^k}. \end{align*}{}
• Einstein tensor: \begin{align*} G_{00} &= 2\grad^2 \Psi + \partial_k \partial_l s^{kl}, \\ G_{0j} &= -\frac{1}{2}\grad^2 w_j + \frac{1}{2} \partial_j \partial_k w^k + 2\partial_0 \partial_j \Psi + \partial_0 \partial_k s{_j}{^k}, \\ G_{ij} &= (\delta_{ij}\grad^2 - \partial_i \partial_j)(\Phi - \Psi) + \delta_{ij} \partial_0 \partial_k w^k - \partial_0 \partial_{(i} w_{j)} + 2\delta_{ij}\partial_0^2 \Psi - \grad^2 s_{ij} + 2\partial_k \partial_{(i} s_{j)}{^k} - \delta_{ij} \partial_k \partial_l s^{kl}. \end{align*}{}
• Gauge transformation: \begin{align*} \Phi &\rightarrow \Phi + \partial_0 \xi^0, \\ w_i &\rightarrow w_i + \partial_0 \xi^i - \partial_i \xi^0, \\ \Psi &\rightarrow \Psi - \frac{1}{3} \partial_i \xi^i, \\ s_{ij} &\rightarrow s_{ij} + \partial_{(i}\xi_{j)} - \frac{1}{3}\partial_k \xi^k \delta_{ij}. \end{align*}{}
• Only $$s_{ij}$$ are free. With $$s_{ij}$$ given,
• $$T_{00}$$ determines $$\Psi$$,
• $$T_{0j}$$ determines $$w^k$$, and
• $$T_{ij}$$ determines $$\Phi$$.
##### Gauge
###### Transverse Gauge
• Gauge condition: \begin{align*} \partial_i s^{ij} = 0 \quad &\Leftarrow \quad \grad^2 \xi^j + \frac{1}{3} \partial_j \partial_i \xi^i = -2\partial_i s^{ij}, \\ \partial_i w^i = 0 \quad & \Leftarrow \quad \grad^2 \xi^0 = \partial_i w^i + \partial_0 \partial_i \xi^i. \end{align*}{}
• Field equations: \begin{align*} G_{00} &= 2\grad^2\Psi = 8\pi G T_{00}, \\ G_{0j} &= -\frac{1}{2} \grad^2 w_j + 2\partial_0 \partial_j \Psi = 8\pi G T_{0j}, \\ G_{ij} &= (\delta_{ij} \grad^2 - \partial_i \partial_j)(\Phi - \Psi) - \partial_0 \partial_{(i} w_{j)} + 2\delta_{ij} \partial^2_0 \Psi - \partial^2 s_{ij} = 8\pi G T_{ij}. \end{align*}{}
###### Synchronous Gauge
• Gauge condition: \begin{align*} \Phi = 0 \quad &\Leftarrow \quad \partial_0 \xi^0 = -\Phi, \\ w^i = 0 \quad & \Leftarrow \quad \partial_0 \xi^i = -w^i + \partial_i \xi^0. \end{align*}{}
• Metric $\dd{s^2} = -\dd{t^2} + (\delta_{ij} + h_{ij}) \dd{x^i} \dd{x^j}.$
###### Lorenz/Harmonic Gauge
• Gauge condition: $\partial_\mu h{^\mu}{_\nu} - \frac{1}{2} \partial_\nu h = 0.$
• No simple expression in terms of decomposed perturbation fields.

Expand

## General Relativity (III)

Field Equations and Black Holes

### Spacetime and Field Equations

#### Spacetime

##### Asymptotically Flat Manifolds
• A manifold $$M$$ is asymptotically simple if it admits a conformal compactification $$\tilde{M}{}$$ such that every null geodesic in $$M$$ has future and past endpoints on the boundary of $$\tilde{M}{}$$.
• A weakly asymptotically simple manifold is a manifold $$M$$ with an open set $$U\in M$$ isometric to a neighbourhood of the boundary of $$\tilde{M}{}$$.
• A manifold $$M$$ is asymptotically flat if it is weakly asymptotically simple and asymptotically empty, i.e. the Ricci tensor vanishes in a neighbourhood of the boundary of $$\tilde{M}{}$$.

The FRW models are not asymptotically flat.

##### Einstein Manifolds
• An Einstein manifold is a pseudo-Riemannian manifold where the Ricci tensor is proportional to the metric.
• Vacuum solution to the field equation with cosmological constant: $R_{\mu\nu} = \frac{2\Lambda}{n-2} g_{\mu\nu}.$
##### Conformal Diagrams
• In a conformal diagram,
• there is a new pair of $$T$$ and $$R$$;
• null-like radial geodesics always satisfies $\dv{T}{R} = \pm 1;$
• infinity is only a finite coordinate value away.
• Generally apply it only to spherically symmetric spaces.
• Boundaries are referred to as conformal infinity.
• The union of the original spacetimes with conformal infinity is the conformal compactification.
##### Minkowski Spacetime

• Original coordinates and metric: $\dd{s^2} = -\dd{t^2} + \dd{r^2} + r^2 \dd{\Omega^2}.$
• Coordinates range: $-\infty < t < \infty;\quad 0\le r < \infty.$
• A series of coordinate transformations: \begin{align*} u &= t-r;\\ v &= t+r. \\ U &= \arctan u; \\ V &= \arctan v. \\ T &= V + U; \\ R &= V - U. \end{align*}
• New coordinates and metric: $\dd{s^2} = \frac{1}{\omega^2(T,R)}\qty(-\dd{T^2} + \dd{R^2} + \sin^2 R \dd{\Omega^2}).$
• Coordinates range: $0 \le R < \pi;\quad \abs{T} < \pi - R.$
• Conformal to $\tilde{\dd{s^2}} = -\dd{T^2} + \dd{R^2} + \sin^2 R \dd{\Omega^2},$ which is $\mathbb{R} \times S^3.$
• Accidentally the Einstein static universe, i.e. a static solution to Einsteins's equation with a perfect fluid and a cosmological constant.
• Notations in the conformal diagram:
• $$i^+$$ for the future time-like infinity;
• $$i^0$$ for the spatial infinity;
• $$i^-$$ for the past time-like infinity;
• $$\mathscr{I}^+$$ for the future null infinity;
• $$\mathscr{I}^-$$ for the past null infinity.
##### De Sitter Space

• The maximally symmetric spacetime with positive curvature is called de Sitter spacetime.
• A hyperboloid given by $-u^2 + x^2 + y^2 + z^2 + w^2 = \alpha^2$ in the spacetime $\dd{s^2} = -\dd{u^2} + \dd{x^2} + \dd{y^2} + \dd{z^2} + \dd{w^2}.$
• Coordinates given by \begin{align*} u &= \alpha \sinh(t/\alpha), \\ w &= \alpha \cosh(t/\alpha) \cos\chi, \\ x &= \alpha \cosh(t/\alpha) \sin\chi \cos\theta, \\ y &= \alpha \cosh(t/\alpha) \sin\chi \sin\theta \cos\phi, \\ z &= \alpha \cosh(t/\alpha) \sin\chi \sin\theta \sin\phi. \end{align*}{}
• Metric on the hyperboloid \begin{align*} \dd{s^2} &= -\dd{t^2} + \alpha^2 \cosh^2(t/\alpha) \qty[\dd{\chi^2} + \sin^2\chi (\dd{\theta^2} + \sin^2 \theta \dd{\phi^2})] \\ &= -\dd{t^2} + \alpha^2 \cosh^2(t/\alpha) \dd{\Omega^2_3}. \end{align*}
• The $$S^3$$ shrinks and then expands.
• New coordinate $\cosh(t/\alpha) = \frac{1}{\cos t'}.$
• New metric $\dd{s^2} = \frac{\alpha^2}{\cos^2 t'} \qty[-\dd{t'^2} + \dd{\chi^2} + \sin^2 \chi \dd{\Omega_2^2}].$
• New range $-\pi/2 < t' < \pi/2.$
##### Anti-de Sitter Space

• The maximally symmetric spacetime with negative curvature is called anti-de Sitter spacetime.
• A hyperboloid given by $-u^2 - v^2 + x^2 + y^2 + z^2 = -\alpha^2$ in the spacetime $\dd{s^2} = -\dd{u^2} - \dd{v^2}+ \dd{x^2} + \dd{y^2} + \dd{z^2}.$
• Coordinates given by \begin{align*} u &= \alpha \sin t' \cosh \rho, \\ w &= \alpha \cos t' \cosh \rho, \\ x &= \alpha \sinh \rho \cos\theta, \\ y &= \alpha \sinh \rho \sin\theta \cos\phi, \\ z &= \alpha \sinh \rho \sin\theta \sin\phi. \end{align*}{}
• Metric on the hyperboloid $\dd{s^2} = \alpha^2\qty(-\cosh^2 \rho \dd{t'^2} + \dd{\rho^2} + \sinh^2 \rho \dd{\Omega^2_2}).$
• New coordinate $\cosh \rho = \frac{1}{\cos \chi}.$
• New metric $\dd{s^2} = \frac{\alpha^2}{\cos^2 \chi} \qty[-\dd{t'^2} + \dd{\chi^2} + \sin^2 \chi \dd{\Omega_2^2}].$
• New range \begin{align*} 0 \le \chi < \pi/2, \\ -\infty < t' < \infty. \end{align*}{}

#### Einstein Field Equations

##### Action
• Total action: $S = S_G + S_M.$
• Einstein-Hilbert action: $S_G = \frac{1}{16\pi G} \int R \sqrt{-g} \dd{^4 x}.$
• Action of matters: $S_M = \int \mathcal{L}_M \sqrt{-g} \dd{^4 x}.$
• Stress-Energy tensor: $T_{\mu\nu} = \frac{-2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}\mathcal{L}_M)}{\delta g^{\mu\nu}} = -2 \frac{\delta \mathcal{L}_M}{\delta g^{\mu\nu}} + g_{\mu\nu} \mathcal{L}_M.$
##### Variations of Geometric Properties
• Variation of covariant derivatives of the Christoffel symbol: $$\delta \Gamma$$ may be treated as a tensor, $\grad_\mu (\delta \Gamma{^\rho}_{\nu\sigma}) = \partial_\mu(\delta \Gamma{^\rho}_{\nu\sigma}) + \Gamma{^\rho}_{\mu\lambda} \delta \Gamma{^\lambda}_{\nu\sigma} - \Gamma{^\lambda}_{\mu\nu} \delta \Gamma{^\rho}_{\lambda\sigma} - \Gamma{^\lambda}_{\mu\sigma} \delta \Gamma{^\rho}_{\nu\lambda}.$
• Variation of the Riemann tensor: $\delta R{^\rho}_{\sigma\mu\nu} = \grad_\mu (\delta \Gamma{^\rho}_{\nu\sigma}) - \grad_\nu (\delta \Gamma{^\rho}_{\mu\sigma}).$
• Variation of the Ricci tensor: $\delta R_{\sigma\nu} = \grad_\rho (\delta \Gamma{^\rho}_{\nu\sigma}) - \grad_\nu (\delta \Gamma{^\rho}_{\rho\sigma}).$
• Variation of the Ricci scalar: $\delta R = R_{\sigma\nu} \delta g^{\sigma\nu} + \grad_\rho(g^{\sigma\nu} \delta \Gamma{^\rho}_{\nu\sigma} - g^{\sigma\rho}\delta \Gamma{^\mu}_{\mu\sigma}).$
• Variation of the determinant: $\delta g = g g^{\mu\nu} \delta g_{\mu\nu}.$
• Variation of the inverse metric: $\delta g^{\mu\nu} = -g^{\mu\rho} g^{\nu\lambda} \delta g_{\rho\lambda.}$
• Stokes' theorem with covariant derivatives: $\sqrt{-g} \grad_\mu V^\mu = \partial_\mu(\sqrt{-g} A^\mu).$
##### Field Equations
• Field equations: $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}.$
• Vacuum solutions are Ricci-flat: taking trace in the field equations, $R_{\mu\nu} = 0.$

Expand
 Quantum Field Theory (IV) Renormalization

## Quantum Field Theory (V)

Applications of Feynman Diagrams

1PIを剃る。そして断面積を求める。

#### Foundations of Field Theory

##### Scattering
• Mandelstam variables: for $$1+2\rightarrow 3+4$$
• $s = (p_1 + p_2)^2 = (p_3 + p_4)^2.$
• $t = (p_1 - p_3)^2 = (p_4 - p_2)^2.$
• $u = (p_1 - p_4)^2 = (p_3 - p_2)^2.$
• $s+t+u = m_1^2 + m_2^2 + m_3^2 + m_4^2.$
$$s$$-channel $$t$$-channel $$u$$-channel
$$\displaystyle \mathcal{M}\propto \frac{1}{s-m_\phi^2}{}$$ $$\displaystyle \mathcal{M}\propto \frac{1}{t-m_\phi^2}{}$$ $$\displaystyle \mathcal{M}\propto \frac{1}{u-m_\phi^2}{}$$
Images from Mandelstam variables.

#### Tricks

##### Spin Sums

\begin{align*} \sum_s u^s(p) \overline{u}^s(p) &= \unicode{x2215}\kern-.5em {p} + m, \\ \sum_s v^s(p) \overline{v}^s(p) &= \unicode{x2215}\kern-.5em {p} - m. \end{align*}

##### Photon Polarization Sums
• Ward Identity: if during a calculation of the amplitude for some QED process involving an external photon with momentum $$k$$, we encounter a invariant matrix of the form $\mathcal{M}(k) = \mathcal{M}^\mu(k)\epsilon^*_\mu(k),$ then we could safely assume that $\color{red} k_\mu \mathcal{M}^\mu(k) = 0.$
• Ward-Takahashi Identity: Let $$\mathcal{M}_0$$ be the diagram without the insertion of photon $$k$$. We have $\sum_{\substack{\mathrm{insertion} \\ \mathrm{points}}} k_\mu \mathcal{M}^\mu(k;p_1\cdots p_n;q_1\cdots q_n) = e\sum_i \qty[{ \mathcal{M}_0(p_1\cdots p_n; q_1\cdots (q_1 - k) \cdots) - \mathcal{M}_0(p_1\cdots (p_i + k) \cdots; q_1\cdots q_n) }].$ This does not contribute to the final $$S$$-matrix.
• Let $S(p) = \frac{i}{\unicode{x2215}\kern-.5em {p} - m - \Sigma(p)}.$ Applying the Ward-Takahashi identity to the case where $$\mathcal{M}_0$$ is a two-point correlation, $-i k_\mu \Gamma^\mu(p+k, p) = S^{-1}(p+k) - S^{-1}(p).$
• Substitution: $\sum_{\text{polarizations}} \epsilon^*_\mu \epsilon_\nu \mathcal{M}^\mu(k) \mathcal{M}^{\nu*}(k) = -g_{\mu\nu} \mathcal{M}^\mu(k) \mathcal{M}^{\nu*}(k).$
• Time-like and longitudinal photons could be omitted.
##### Trace Technology
• Traces of products: \begin{align*} \tr(\mathbb{1}) &= 4, \\ \tr(\text{odd number of } \gamma) &= 0, \\ \tr{\gamma^\mu \gamma^\nu} &= 4g^{\mu\nu}, \\ \tr(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma) &= 4(g^{\mu\nu} g^{\rho\sigma} - g^{\mu\rho}g^{\nu\sigma} + g^{\mu\sigma}g^{\nu\rho}), \\ \tr(\gamma^5) &= 0, \\ \tr(\gamma^\mu \gamma^\nu \gamma^5) &= 0, \\ \tr{\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^5} &= -4i \epsilon^{\mu\nu\rho\sigma}. \end{align*}{}
• Order be reversed: $\tr(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \cdots) = \tr(\cdots \gamma^\sigma \gamma^\rho \gamma^\nu \gamma^\mu ).$
• Contraction: \begin{align*} \gamma^\mu \gamma_\mu &= 4, \\ \gamma^\mu \gamma^\nu \gamma_\mu &= -2\gamma^\nu, \\ \gamma^\mu \gamma^\nu \gamma^\rho \gamma_\mu &= 4g^{\nu\rho}, \\ \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma_\mu &= -2\gamma^\sigma \gamma^\rho \gamma^\nu. \end{align*}{}
##### Crossing Symmetry
• Crossing symmetry: $\mathcal{M}(\phi(p) + \cdots \rightarrow \cdots) = \mathcal{M}(\cdots \rightarrow \cdots + \overline{\phi}(k)).$
• A particle of momentum $$p$$ on the LHS moved to an anti-particle of momentum $$k=-p$$ on the RHS.
• Only in the sense of analytic continuation.
• There is an overall minus sign and should be manually removed.

#### General Procedures

##### How do We Obtain Cross Sections?
1. Draw the diagrams for the desired process.
2. Use the Feynman rules to write down the amplitude $$\mathcal{M}$$.
3. Square the amplitude and average or sum over spins, using the spin sum relations.
4. Evaluate traces using the trace identities.
5. Specialize to a particular frame of reference, and draw a picture of the kinematic variables in that frame. Express all 4-momentum vectors in terms of a suitably chosen set of variables, e.g. $$E$$ and $$\theta$$.
6. Plug the resulting expression for $$\abs{\mathcal{M}}^2$$ into the cross-section formula $\dd{\sigma} = \frac{1}{2E_A 2E_B \abs{v_a - v_B}} \qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3} \frac{1}{2E_f})\times \abs{\mathcal{M}(p_A,p_B \rightarrow \qty{p_f})}^2 (2\pi)^4 \delta^{(4)}(p_A + p_B - \sum p_f).$ Then we integrate over phase-space variables to obtain a cross section in the desire form.
##### Scattering of a Specific Helicity?

Projection onto left-helicity of $$u$$ is onto right-helicity of $$v$$, and vice versa.

• Projection of helicity:
• Onto left-handed (for $$u$$, right-handed for $$v$$): $\frac{1+\gamma^5}{2} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$
• Onto right-handed (for $$u$$, left-handed for $$v$$): $\frac{1-\gamma^5}{2} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.$
##### Feynman Parameters: Loop Diagrams
1. Draw the diagrams and write down the amplitude.
2. Sqeezing the denominator: \begin{align*} \frac{1}{AB^n} &= \int_0^1 \dd{x} \dd{y} \delta(x+y-1) \frac{ny^{n-1}}{[xA + yB]^{n+1}}; \\ \frac{1}{A_1 \cdots A_n} &= \int_0^1 \dd{x_1} \cdots \dd{x_n} \delta(\sum x_i - 1) \frac{(n-1)!}{[x_1A_1 + \cdots + x_n A_n]^n}; \\ \frac{1}{A_1^{m_1} \cdots A_n^{m_n}} &= \int_0^1 \dd{x_1} \cdots \dd{x_n} \delta(\sum x_i - 1) \frac{\prod x_i^{m_i - 1}}{\qty[\sum x_i A_i]^{\sum m_i}} \frac{\Gamma(m_1 + \cdots + m_n)}{\Gamma(m_1) \cdots \Gamma(m_n)}. \end{align*}
3. Completing the square in the new denominator by shifting to a new loop momentum $$l$$. Therefore, the denominator is of the form $$D^m$$ where $$D$$ is an even function of $$l$$, $D = l^2 - \Delta + i\epsilon.$
4. Eliminate some terms in the numerator by symmetry, e.g. \begin{align*} \int \frac{\dd{^4 l}}{(2\pi)^4} \frac{l^\mu}{D^m} &= 0; \\ \int \frac{\dd{^4 l}}{(2\pi)^4} \frac{l^\mu l^\nu}{D^m} &= \int \frac{\dd{^4 l}}{(2\pi)^4} \frac{g^{\mu\nu} l^2/4}{D^m}; \\ \int \frac{\dd{^d l}}{(2\pi)^d} \frac{l^\mu l^\nu}{D^m} &= \int \frac{\dd{^4 l}}{(2\pi)^4} \frac{g^{\mu\nu} l^2/d}{D^m}. \end{align*} Note that contraction identities should be modified in the case of dimensional regularization to \begin{align*} \gamma^\mu \gamma^\nu \gamma_\mu &= -(2-\epsilon) \gamma^\nu; \\ \gamma^\mu \gamma^\nu \gamma^\rho \gamma_\mu &= 4g^{\nu\rho} - \epsilon \gamma^\nu \gamma^\rho; \\ \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma_\mu &= -2\gamma^\sigma \gamma^\rho \gamma^\nu + \epsilon \gamma^\nu \gamma^\rho \gamma^\sigma. \end{align*}{} In this way, we rewrite the integral in a isotropic form.
• We may also write the numerator in the form $\gamma^\mu \cdot A + (p'^\mu + p^\mu) \cdot B + q^\mu \cdot C.$
1. Regularize the integral using, for example, Pauli-Villars prescription, or dimensional regularization.
• How do we choose the regulator?

The question of which regulator to use has no a priori answer in quantum field theory. Often the choice has no effect on the predictions of the theory. When two regulators gives different answers for observable quantities, it is generally because some symmetry (such as the Ward idensity) is being violated by one (or both) of them. In this case we take the symmetry to be fundamental anddemand that it be preserved by the regulator.

2. Apply Wick rotation:
• For the integration in $$l^0$$, the poles lies at $l^0 = \pm\qty(-\sqrt{\vb*{l}^2 + \Delta} + i\epsilon).$ Therefore, we are entitled to rotate the path by $$\pi/2$$.
• The integration is rewritten as $\int \frac{\dd{^4 l}}{(2\pi)^4} \frac{1}{[l^2 - \Delta]^m} = \frac{i(-1)^m}{(2\pi)^4} \int \dd{\Omega_4} \int_0^\infty \dd{l_E} \frac{l_E^3}{[l_E^2 + \Delta]^m}.$
• Examples: \begin{align*} \int \frac{\dd{^4 l}}{(2\pi)^4} \frac{1}{[l^2 - \Delta]^m} &= \frac{i(-1)^m}{(4\pi)^2} \frac{1}{(m-1)(m-2)} \frac{1}{\Delta^{m-2}}, \\ \int \frac{\dd{^4 l}}{(2\pi)^4} \frac{l^2}{[l^2 - \Delta]^m} &= \frac{i(-1)^{m-1}}{(4\pi)^2} \frac{2}{(m-1)(m-2)(m-3)} \frac{1}{\Delta^{m-3}}. \end{align*}
##### Physical Discontinuity
1. Cut through the diagrams in all possible ways such that the cut propagators can simultaneously be put on shell.
2. For each cut, replace $\frac{1}{p^2 - m^2 + i\epsilon} \rightarrow -2\pi i \delta(p^2 - m^2)$ in each cut propagator, and then perform the loop integrals.
3. Sum the contributions of all possible cuts.
##### Dimensional Regularization
• Area of a $$d$$-dimensional unit sphere: $\int \dd{\Omega_d} = \frac{2\pi^{d/2}}{\Gamma(d/2)}.$
• Propagators: \begin{align*} \int \frac{\dd{^d l_E}}{(2\pi)^d} \frac{1}{(l^2_E + \Delta)^n} &= \frac{1}{(4\pi)^{d/2}} \frac{\Gamma(n - d/2)}{\Gamma(n)} \qty(\frac{1}{\Delta})^{n-d/2}; \\ \int \frac{\dd{^d l_E}}{(2\pi)^d} \frac{l^2_E}{(l^2_E + \Delta)^n} &= \frac{1}{(4\pi)^{d/2}} \frac{d}{2} \frac{\Gamma(n-d/2-1)}{\Gamma(n)}\qty(\frac{1}{\Delta})^{n-d/2-1}. \end{align*}

#### Diagrams

##### 1PI Diagrams
• 1PI diagram:
• one-particle irreducible diagram,
• cannot be split into two by removeing a single line.

#### φ⁴ Theory: Loop

##### One-Loop
1. Cutting the diagram: Working in the COMF, with $$k = (k^0, \vb{0})$$, and now we evaluate the integral $i\delta \mathcal{M} = \frac{\lambda^2}{2} \int \frac{\dd{^4 q}}{(2\pi)^4} \frac{1}{(k/2 - q)^2 - m^2 + i\epsilon} \frac{1}{(k/2 + q)^2 - m^2 + i\epsilon}.$
2. Completing the path below, and replacing $\frac{1}{(k/2 + q)^2 - m^2 + i\epsilon} \rightarrow -2\pi i \delta((k/2 + q)^2 - m^2)$ to pick up the residue at $q^0 = -\frac{1}{2}k^0 + E_{\vb*{q}},$ we find $i\delta M = -2\pi i \frac{\lambda^2}{2} \frac{4\pi}{(2\pi)^4} \int_m^\infty \dd{E_{\vb*{q}}} E_{\vb*{q}} \abs{\vb*{q}} \frac{1}{2E_{\vb*{q}}} \frac{1}{k^0 (k^0 - 2E_{\vb*{q}})}.$ The discontinuity is given by replacing the pole with a delta function, i.e. another replacement $\frac{1}{(k/2 - q)^2 - m^2 + i\epsilon} \rightarrow -2\pi i \delta((k/2 - q)^2 - m^2).$ With the result to the first order for the $$\phi^4$$-theory, i.e. $$\mathcal{M}(k) = -\lambda$$, we verified to $$\lambda^2$$ that (cf. the optical theorem) \begin{align*} \operatorname{Disk} \mathcal{M}(k) &= 2i \Im \mathcal{M}(k) \\ &= \frac{i}{2} \int \frac{\dd{^3 p_1}}{(2\pi)^3} \frac{1}{2E_1} \frac{\dd{^3 p_2}}{(2\pi)^3} \frac{1}{2E_2} \abs{\mathcal{M}(k)}^2 (2\pi)^4 \delta^{(4)}(p_1 + p_2 - k). \end{align*}{}
3. We don't have other contributions yet.

#### Quantum Electrodynamics: Tree Level

##### Electron–Positron Annihilation: to Muon
###### Unpolarized Cross Section
1. Feynman diagram:
2. Amplitude: \begin{align*} i\mathcal{M} &= \overline{v}^{s'}(p')(-ie\gamma^\mu) u^s(p) \qty(\frac{-ig_{\mu\nu}}{q^2})\overline{u}^r(k)(-ie\gamma^\nu) v^{r'}(k') \\ &= \frac{ie^2}{q^2}\qty({\overline{v}(p')\gamma^\mu u(p)})\qty({\overline{u}(k)\gamma_\mu v(k')}). \end{align*}{}
3. Amplitude squared and averaged: \begin{align*} \frac{1}{4}\sum_{s,s',r,r'} \abs{\mathcal{M}(s,s' \rightarrow r,r')}^2 &= \frac{e^4}{4q^4} \tr\qty[(\unicode{x2215}\kern-.5em {p}' - m_e)\gamma^\mu (\unicode{x2215}\kern-.5em {p}+m_e)\gamma^\nu] \tr\qty[(\unicode{x2215}\kern-.5em {k} + m_\mu)\gamma_\mu (\unicode{x2215}\kern-.5em {k}' - m_\mu)\gamma_\nu]. \end{align*}{}
4. Traces evaluated: $\frac{1}{4}\sum_{s,s',r,r'} \abs{\mathcal{M}(s,s' \rightarrow r,r')}^2 = \frac{8e^4}{q^4}\qty[(p\cdot k)(p'\cdot k')+(p\cdot k')(p'\cdot k) + m_\mu^2 (p\cdot p')].$
5. New kinematic variables: \begin{align*} p &= (E,E\hat{\vb*{z}}), \\ p' &= (E, -E\hat{\vb*{z}}), \\ k &= (E,\vb*{k}), \\ k' &= (E,-\vb*{k}). \end{align*}{}
• Dot products: $\begin{gather*} q^2 = (p+p')^2 = 4E^2, \\ p\cdot p' = 2E^2, \\ p\cdot k = p'\cdot k' = E^2 - E\abs{\vb*{k}} \cos\theta, \\ p\cdot k' = p'\cdot k = E^2 + E\abs{\vb*{k}} \cos\theta. \end{gather*}{}$
• Amplitude simplified: $\frac{1}{4}\sum_{s,s',r,r'} \abs{\mathcal{M}(s,s' \rightarrow r,r')}^2 = e^4 \qty[\qty(1+\frac{m_\mu^2}{E^2}) + \qty(1-\frac{m_\mu^2}{E^2})\cos^2\theta].$
• With Mandelstam variables: in massless limit, $\frac{1}{4}\sum_{s,s',r,r'} \abs{\mathcal{M}(s,s' \rightarrow r,r')}^2 = \frac{8e^4}{s^2}\qty[\qty(\frac{t}{2})^2 + \qty(\frac{u}{2})^2].$
6. Differential cross section: \begin{align*} \dv{\sigma}{\Omega} &= \frac{1}{2E_{\mathrm{cm}}^2} \frac{\abs{\vb*{k}}}{16\pi^2 E_{\mathrm{cm}}} \cdot \frac{1}{4}\sum_{s,s',r,r'} \abs{\mathcal{M}(s,s' \rightarrow r,r')}^2 \\ &= \frac{\alpha^2}{4E_{\mathrm{cm}}^2}\sqrt{1-\frac{m_\mu^2}{E^2}}\qty[\qty(1+\frac{m_\mu^2}{E^2}) + \qty(1-\frac{m_\mu^2}{E^2})\cos^2\theta]. \end{align*}{} Total cross section: $\sigma = \frac{4\pi \alpha^2}{3E_{\mathrm{cm}}^2} \sqrt{1-\frac{m_\mu^2}{E^2}}\qty(1+\frac{1}{2}\frac{m_\mu^2}{E^2}).$
###### Helicity Structure

Relativistic limit: assuming massless. $e^-_R e^+_L \rightarrow \mu^-_R \mu^+_L.$

1. Feynman diagram: in the previous example.
2. Amplitude: we should apply projections onto specific helicities to $$u$$, $$v$$, etc. We apply it only to one of the incident particles and one of the outcoming particles, since $\overline{v}\gamma^\mu\qty(\frac{1+\gamma^5}{2})u = \overline{v}\gamma^\mu\qty(\frac{1+\gamma^5}{2})^2 u = {v}^\dagger \qty(\frac{1+\gamma^5}{2}) \gamma^0 \gamma^\mu\qty(\frac{1+\gamma^5}{2})u.$ We see the opposite helicities automatically applies to $$u$$ and $$v$$.
3. We hypocritically sum over all spins again, only to exploit the spin sum identities, although we know that some helicities are effectively thrown out.
4. Spin sum carried out. Trace Evaluated. \begin{align*} \sum_{s,s'} \abs{\overline{v}(p')\gamma^\mu \qty(\frac{1+\gamma^5}{2})u(p)}^2 &= \tr\qty[\unicode{x2215}\kern-.5em {p}'\gamma^\mu \unicode{x2215}\kern-.5em {p} \gamma^\nu \qty(\frac{1+\gamma^5}{2})] \\ &= 2(p'^\mu p^\nu + p'^\nu p^\mu - g^{\mu\nu}p\cdot p' - i\epsilon^{\alpha\mu\beta\nu} p'_\alpha p_\beta). \\ \sum_{r,r'} \abs{\overline{u}(k)\gamma^\mu \qty(\frac{1+\gamma^5}{2})v(k')}^2 &= 2(k_\mu k'_\nu - k_\nu k'_\mu - g_{\mu\nu} k\cdot k' - i\epsilon_{\rho\mu\sigma\nu} k^\rho k'^\sigma). \end{align*}{}
5. Using new kinematic variables: $\abs{\mathcal{M}}^2 = \qty(1+\cos\theta)^2.$
6. Differential cross section: \begin{align*} \dv{\sigma}{\Omega}\qty(e^-_R e^+_L \rightarrow \mu^-_R \mu^+_L) = \dv{\sigma}{\Omega}\qty(e^-_L e^+_R \rightarrow \mu^-_L \mu^+_R) = \frac{\alpha^2}{4E_{\mathrm{cm}}^2}\qty(1-\cos\theta)^2, \\ \dv{\sigma}{\Omega}\qty(e^-_R e^+_L \rightarrow \mu^-_L \mu^+_R) = \dv{\sigma}{\Omega}\qty(e^-_L e^+_R \rightarrow \mu^-_R \mu^+_L) = \frac{\alpha^2}{4E_{\mathrm{cm}}^2}\qty(1-\cos\theta)^2. \end{align*}

Brute-force approach: electrons and muons in relativistic limit,

• Amplitude $\mathcal{M} = \frac{e^2}{q^2}\qty(\overline{v}\qty{p'}\gamma^\mu u\qty{p})\qty(\overline{u}\qty(k)\gamma_\mu v\qty(k')).$
• $$u$$ and $$v$$ in relativistic limit: \begin{align*} u(p) &= \sqrt{2E}\cdot \frac{1}{2} \begin{pmatrix} (1-\hat{p}\cdot \vb*{\sigma}) \xi \\ (1+\hat{p}\cdot \vb*{\sigma}) \xi \end{pmatrix}, \\ v(p) &= \sqrt{2E}\cdot \frac{1}{2} \begin{pmatrix} (1-\hat{p}\cdot \vb*{\sigma}) \xi \\ -(1+\hat{p}\cdot \vb*{\sigma}) \xi \end{pmatrix}. \end{align*}{}
• Electron half by brute-force: $\overline{v}(p') \gamma^\mu u(p) = -2E(0,1,i,0).$
• Muon half by rotation: $\overline{u}(k) \gamma^\mu v(k') = -2E(0,\cos\theta,-i,-\sin\theta).$
• Amplitude $\mathcal{M}\qty(e^-_R e^+_L \rightarrow \mu^-_R \mu^+_L) = -e^2(1+\cos\theta).$

Brute-force approach: electrons in relativistic limit, muons in nonrelativistic limit,

• Amplitude: in the previous example.
• $$p$$ in relativistic limit: in the previous example. $$k$$ in nonrelativistic limit: $u(k) = \sqrt{m}\begin{pmatrix} \xi \\ \xi \end{pmatrix},\quad v(k') = \sqrt{m} \begin{pmatrix} \xi' \\ -\xi' \end{pmatrix}.$
• Electron half: in the previous example.
• Muon half: $\overline{u}(k) \gamma^\mu v(k') = m \begin{pmatrix} \xi^\dagger & \xi^\dagger \end{pmatrix}\begin{pmatrix} \overline{\sigma}^\mu & 0 \\ 0 & \sigma^\mu \end{pmatrix}\begin{pmatrix} \xi' \\ -\xi' \end{pmatrix} = \begin{cases} 0, & \text{if } \mu = 0, \\ -2m\xi^\dagger \sigma^i \xi', & \text{if } \mu = i. \end{cases}$
• Amplitude: $\mathcal{M}(e_R^- e_L^+ \rightarrow \mu^+ \mu^-) = -2e^2 \xi^\dagger \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \xi'.$
• Muon spins summed over: all four particles have spin $$\uparrow$$, \begin{align*} \mathcal{M}^2 &= 4e^4, \\ \dv{\sigma}{\Omega}(e_R^- e_L^+ \rightarrow \mu^+ \mu^-) &= \frac{\alpha^2}{E_{\mathrm{cm}}^2} \frac{\abs{\vb*{k}}}{E}. \end{align*}
##### Electron-Position Annihilation: to Bound States
• Amplitude of general spin configurations: $i\mathcal{M}(\text{something} \rightarrow \vb*{k}, \vb*{k}') = \xi^\dagger [\Gamma\qty(\vb*{k})] \xi = \tr(\xi' \xi^\dagger \Gamma(\vb*{k})) = \tr\qty(\frac{\vb*{n}^*\cdot \vb*{\sigma}}{\sqrt{2}}\Gamma\qty(\vb*{k})).$
• Substitution: $\xi' \xi^\dagger \rightarrow \frac{1}{\sqrt{2}}\vb*{n}^*\cdot \vb*{\sigma}.$
• $\uparrow\uparrow \quad\mathbin{\leftrightarrow}\quad \vb*{n} = \frac{\hat{\vb*{x}} + i\hat{\vb*{y}}}{\sqrt{2}},$
• $\downarrow\downarrow \quad\mathbin{\leftrightarrow}\quad \vb*{n} = \frac{\hat{\vb*{x}} - i\hat{\vb*{y}}}{\sqrt{2}},$
• $\frac{\uparrow\downarrow + \downarrow\uparrow}{\sqrt{2}} \quad\mathbin{\leftrightarrow}\quad \vb*{n} = \hat{\vb*{z}}.$
• Scattering into bound states: see bound states for the expansion of a bound state. \begin{align*} \mathcal{M}(\uparrow \uparrow \rightarrow B) &= \sqrt{2M} \int \frac{\dd{^3 k}}{(2\pi)^3} \tilde{\psi}^*(\vb*{k}) \frac{1}{\sqrt{2m}} \frac{1}{\sqrt{2m}} \mathcal{M}(\uparrow \uparrow \rightarrow \vb*{k}\uparrow, -\vb*{k}\uparrow) \\ &= \sqrt{\frac{2}{M}}(-2e^2)\psi^*(0). \end{align*}{}
• For general spin configurations, $i\mathcal{M}(\text{something}\rightarrow B) = \sqrt{\frac{2}{M}} \int \frac{\dd{^3 k}}{(2\pi)^3} \tilde{\psi}^*(\vb*{k}) \tr\qty(\frac{\vb*{n}^* \cdot \vb*{\sigma}}{\sqrt{2}}\Gamma\qty(\vb*{k})).$
• For the current case: $\mathcal{M}(e_R^- e_L^+ \rightarrow B) = \sqrt{\frac{2}{M}}(-2e^2)(\vb*{n}^* \cdot \vb*{\epsilon}_+) \psi^*(0).$
• Polarization states of incidental electrons: $\vb*{\epsilon}_+ = \frac{\hat{\vb*{x}} + i\hat{\vb*{y}}}{\sqrt{2}}.$
• Polarization of outcoming muons: $$\vb*{n}^*$$.
• Total cross section: averaged over all possible direction of $$\vb*{n}$$, \begin{align*} \sigma(e^+ e^- \rightarrow B) &= \frac{1}{2} \frac{1}{2m} \frac{1}{2m} \int \frac{\dd{^3 K}}{(2\pi)^3} \frac{1}{2E_{\vb*{K}}}(2\pi)^4 \delta^{(4)}(p+p'-K)\cdot \frac{2}{M}(4e^4)\frac{1}{2}\abs{\psi(0)}^2 \\ &= 64\pi^3 \alpha^2 \frac{\abs{\psi(0)}^2}{M^3}\delta(E^2_{\mathrm{cm}} - M^2). \end{align*}{}
##### Bound States Decay: to Electron-Position Pairs
• Decay of bound states: \begin{align*} \Gamma(B \rightarrow e^+ e^-) &= \frac{1}{2M} \int \dd{\Pi_2} \abs{\mathcal{M}}^2 \\ &= \frac{1}{2M}\int \qty(\frac{1}{8\pi} \frac{\dd{\cos\theta}}{2}) \frac{8e^4}{M}\abs{\psi(0)}^2\qty(\abs{\vb*{n}\cdot \vb*{\epsilon}}^2 + \abs{\vb*{n}\cdot \vb*{\epsilon}^*}^2). \end{align*}
• Averaged over all three directions of $$\vb*{n}$$: $\Gamma(B\rightarrow e^+ e^-) = \frac{16\pi \alpha^2}{3} \frac{\abs{\psi(0)}^2}{M^2}.$
##### Electron-Muon Scattering
1. Feynman diagram:
2. Amplitude: $i\mathcal{M} = \frac{ie^2}{q^2} \overline{u}(p'_1)\gamma^\mu u(p_1) \overline{u}(p'_2) \gamma_\mu u(p_2).$
3. Amplitude squared and averaged: $\frac{1}{4}\sum_{\text{spins}} \abs{\mathcal{M}}^2 = \frac{e^4}{4q^4}\tr\qty[(\unicode{x2215}\kern-.5em {p}'_1 + m_e)\gamma^\mu (\unicode{x2215}\kern-.5em {p}_1 + m_e)\gamma^\nu]\tr\qty[(\unicode{x2215}\kern-.5em {p}'_2 + m_\mu)\gamma_\mu (\unicode{x2215}\kern-.5em {p}_2 + m_\mu)\gamma_\nu].$
4. Traces evaluated $\frac{1}{4}\sum_{\text{spins}} \abs{\mathcal{M}}^2 = \frac{8e^4}{q^4}\qty[(p_1 \cdot p'_2)(p'_1 \cdot p_2) + (p_1\cdot p_2)(p'_1\cdot p'_2) - m_\mu^2 (p_1\cdot p'_1)].$
5. New kinematic variables: \begin{align*} p_1 &= (k,k\hat{\vb*{z}}), \\ p_2 &= (E, -k\hat{\vb*{z}}), \\ p'_1 &= (k,\vb*{k}), \\ p'_2 &= (E,-\vb*{k}). \end{align*}{}
• Dot products: $\begin{gather*} p_1 \cdot p_2 = p'_1 \cdot p'_2 = k(E+k), \\ p'_1 \cdot p_2 = p_1 \cdot p'_2 = k(E+k\cos\theta), \\ p_1 \cdot p'_1 = k^2(1-\cos\theta), \\ q^2 = -2p_1\cdot p'_1 = -2k^2(1-\cos\theta). \end{gather*}$
• Amplitude simplified: $\frac{1}{4}\sum_{\text{spins}} \abs{\mathcal{M}}^2 = \frac{2e^4}{k^2(1-\cos\theta)^2}\qty(\qty(E+k)^2 + \qty(E+k\cos\theta)^2 - m_\mu^2\qty(1-\cos\theta)).$
• With Mandelstam variables: in massless limit, $\frac{1}{4}\sum_{\text{spins}} \abs{\mathcal{M}}^2 = \frac{8e^4}{s^2}\qty[\qty(\frac{s}{2})^2 + \qty(\frac{u}{2})^2].$
6. Differential cross section: $\qty(\dv{\sigma}{\Omega})_{\mathrm{CM}} = \frac{\alpha^2}{2k^2(E+k)^2(1-\cos\theta)^2} \qty(\qty(E+k)^2 + \qty(E+k\cos\theta)^2 - m_\mu^2\qty(1-\cos\theta)).$
• Note the $$\theta^{-4}$$ singularity as $$\theta$$ approaches $$0$$.
##### Compton Scattering
1. Feynman diagrams:

$$+$$

1. Amplitude: \begin{align*} i\mathcal{M} &= \overline{u}(p')(-ie\gamma^\mu)\epsilon^*_\mu(k') \frac{i( \unicode{x2215}\kern-.5em {p} + \unicode{x2215}\kern-.5em {k} + m )}{(p+k)^2 - m^2}(-ie\gamma^\nu)\epsilon_\nu(k)u(p) + \overline{u}(p')(-ie\gamma^\nu)\epsilon_\nu(k) \frac{i( \unicode{x2215}\kern-.5em {p} - \unicode{x2215}\kern-.5em {k}' + m )}{(p-k')^2 - m^2}(-ie\gamma^\mu)\epsilon^*_\mu(k')u(p) \\ &= -ie^2 \epsilon^*_\mu(k')\epsilon_\nu(k)\overline{u}(p')\qty[\frac{\gamma^\mu \unicode{x2215}\kern-.5em {k} \gamma^\nu + 2\gamma^\mu p^\nu}{2p\cdot k} + \frac{-\gamma^\nu \unicode{x2215}\kern-.5em {k}' \gamma^\mu + 2\gamma^\nu p^\mu}{-2p\cdot k'}]u(p). \end{align*}{}
2. Amplitude squared and averaged: with the substitution applied, \begin{align*} \frac{1}{4}\sum_{\text{spins}}\abs{\mathcal{M}}^2 &= \frac{e^4}{4}g_{\mu\rho} g_{\nu\sigma} \cdot \tr\qty{ (\unicode{x2215}\kern-.5em {p}' + m)\qty[\frac{\gamma^\mu \unicode{x2215}\kern-.5em {k} \gamma^\nu + 2\gamma^\mu p^\nu}{2p\cdot k} + \frac{-\gamma^\nu \unicode{x2215}\kern-.5em {k}' \gamma^\mu + 2\gamma^\nu p^\mu}{-2p\cdot k'}] (\unicode{x2215}\kern-.5em {p} + m)\qty[\frac{\gamma^\sigma \unicode{x2215}\kern-.5em {k} \gamma^\rho + 2\gamma^\rho p^\sigma}{2p\cdot k} + \frac{-\gamma^\rho \unicode{x2215}\kern-.5em {k}' \gamma^\sigma + 2\gamma^\sigma p^\rho}{-2p\cdot k'}] } \\ &= \frac{e^4}{4}\qty{\frac{\vb{I}}{(2p\cdot k)^2} + \frac{\vb{II}}{(2p\cdot k)(2p\cdot k')} + \frac{\vb{III}}{(2p\cdot k')(2p\cdot k)} + \frac{\vb{IV}}{(2p\cdot k')^2}}. \end{align*}{}
3. Trace evaluated: $\frac{1}{4}\sum_{\text{spins}}\abs{\mathcal{M}}^2 = 2e^4 \qty[\frac{p\cdot k'}{p\cdot k} + \frac{p\cdot k}{p\cdot k'} + 2m^2\qty(\frac{1}{p\cdot k} - \frac{1}{p\cdot k'}) + m^4 \qty(\frac{1}{p\cdot k} - \frac{1}{p\cdot k'})^2].$
4. New kinematic variables: in the lab frame, \begin{align*} k &= (\omega, \omega \hat{\vb*{z}}), \\ p &= (m, 0), \\ k' &= (\omega', \omega' \sin\theta, 0, \omega' \cos\theta), \\ p' &= (E', \vb*{p}'). \end{align*}
• Dot products: $p\cdot k = m\omega,\quad p\cdot k' = m\omega'.$
• Compton's formula: $\omega' = \frac{\omega}{\displaystyle 1 + \frac{\omega}{m}(1-\cos\theta)}.$
5. Phase space integral: \begin{align*} \int \dd{\Pi_2} &= \int \frac{\dd{^3 k}}{(2\pi)^3} \frac{1}{2\omega'} \frac{\dd{^3 p'}}{(2\pi)^3} \frac{1}{2E'} (2\pi)^4 \delta^{(4)}(k'+p'-k-p) \\ &= \frac{1}{8\pi} \int \dd{\cos\theta} \frac{(\omega')^2}{\omega m}. \end{align*}{} Differential cross section: \begin{align*} \dv{\sigma}{\cos\theta} &= \frac{1}{2\omega} \frac{1}{2m} \cdot \frac{1}{8\pi} \frac{(\omega')^2}{\omega m}\cdot \qty(\frac{1}{4} \sum_{\text{spins}} \abs{\mathcal{M}}^2) \\ &= \frac{\pi \alpha^2}{m^2}\qty(\frac{\omega'}{\omega})^2 \qty[\frac{\omega'}{\omega} + \frac{\omega}{\omega'} - \sin^2\theta]. \end{align*}{}

High-energy behavior:

1. New kinematic variables: in the center-of-mass frame, \begin{align*} k &= (\omega, \omega \hat{\vb*{z}}), \\ p &= (E, -\omega \hat{\vb*{z}}), \\ k' &= (\omega, \omega \sin\theta, 0, \omega\cos\theta), \\ p' &= (E', -\omega \sin\theta, 0, -\omega\cos\theta). \end{align*}{}
• Amplitude simplified: for $$E\gg m$$ and $$\theta\approx \pi$$, $\frac{1}{4}\sum_{\text{spins}}\abs{\mathcal{M}}^2 = 2e^4\cdot \frac{E + \omega}{E + \omega \cos\theta}.$
2. Differential cross section: \begin{align*} \dv{\sigma}{\cos\theta} &= \frac{1}{2}\cdot \frac{1}{2E} \cdot \frac{1}{2\omega} \cdot \frac{\omega}{(2\pi)4(E+\omega)} \cdot \frac{2e^4 (E+\omega)}{E+\omega\cos\theta} \\ &\approx \frac{2\pi \alpha^2}{2m^2 + s(1+\cos\theta)}. \end{align*}{} Total cross section $\sigma \approx \frac{2\pi\alpha^2}{s}\log\qty(\frac{s}{m^2}).$
###### Helicity Structure
• $$u$$-channel diagram:
• Invariant matrix: $i\mathcal{M} = ie^2 \epsilon_\mu(k)\epsilon^*_\nu(k') \overline{u}(p')\gamma^\mu \frac{\unicode{x2215}\kern-.5em {p} - \unicode{x2215}\kern-.5em {k}' + m}{(p-k')^2 - m^2}\gamma^\nu u(p).$
• $\chi = \pi - \theta.$
• Region: $\frac{m}{\omega} \ll \chi \ll 1.$ Contribution from $(\unicode{x2215}\kern-.5em {p} - \unicode{x2215}\kern-.5em {k}').$
• Initial electron right-handed: final electron right-handed, initial photon right-handed, final photon right-handed, \begin{align*} \mathcal{M}(e^-_R \gamma_R \rightarrow e^-_R \gamma_R) &= -e^2 \epsilon_\mu(k) \epsilon^*_\nu(k')u^\dagger_R(p')\sigma^\mu \frac{\overline{\sigma}\cdot (p-k')}{-(\omega^2 \chi^2 + m^2)}\sigma^\nu u_R(p) &\approx \frac{4e^2\chi}{\chi^2 + m^2/\omega^2}. \end{align*}{}
• Equal contribution from left-handed initial electron.
• Region: direct backward scattering, $\chi = 0.$ Contribution from $$m$$.
• $\mathcal{M}(e^-_R \gamma_L \rightarrow e^-_L \gamma_R) = \frac{4e^2 m/\omega}{\chi^2 + m^2/\omega^2}.$
• Equal contribution from left-handed initial electron.
• Differential cross section: \begin{align*} \dv{\sigma}{\cos\theta} &= \frac{1}{2} \frac{1}{2E} \frac{1}{2\omega} \frac{\omega}{(2\pi)4(E+\omega)}\qty[\mathcal{M}(e^-_R \gamma_R \rightarrow e^-_R \gamma_R) + \mathcal{M}(e^-_R \gamma_L \rightarrow e^-_L \gamma_R)] \\ &= \frac{4\pi\alpha^2}{s(\chi^2 + 4m^2/s)}. \end{align*}{}
##### Electron-Position Annihilation: to Photons
1. Feynman diagrams:

$$+$$

1. Making replacements from the Compton scattering: $p \rightarrow p_1, \quad p' \rightarrow -p_2,\quad k\rightarrow -k_1,\quad k'\rightarrow -k_2.$
2. Amplitude squared and averaged, and trace evaluated: $\frac{1}{4}\sum_{\text{spins}}\abs{\mathcal{M}}^2 = -2e^4\qty[\frac{p_1 \cdot k_2}{p_1 \cdot k_1} + \frac{p_1 \cdot k_1}{p_1\cdot k_2} + 2m^2\qty(\frac{1}{p_1\cdot k_1} + \frac{1}{p_1 \cdot k_2}) - m^4 \qty(\frac{1}{p_1 \cdot k_1} + \frac{1}{p_1 \cdot k_2})^2].$
• Overall minus sign from the crossing relation and should be removed.
3. New kinematic variables: \begin{align*} p_1 &= (E,p\hat{\vb*{z}}), \\ p_2 &= (E,-p\hat{\vb*{z}}), \\ k_1 &= (E,E\sin\theta, 0, E\cos\theta), \\ k_2 &= (E, -E\sin\theta, 0, -E\cos\theta). \end{align*}{}
4. Differential cross section: $\dv{\sigma}{\cos\theta} = \frac{2\pi\alpha^2}{s}\qty(\frac{E}{p})\qty[\frac{E^2 + p^2 \cos^2\theta}{m^2 + p^2\sin^2\theta} + \frac{2m^2}{m^2 + p^2\sin^2\theta} - \frac{2m^4}{(m^2 + p^2\sin^2\theta)^2}].$

#### Quantum Electrodynamics: Loop

##### Bremsstrahlung: Classical Computation
• Classical process: an electron get a kicked at $$t=0$$ and $$\vb*{x} = 0$$, and undergoes a momentum change $p \rightarrow p'.$
• The energy radiated is given by \begin{align*} \mathrm{Energy} &= \int \frac{\dd{^3 k}}{(2\pi)^3} \sum_{\lambda=1,2} \frac{e^2}{2}\abs{\vb*{\epsilon}_\lambda(\vb*{k})\cdot \qty(\frac{\vb*{p}'}{k\cdot p'} - \frac{\vb*{p}}{k\cdot p})}^2 \\ &= \frac{e^2}{(2\pi)^2} \int \dd{k} \mathcal{I}(\vb*{v}, \vb*{v}') \\ &= \frac{\alpha}{\pi} \cdot k_{\mathrm{max}} \cdot \mathcal{I}(\vb*{v}, \vb*{v}'), \end{align*} where
• we have carefully picked a frame such that $$p^0 = p'^0$$, and defined $p^\mu = E(1,\vb*{v}), \quad p'^\mu = E(1,\vb*{v}');$
• differential intensity $\mathcal{I}(\vb*{v}, \vb*{v}') = \int \frac{\dd{\Omega_{\hat{\vb*{k}}}}}{4\pi} \qty({ \frac{2(1-\vb*{v}\cdot \vb*{v}')}{(1 - \hat{\vb*{k}}\cdot \vb*{v})(1 - \hat{\vb*{k}} \cdot \vb*{v}')} - \frac{m^2/E^2}{(1-\hat{\vb*{k}}\cdot \vb*{v}')^2} - \frac{m^2/E^2}{(1-\hat{\vb*{k}}\cdot \vb*{v})^2} });$
• cutoff at $$k_{\mathrm{max}}$$.
• In extreme relativistic limit: $$E\gg m$$,
• Differential intensity $\mathcal{I}(\vb*{v}, \vb*{v}') \approx 2 \log\qty(\frac{-q^2}{m^2}),$ where $q^2 = (p - p')^2.$
• Radiated energy $\mathrm{Energy} = \frac{2\alpha}{\pi} \int_0^{k_{\mathrm{max}}} \dd{k} \log\qty(\frac{-q^2}{m^2}).$
• Number of photons radiated $\mathrm{Number\ of\ photons} = \frac{\alpha}{\pi} \int_0^{k_{\mathrm{max}}} \dd{k} \frac{1}{k} \mathcal{I}(\vb*{v}, \vb*{v}').$
##### Bremsstrahlung: Quantum Computation
• Quantum process: one photon radiated during the scattering of an electron, before and after the interaction with the external field.

• Notations and assumptions:
• $$\mathcal{M}_0$$ the part of the amplitude that comes from the electrons's interaction with the external field.
• Soft radiation: $\abs{\vb*{k}} \ll \abs{\vb*{p}' - \vb*{p}}.$
• Approximation: $\mathcal{M}_0(p', p-k) \approx \mathcal{M}_0(p'+k, p) \approx \mathcal{M}_0(p', p).$
• $$\unicode{x2215}\kern-.5em {k}$$ ignored in the numerator.
• Amplitude: $i\mathcal{M} \approx \underbrace{\overline{u}(p')[\mathcal{M}_0(p',p)]u(p)}_{\text{elastic scattering}}\cdot \underbrace{\qty[{ e\qty({ \frac{p'\cdot \epsilon^*}{p'\cdot k} - \frac{p\cdot \epsilon^*}{p\cdot k} }) }]}_{\text{emission of photon}}.$
• Differential cross section: $\dd{\sigma(p\rightarrow p' + \gamma)} = \dd{\sigma(p\rightarrow p')} \cdot \int \dd{(\mathrm{prob})},$ where $\dd{(\mathrm{prob})} = \frac{\dd{^3 k}}{(2\pi)^3} \sum_\lambda \frac{e^2}{2k} \abs{\vb*{\epsilon}_\lambda \cdot \qty({ \frac{\vb*{p}'}{p'\cdot k} - \frac{\vb*{p}}{p\cdot k} })}^2$ gives the probability of emission of a single photon. This should be re-interpreted as the number of photon radiated.
• Total probability: photon mass $$\mu$$, $\mathrm{Total\ probability} \approx \frac{\alpha}{\pi} \int_{\mu}^{\sim \abs{\vb*{q}}} \dd{k} \frac{1}{k} \mathcal{I}(\vb*{v}, \vb*{v}').$ Therefore, $\dd{\sigma}(p\rightarrow \gamma(k)) \approx \dd{\sigma}(p\rightarrow p')\cdot \frac{\alpha}{\pi} \log\qty(\frac{-q^2}{\mu^2})\log\qty(\frac{-q^2}{m^2}).$
##### Electron Vertex Function: Formal Structure

$$=$$ $$+$$ $$+$$ $$+ \cdots$$

• Sum of vertex diagrams: $\begin{gather*} i\mathcal{M} (2\pi) \delta(p'^0 - p^0) = -ie \overline{u}(p') {\color{red} \gamma^\mu} u(p) \cdot \tilde{A}^{\mathrm{cl}}_\mu(p' - p) \\ \Downarrow \text{Vectex corrections} \\ i\mathcal{M} (2\pi) \delta(p'^0 - p^0) = -ie \overline{u}(p') {\color{red} \Gamma^\mu(p', p)} u(p) \cdot \tilde{A}^{\mathrm{cl}}_\mu(p' - p). \end{gather*}$
• Assuming perturbation of the form $\Delta H_{\mathrm{int}} = \int \dd{^3 x} eA^{\mathrm{cl}}_\mu \overline{\psi} \gamma^\mu \psi.$
• Form of $$\Gamma^\mu$$: $\color{red} \Gamma^\mu(p',p) = \gamma^\mu F_1(q^2) + \frac{i\sigma^{\mu\nu} q_\nu}{2m} F_2(q^2).$
• To the lowest order, $F_1 = 1, \quad F_2 = 0.$
###### Correction to Electron Charge

The amplitude of electron scattering from an electric field is given by $i\mathcal{M} = -ie F_1(0)\tilde{\phi}(\vb*{q}) \cdot 2m \xi'^\dagger \xi$ under non-relativistic limit for slowly varying field, which indicate a potential given by $V(\vb*{x}) = eF_1(0) \phi(\vb*{x}).$

###### Correction to Electron Magnetic Moment

The amplitude of electron scattering in an magnetic field is given by \begin{align*} i\mathcal{M} &= +ie\qty[{ \overline{u}(p') \qty({ \gamma^i F_1 + \frac{i\sigma^{i\nu} q_\nu}{2m} F_2 }) u(p) }] \tilde{A}^i_{\mathrm{cl}}(\vb*{q}) &= -i(2m) \cdot e \xi'^\dagger \qty({ \frac{-1}{2m} \sigma^k \qty[F_1(0) + F_2(0)] }) \xi \tilde{B}^k(\vb*{q}). \end{align*} This is equivalent to the potential $V(\vb*{x}) = -\langle \vb*{\mu} \rangle \cdot \vb*{B}(\vb*{x}),$ where \begin{align*} \langle \vb*{\mu} \rangle &= \frac{e}{m} \qty[F_1(0) + F_2(0)] \xi'^\dagger \frac{\vb*{\sigma}}{2} \xi; \\ \vb*{\mu} &= g\qty(\frac{e}{2m}) \vb*{S}; \\ g &= 2\qty[F_1(0) + F_2(0)] = 2 + 2F_2(0). \end{align*}

##### Electron Vertex Function: Evaluation

1. Tree-level correction plus loop: $\Gamma^\mu = \gamma^\mu + \delta \Gamma^\mu$ where $\overline{u}(p') \delta \Gamma^\mu(p',p) u(p) = 2ie^2 \int \frac{\dd{^4 k}}{(2\pi)^4} \frac{\overline{u}(p')\qty[ \unicode{x2215}\kern-.5em {k} \gamma^\mu \unicode{x2215}\kern-.5em {k} + m^2\gamma^\mu - 2m(k+k') ]u(p)}{((k-p)^2 + i\epsilon)(k'^2 - m^2 + i\epsilon)(k^2 - m^2 + i\epsilon)}.$
2. Feynman parameters: $\frac{1}{((k-p)^2 + i\epsilon)(k'^2 - m^2 + i\epsilon)(k^2 - m^2 + i\epsilon)} = \int_0^1 \dd{x} \dd{y} \dd{z} \delta(x+y+z-1) \frac{2}{D^3}.$
3. Completing the square: $D = l^2 - \Delta + i\epsilon$ and $\Delta = -xyq^2 + (1-z)^2 m^2.$
4. Simplifying the numerator: $\overline{u}(p') \delta \Gamma^\mu(p',p) u(p) = 2ie^2 \int \frac{\dd{^4 l}}{(2\pi)^4} \int_0^1 \dd{x} \dd{y} \dd{z} \delta(x+y+z-1) \frac{2}{D^3} \overline{u}(p')\qty[\mathrm{Numerator}] u(p)$ where $[\mathrm{Numerator}] = \gamma^\mu \cdot \qty({-\frac{1}{2} l^2 + (1-x)(1-y) q^2 + (1-4z+z^2)m^2 }) + \frac{i\sigma^{\mu\nu}q_\nu}{2m}(2m^2 z(1-z)).$
5. Regularizing the integral: introduce a large $$\Lambda$$ and replace the photon propagator $\frac{1}{(k-p)^2 + i\epsilon} \rightarrow \frac{1}{(k-p)^2 + i\epsilon} - \frac{1}{(k-p)^2 - \Lambda^2 + i\epsilon}.$ For the $$\Lambda$$ term, the denominator is altered by $\Delta \rightarrow \Delta_\Lambda = -xyq^2 + (1-z)^2 m^2 + z\Lambda^2.$ This replacement introduces a correction $\int \frac{\dd{^4 l}}{(2\pi)^4} \qty({ \frac{l^2}{[l^2 - \Delta]^3} - \frac{l^2}{[l^2 - \Delta_\Lambda]^3} }) = \frac{i}{(4\pi)^2} \log \qty(\frac{\Delta_\Lambda}{\Delta}) \approx \frac{i}{(4\pi)^2} \log \qty(\frac{z\Lambda^2}{\Delta}).$ We are counting on this correction to fix the divergence of the $$l^2$$-term in the numerator.
6. Applying Wick rotation, we obtain $\overline{u}(p') \delta \Gamma^\mu(p',p) u(p) = \frac{\alpha}{2\pi} \int_0^1 \dd{x}\dd{y}\dd{z} \delta(x+y+z-1) \overline{u}(p') \qty(\mathrm{Mess}) u(p)$ where $(\mathrm{Mess}) = \gamma^\mu \qty[{ \log \frac{z\Lambda^2}{\Delta} + \frac{1}{\Delta}\qty({ (1-x)(1-y)q^2 + (1-4z+z^2)m^2 }) }] + \frac{i\sigma^{\mu\nu} q_\nu}{2m}\qty[{ \frac{1}{\Delta} 2m^2 z(1-z) }].$

A few notes are in order:

• The correction to $$F_1$$ is nonzero at $$q = 0$$. Therefore, we manually set $\delta F_1(q^2) \rightarrow \delta F_1(q^2) - \delta F_1(0),$ which is justified later.
• $$\Delta$$ in the denominator introduces a divergence. We set the photon mass to a nonzero $$\mu$$, and therefore $\Delta \rightarrow \Delta + z\mu^2.$
• We are now able to obtain $F_2(q^2=0) = \frac{\alpha}{2\pi},$ and therefore the anomalous magnetic moment of electron $a_e = \frac{g-2}{2} = \frac{\alpha}{2\pi}.$
##### Electron Vertex Function: Electron Self-Energy Correction
• With the electron self-energy correction, the exact vertex function should be $Z_2 \Gamma^\mu(p',p) = \gamma^\mu F_1(q^2) + \frac{i\sigma^{\mu\nu} q_\nu}{2m} F_2(q^2).$
• $$F_2$$ is still of order $$\alpha$$.
• $$F_1(q^2)$$ receives a correction from $$Z_2$$, $F_1(q^2) = 1 + \delta F_1(q^2) + \delta Z_2 = 1 + [\delta F_1(q^2) - \delta F_1(0)]$ since $\delta Z_2 = -\delta F_1(0).$
##### Electron Vertex Function: Infrared Divergence
• Infrared divergence: only in $$F_1$$ near $$z=1$$ and $$x=y=0$$. Retaining only the terms divergent when $$\mu\rightarrow 0$$, and set $$x=y=0$$ and $$z=1$$ in the denominator, with the substitution $y = (1-z) \xi,\quad w = (1-z)$ we find \begin{align*}F_1(q^2) &= \frac{\alpha}{4\pi} \int_0^1 \dd{\xi} \qty[{ \frac{-2m^2 + q^2}{m^2 - q^2\xi(1-\xi)}\log\qty({ \frac{m^2 - q^2 \xi(1-\xi)}{\mu^2} }) + 2\log\qty(\frac{m^2}{\mu^2}) }]\\ &= 1 - \frac{\alpha}{2\pi} f_{\mathrm{IR}}(q^2)\log\qty({ \frac{-q^2\ \mathrm{or}\ m^2}{\mu^2} }) + \mathcal{O}(\alpha^2), \end{align*} where $f_{\mathrm{IR}}(q^2) = \int_0^1 \dd{\xi} \qty(\frac{m^2 - q^2/2}{m^2 - q^2 \xi(1-\xi)}) - 1.$
• Easily verified that $\color{red} \mathcal{I}(\vb*{v}, \vb*{v}') = 2f_{\mathrm{IR}}(q^2).$
• Divergence behavior of $$F_1$$: $F_1(-q^2 \rightarrow \infty) = 1 - \frac{\alpha}{2\pi} \log \qty(\frac{-q^2}{m^2}) \log \qty(\frac{-q^2}{\mu^2}) + \mathcal{O}(\alpha^2).$
• Since infrared divergence only affects $$F_1$$, the correction to the amplitude of electron scattering is given by replacing $e \rightarrow e\cdot F_1(q^2).$
###### Dawn: It Cancels
• Correction from the vertex factor: $\dv{\sigma}{\Omega}(p\rightarrow p') = \qty(\dv{\sigma}{\Omega})_0\qty[{ 1 - \frac{\alpha}{\pi}\log\qty(\frac{-q^2}{m^2})\log\qty(\frac{-q^2}{\mu^2}) + \mathcal{O}(\alpha^2) }];$
• Correction from Bremsstrahlung: $\dv{\sigma}{\Omega}(p\rightarrow p' + \gamma) = \qty(\dv{\sigma}{\Omega})_0\qty[{+\frac{\alpha}{\pi}\log\qty(\frac{-q^2}{m^2})\log\qty(\frac{-q^2}{\mu^2}) + \mathcal{O}(\alpha^2) }].$
###### Measurement
• Events are identified as $p \rightarrow p'$ if the Bremsstrahlung $$\gamma$$ is lower than a certain threshold. From the formula for the total probability we find \begin{align*} \qty(\dv{\sigma}{\Omega})_{\mathrm{measured}} &= \dv{\sigma}{\Omega}(p\rightarrow p') + \dv{\sigma}{\Omega}(p\rightarrow p' + \gamma(k < E_l)) \\ &= \qty(\dv{\sigma}{\Omega})_0\qty[{ 1 - \frac{\alpha}{\pi} f_{\mathrm{IR}}(q^2) \log\qty(\frac{-q^2\ \mathrm{or}\ m^2}{E_l^2}) + \mathcal{O}(\alpha^2) }]. \end{align*}
• In the $$-q^2 \gg m^2$$ limit we find $\qty(\dv{\sigma}{\Omega})_{\mathrm{measured}} = \qty(\dv{\sigma}{\Omega})_0\qty[{ 1 - \frac{\alpha}{\pi} \log\qty(\frac{-q^2}{m^2}) \log\qty(\frac{-q^2}{E_l^2}) + \mathcal{O}(\alpha^2) }].$
##### To Wrap Up: Beyond One Photon
• To get a singular denominator in an electron propagator, we need the momentum to be near the mass shell. Therefore, we consider only the diagrams in which an arbitrary hard process is modified by the addition of soft real and virtual photons on the electron legs.
• Emission at the outgoing leg: killing $$\unicode{x2215}\kern-.5em {k}$$ in the numerator and $$\mathcal{O}(k^2)$$ terms in the denominator,

and taking into account all permutations, the amplitude is given by \begin{align*} & \phantom{{}={}} \sum_{\mathrm{permutations\ on\ }(1,\cdots,n)} \overline{u}(p') \qty(e\frac{p'^{\mu_1}}{p'\cdot k_1}) \qty(e\frac{p'^{\mu_2}}{p'\cdot (k_1 + k_2)}) \cdots \qty(e\frac{p'^{\mu_2}}{p'\cdot (k_1 + \cdots + k_n)}) \cdots \\ &= \overline{u}(p')\qty(e\frac{p'^{\mu_1}}{p'\cdot k_1})\cdots \qty(e\frac{p'^{\mu_n}}{p'\cdot k_n}). \end{align*}
• Emission at the incoming leg: killing $$\unicode{x2215}\kern-.5em {k}$$ in the numerator and $$\mathcal{O}(k^2)$$ terms in the denominator,

the amplitude is obtained similarly.
• All possible combinations: $$p_i$$ may be on the incoming or outgoing leg,

the amplitude is given by $\overline{u}(p') i\mathcal{M}_{\mathrm{hard}} u(p) \cdot e\qty({ \frac{p'^{\mu_1}}{p'\cdot k_1}-\frac{p^{\mu_1}}{p\cdot k_1} })\cdots e\qty({ \frac{p'^{\mu_n}}{p'\cdot k_n}-\frac{p^{\mu_n}}{p\cdot k_n} }).$
• Virtual photons:
1. Pick two photon momenta $$k_i$$ and $$k_j$$, set $$k_j = -k_i = k$$;
2. Multiply by the photon propagator;
3. Integrate over $$k$$. This introduces a factor $\vb{X} = \frac{e^2}{2} \int \frac{\dd{^4 k}}{(2\pi)^4} \frac{-i}{k^2 + i\epsilon} \qty({ \frac{p'}{p'\cdot k} - \frac{p}{p\cdot k} }) \cdot \qty({ \frac{p'}{-p'\cdot k} - \frac{p}{-p\cdot k} }) = -\frac{\alpha}{2\pi} f_{\mathrm{IR}}(q^2) \log\qty(\frac{-q^2}{\mu^2})$ in the amplitude.
• Real photons: we apply Ward's identity here,
1. Multiply by its polarization vector;
2. Sum over polarizations;
3. Integrate the squared matrix element over the photon's phase space. This introduces a factor $\vb{Y} = \int \frac{\dd{^3 k}}{(2\pi)^3} \frac{1}{2k} e^2(-g_{\mu\nu}) \qty({ \frac{p'^\mu}{p'\cdot k}-\frac{p^{\mu}}{p\cdot k} }) \qty({ \frac{p'^\nu}{p'\cdot k}-\frac{p^{\nu}}{p\cdot k} }) = \frac{\alpha}{\pi} f_{\mathrm{IR}}(q^2) \log\qty(\frac{E^2_l}{\mu^2})$ in the cross section.
• To wrap up: \begin{align*} & \phantom{{}={}} \qty(\dv{\sigma}{\Omega})_{\mathrm{measured}}\qty(\vb*{p} \rightarrow \vb*{p}' + \mathrm{any\ number\ of\ photons\ with\ }k<E_l) \\ &= \qty(\dv{\sigma}{\Omega})_0 \times \qty(\sum_{m=0}^\infty \frac{\vb{X}^m}{m!})^2 \sum_{n=0}^\infty \frac{\vb{Y}^n}{n!} \\ &= \qty(\dv{\sigma}{\Omega})_0 \times \exp\qty[-\frac{\alpha}{\pi} f_{\mathrm{IR}}(q^2) \log \qty(\frac{-q^2}{E_l^2})]. \end{align*}
##### Electron Self-Energy

$$+$$

1. Free propagator $\frac{i(\unicode{x2215}\kern-.5em {p} + m_0)}{p^2 - m_0^2 + i\epsilon}$ plus one-loop correction $\frac{i(\unicode{x2215}\kern-.5em {p} + m_0)}{p^2 - m_0^2}[-i\Sigma_2(p)]\frac{i(\unicode{x2215}\kern-.5em {p} + m_0)}{p^2 - m_0^2 + i\epsilon}$ where $-i\Sigma_2(p) = (-ie)^2 \int \frac{\dd{^4 k}}{(2\pi)^4} \gamma^\mu \frac{i(\unicode{x2215}\kern-.5em {k} + m_0)}{k^2 - m_0^2 + i\epsilon} \gamma_\mu \frac{-i}{(p-k)^2 - \mu^2 + i\epsilon}.$
2. Feynman parameters: $\frac{1}{k_2 - m_0^2 + i\epsilon} \frac{1}{(p-k)^2 - \mu^2 + i\epsilon} = \int_0^1 \dd{x} \frac{1}{D^2}.$
3. Completing the square: $D = l^2 - \Delta + i\epsilon$ and $\Delta = -x(1-x) p^2 + x\mu^2 + (1-x) m_0^2.$
4. Simplifying the numerator: $-i\Sigma_2(p) = -e^2 \int_0^1 \dd{x} \int \frac{\dd{^4 l}}{(2\pi)^4} \frac{-2 x\unicode{x2215}\kern-.5em {p} + 4m_0}{D^2}.$
5. Regularizing the integral: introduce a large $$\Lambda$$ and replace the photon propagator $\frac{1}{(p-k)^2 + i\epsilon} \rightarrow \frac{1}{(p-k)^2 + i\epsilon} - \frac{1}{(k-p)^2 - \Lambda^2 + i\epsilon}.$ For the $$\Lambda$$ term, the denominator is altered by $\Delta \rightarrow \Delta_\Lambda = -x(1-x)p^2 + x\mu^2 + (1-x)m_0^2.$ This replacement introduces a correction which gives $\frac{i}{(4\pi)^2} \int_0^\infty \dd{l_E^2} \qty({ \frac{l_E^2}{[l_E^2 + \Delta]^2} - \frac{l_E^2}{[l_E^2 + \Delta_\Lambda]^2} }) = \frac{i}{(4\pi)^2} \log\qty(\frac{\Delta_\Lambda}{\Delta}).$
6. Applying Wick rotation, we obtain (with $$\Lambda \rightarrow \infty$$) $\Sigma_2(p) = \frac{\alpha}{2\pi} \int_0^1 \dd{x} (2m_0 - x\unicode{x2215}\kern-.5em {p}) \log \qty(\frac{x\Lambda^2}{(1-x) m_0^2 + x\mu^2 - x(1-x)p^2}).$
• Analytic Behavior of $$\Sigma_2$$: branch cut occurs when $p^2 = (m_0 + \mu)^2.$
• 1PI diagrams: denoted by $$-i\Sigma(p)$$, which equals to $$-i\Sigma_2(p)$$ to the first order of $$\alpha$$.
• Two-point correlation with correction \begin{align*} \int \dd{^4 x} \bra{\Omega} T\psi(x)\overline{\psi}(0)\ket{\Omega} e^{-ip\cdot x} &= \frac{i(\unicode{x2215}\kern-.5em {p} + m_0)}{p^2 - m_0^2} + \frac{i(\unicode{x2215}\kern-.5em {p} + m_0)}{p^2 - m_0^2} (-i\Sigma) \frac{i(\unicode{x2215}\kern-.5em {p} + m_0)}{p^2 - m_0^2} + \cdots \\ &= \frac{i}{\unicode{x2215}\kern-.5em {p} - m_0 - \Sigma(\unicode{x2215}\kern-.5em {p})}. \end{align*}
• Position of pole determined by $\qty[\unicode{x2215}\kern-.5em {p} - m_0 - \Sigma(\unicode{x2215}\kern-.5em {p})]\vert_{\unicode{x2215}\kern-.5em {p} = m} = 0.$
• $$Z_2$$ determined by $Z_2^{-1} = 1 - \left.\dv{\Sigma}{\unicode{x2215}\kern-.5em {p}}\right|_{\unicode{x2215}\kern-.5em {p} = m}.$
• To order $$\alpha$$, $\delta m = m - m_0 \approx \Sigma_2(\unicode{x2215}\kern-.5em {p} = m_0) \xrightarrow{\Lambda \rightarrow \infty} \frac{3\alpha}{4\pi} m_0 \log\qty(\frac{\Lambda^2}{m_0^2}).$
• Not hard to show that $\delta F_1(0) + \delta Z_2 = 0.$
##### Renormalization of Charge

1. One-loop correction \begin{align*}i\Pi^{\mu\nu}_2(q) &= -(-ie)^2 \int \frac{\dd{^4 k}}{(2\pi)^4} \tr\qty[{ \gamma^\mu \frac{i(\unicode{x2215}\kern-.5em {k} + m)}{k^2 - m^2} \gamma^\nu \frac{i(\unicode{x2215}\kern-.5em {k} + \unicode{x2215}\kern-.5em {q} + m)}{(k+q)^2 - m^2} }] \\ &= -4e^2 \int \frac{\dd{^4 k}}{(2\pi)^4} \frac{k^\mu (k+q)^\nu + k^\nu (k+q)^\mu - g^{\mu\nu}(k\cdot (k+q) - m^2)}{(k^2 - m^2)((k+q)^2 - m^2)}. \end{align*}{}
2. Feynman parameters: $\frac{1}{(k^2 - m^2)((k+q)^2 - m^2)} = \int_0^1 \dd{x} \frac{1}{D^2}.$
3. Completing the square: $D = l^2 - \Delta + i\epsilon$ and $\Delta = m^2 - x(1-x) q^2.$
4. Simplifying the numerator: $\mathrm{Numerator} = 2l^\mu l^\nu - g^{\mu\nu}l^2 - 2x(1-x) q^\mu q^\nu + g^{\mu\nu} \qty(m^2 + x(1-x) q^2) + (\mathrm{terms\ linear\ in\ }l).$
5. Applying Wick rotation, $i\Pi^{\mu\nu}_2(q) = -4ie^2 \int_0^1 \dd{x} \int \frac{\dd{^4 l_E}}{(2\pi)^4} \frac{-(1/2)g^{\mu\nu} l_E^2 + g^{\mu\nu} l_E^2 - 2x(1-x)q^\mu q^\nu + g^{\mu\nu}(m^2 + x(1-x) q^2)}{(l_E^2 + \Delta)^2}.$
6. Regularizing the integral: using dimensional regularization. \begin{align*} i\Pi_2^{\mu\nu}(q) &= -4ie^2 \int_0^1 \dd{x} \frac{1}{(4\pi)^{d/2}} \frac{\Gamma(2-d/2)}{\Delta^{2-d/2}} \times \qty[{ g^{\mu\nu}(-m^2 + x(1-x) q^2) + g^{\mu\nu}(m^2 + x(1-x)q^2) - 2x(1-x)q^\mu q^\nu }] \\ &= (q^2 g^{\mu\nu} - q^\mu q^\nu) \cdot i\Pi_2(q^2) \end{align*} where \begin{align*} \Pi_2(q^2) &= \frac{-8e^2}{(4\pi)^{d/2}} \int_0^1 \dd{x} x(1-x) \frac{\Gamma(2-d/2)}{\Delta^{2-d/2}} \\ &\xrightarrow{d\rightarrow 4} -\frac{2\alpha}{\pi} \int_0^1 \dd{x} x(1-x) \qty(\frac{2}{\epsilon} - \log \Delta - \gamma + \log(4\pi)). \end{align*}
• Ward identity violated without regularization.
• The observed difference is given by $\hat{\Pi}_2(q^2) = \Pi_2(q^2) - \Pi_2(0) = -\frac{2\alpha}{\pi} \int_0^1 \dd{x} x(1-x) \log\qty(\frac{m^2}{m^2 - x(1-x) q^2}).$
• Branch cut of $$\Pi_2$$ lies at $q^2 > 4m^2.$
• Discontinuity by $\Im[\hat{\Pi}_2(q^2 \pm i\epsilon)] = \mp \frac{\alpha}{3} \sqrt{1-\frac{4m^2}{q^2}}\qty(1+\frac{2m^2}{q^2}).$
• Let $$i\Pi^{\mu\nu}(q)$$ denote the 1PI diagram of photon. With the Ward identity we identify $\color{red} \Pi^{\mu\nu}(q) = (q^2 g^{\mu\nu} - q^\mu q^\nu) \Pi(q^2).$
• Exact two-point function: \begin{align*} &\phantom{{}={}} \frac{-ig_{\mu\nu}}{q^2} + \frac{-ig_{\mu\nu}}{q^2}\qty[i(q^2 g^{\rho\sigma} - q^\rho q^\sigma)\Pi(q^2)]\frac{-ig_{\sigma\nu}}{q^2} + \cdots \\ &= \frac{-i}{q^2(1-\Pi(q^2))}\qty(g_{\mu\nu} - \frac{q_\mu q_\nu}{q^2}) + \frac{-i}{q^2}\qty(\frac{q_\mu q_\nu}{q^2}) \\ &\doteq \frac{-i g_{\mu\nu}}{q^2(1-\Pi(q^2))}. \end{align*}
• The last equality holds because of Ward identity, i.e. terms proportional to $$q_\mu$$ sums to zero.
• The residue of the $$q^2 = 0$$ pole: $\frac{1}{1-\Pi(0)} = Z_3.$
• Renormalization of charge: $e = \sqrt{Z_3} e_0.$
• $$Z_3 = 1$$ to the lowest order.
• For $$q^2 \neq 0$$, replace $\alpha_0 \rightarrow \alpha_{\mathrm{eff}}(q^2) = \frac{e_0^2/4\pi}{1-\Pi(q^2)} \approx \frac{\alpha}{1-[\Pi_2(q^2) - \Pi_2(0)]}.$

Uehling potential: recalculate the tree level $e^- + e^- \rightarrow e^- + e^-$ we find, for $$r\gg 1/m$$, \begin{align*} V(\vb*{x}) &= \int \frac{\dd{^3 q}}{(2\pi)^3} e^{i\vb*{q} \cdot \vb*{x}} \frac{-e^2}{\abs{\vb*{q}}^2 [1 - \hat{\Pi}_2(-\abs{\vb*{q}}^2)]} \\ &\approx -\frac{\alpha}{r}\qty(1 + \frac{\alpha}{4\sqrt{\pi}} \frac{e^{-2mr}}{(mr)^{3/2}}). \end{align*} However, at small distance, or $$-q^2 \gg m^2$$, $\alpha_{\mathrm{eff}}(q^2) = \frac{\alpha}{\displaystyle 1-\frac{\alpha}{3\pi} \log\qty(\frac{-q^2}{Am^2})}$ where $$A = \exp(5/3).$$

#### Miscellaneous

##### Asymptotic Series
 Index Index

Expand

## Advanced Solid State Physics (II)

Strongly Correlated Systems

#### Solid State Physics

##### Wannier Functions
• Bloch to Wannier: $\psi_{n\vb*{k}} = N^{-1/2} \sum_{\vb*{r}} e^{i\vb*{k}\cdot \vb*{l}} a_n(\vb*{r} - \vb*{l}).$
• Annihilation operators: $c_{n\vb*{k}} = \frac{1}{\sqrt{N}} \sum_{\vb*{l}} c_{n\vb*{l}} e^{-i\vb*{k}\cdot \vb*{l}}.$
• Orthogonality: $\int \dd{\vb*{r}} a^*_{n}(\vb*{r} - \vb*{l}) a_{n'}(\vb*{r} - \vb*{l}') = \delta_{nn'}\delta_{ll'}.$
• Completeness: $\sum_{n} \sum_{l} a^*_n(\vb*{r} - \vb*{l}) a_n(\vb*{r}' - \vb*{l}) = \delta(\vb*{r} - \vb*{r}').$
##### Statistical Mechanics
• The trace on a grand canonical ensemble is defined by $\langle \cdots \rangle = \frac{\operatorname{Tr}{\qty[e^{-\beta(H-\mu N)} \cdots]}}{\operatorname{Tr}{\qty[e^{-\beta(H-\mu N)}]}}.$

#### Green's Functions

##### General Formulation
• Retarded Green's function: $\color{orange}\langle\langle A(t); B(t') \rangle\rangle = -i\Theta(t-t') \langle \qty{A(t), B(t')} \rangle.$
• Fourier transform: $\langle\langle A|B \rangle\rangle (\omega + i\epsilon) = \int_{-\infty}^{+\infty} \langle\langle A(t); B(0)\rangle\rangle e^{i(\omega + i\epsilon) t}.$
• Denoted by $$G_{\mathrm{r}}(\omega)$$.
• Spectral theorem: $\color{red} \langle BA \rangle = \frac{i}{2\pi} \int_{-\infty}^{+\infty} \dd{\omega} \frac{\qty{\langle\langle A|B\rangle\rangle(\omega + i\eta) - \langle\langle A|B\rangle\rangle(\omega - i\eta)}}{e^{\beta(\omega - \mu)} - 1}.$
• $\langle BA \rangle = \sum \operatorname*{Res}_{\omega \in \mathbb{R}} \langle \langle A|B \rangle \rangle(\omega) f(\omega).$
• Fluctuation-Dissipation Theorem: for canonical ensembles, $\color{red} \langle BA \rangle = \frac{1}{2\pi} \int_{-\infty}^{\infty} \dd{\omega} \frac{\qty{-2\Im G_{\mathrm{r}}(\omega)}}{e^{\beta\hbar\omega} \pm 1}.$
• Equation of motion: \begin{gather*} i\dv{}{t} \langle\langle A(t);B(t') \rangle\rangle = \delta(t-t') \langle \qty{A(t),B(t)} \rangle + \langle\langle [A(t),H]; B(t')\rangle\rangle \\ \Updownarrow \\ \color{red}\langle\langle A|B\rangle\rangle(\omega) = \begin{cases} \langle \qty{A,B} \rangle + \langle\langle [A,H]|B \rangle\rangle(\omega), \\ \langle \qty{A,B} \rangle - \langle\langle A|[B,H] \rangle\rangle(\omega). \end{cases} \end{gather*}
• Population of electrons: \begin{align*} \color{darkcyan} n_\sigma &= \frac{1}{N} \sum_j \langle c^\dagger_{j\sigma} c_{j\sigma}\rangle \\ &\color{darkcyan}= \int_{-\infty}^{+\infty} \dd{\omega} f(\omega) \rho_\sigma(\omega). \end{align*}{}
• Local density of states: \begin{align*} \color{orange} \rho_\sigma(\omega) &\color{orange}= \frac{i}{2\pi N} \sum_j \qty{G^\sigma_{jj}(\omega + i\eta) - G^\sigma_{jj}(\omega - i\eta)} \\ &\color{orange}= \frac{i}{2\pi N} \sum_{\vb*{k}} \qty{G^\sigma_{\vb*{k}}(\omega + i\eta) - G^\sigma_{\vb*{k}}(\omega - i\eta)}. \end{align*}
##### Green's Function of the Field Operator
• The Green's functions solve the equation $(-i\hbar\partial_t + H) G(\vb*{r},t;\vb*{r}',t') = -\hbar \delta(\vb*{r} - \vb*{r}')\delta(t-t').$
• The field operator is defined by $\psi^\dagger_\sigma(\vb*{r},t) = \sum_{\vb*{p},\sigma} u_{\vb*{p}}(\vb*{r}) a^\dagger_{\vb*{p},\sigma}.$
• $G_{nn} = \frac{1}{2\pi} \int \dd{E} \bra{n}G\ket{n} e^{-iEt/\hbar} = \begin{cases} -ie^{-iE_nt/\hbar}, & \text{if } t>0, \\ 0, & \text{if } t\le 0. \end{cases}$
• The Green's function is given by \begin{align*} G_\sigma(\vb*{r},t;\vb*{r}',t') &= -i \operatorname{T}\langle \psi_\sigma(\vb*{r}, t) \psi^\dagger_\sigma(\vb*{r}',t') \rangle \\ &= \Theta(t-t') G^>_\sigma(\vb*{r},t;\vb*{r}',t') + \Theta(t'-t) G^<_\sigma(\vb*{r},t;\vb*{r}',t'). \end{align*}
• $G^> = -i \langle \psi_\sigma(\vb*{r},t) \psi^\dagger_\sigma(\vb*{r}',t') \rangle.$
• $G^< = i \langle \psi^\dagger_\sigma(\vb*{r'},t') \psi_\sigma(\vb*{r},t) \rangle.$
• Retarded Green's function: $G^{\mathrm{R}}_\sigma(\vb*{r},t;\vb*{r}',t') = -i\Theta(t-t')\langle \qty{\psi_\sigma(\vb*{r},t), \psi^\dagger_\sigma(\vb*{r}',t)} \rangle.$
• Non-zero only for $$t>t'$$.
• Advanced Green's function: $G^{\mathrm{A}}_\sigma(\vb*{r},t;\vb*{r'},t') = i\Theta(t'-t)\langle \qty{\psi_\sigma(\vb*{r},t), \psi^\dagger_\sigma(\vb*{r}',t')} \rangle.$
• Non-zero only for $$t<t'$$.

#### Hubbard Model

• Assumptions:
• The system admits translational invariance.
• Hamiltonian: single conduction band, $\color{orange} H = \underbrace{\sum_{i,j} \sum_\sigma T_{ij} c^\dagger_{i\sigma} c_{j\sigma}}_{H_0} + \frac{U}{2} \sum_i \sum_\sigma n_{i\sigma}n_{i, -{\sigma}}.$
• Creation and annihilation in the Wannier basis.
• $H_0 = \sum_{\vb*{k},\sigma} E_{\vb*{k}} c^\dagger_{\vb*{k}\sigma} c_{\vb*{k}\sigma}.$
• $T_{ij} = N^{-1} \sum_{\vb*{k}} e^{i\vb*{k}\cdot (\vb*{R}_i - \vb*{R}_j)} E_{\vb*{k}}.$
• $T_0 = T_{ii}.$
• $T_1 = T_{i,i+1} < 0.$
• $$i+1$$ denotes the neighbours of $$i$$.
• $$T_1 \sim \Delta$$ (width of the band): $E_{\vb*{k}} \approx T_0 + T_1 \sum_{\text{n. n.}} e^{-i\vb*{k}\cdot \vb*{R}_{\mathrm{n}}}.$
• $U = \bra{ii} v\ket{ii} = e^2 \int \frac{a^*(\vb*{r} - \vb*{R}) a^*(\vb*{r'} - \vb*{R}) a(\vb*{r} - \vb*{R}_i)a(\vb*{r}' - \vb*{R}_i)}{\abs{\vb*{r} - \vb*{r}'}}\dd{\vb*{r}}\dd{\vb*{r}'}.$
##### Zero Band Width
• Hamiltonian: $$T_{ij} = T_0 \delta_{ij}$$, $H = T_0 \sum_{i,\sigma} n_{i\sigma} + \frac{1}{2}U\sum_{i,\sigma} n_{i\sigma}n_{i,-\sigma}.$
• Single-particle Green's function: $\color{orange} G^\sigma_{ij}(\omega) = \langle\langle c_{i\sigma} | c^\dagger_{j\sigma}\rangle\rangle(\omega).$

$\delta(\omega - \omega_0) = -\frac{1}{\pi} \lim_{\epsilon\rightarrow 0} \Im \frac{1}{\omega - \omega_0 + i\epsilon}.$

• Solution (exact): $\color{darkcyan} G^\sigma_{ij}(\omega) = \delta_{ij} \qty{\frac{1-\langle n_{i,-\sigma} \rangle}{\omega - T_0} + \frac{\langle n_{i,-\sigma}\rangle}{\omega - T_0 - U}}.$
• Local density of states: $\color{darkcyan} \rho_\sigma(\omega) = (1-\langle n_{-\sigma}\rangle)\delta(\omega - T_0) + \langle n_{-\sigma}\rangle \delta(\omega - T_0 - U).$
##### Small Band Width
• Hamiltonian: $$T_{ij} \neq 0$$.
• Green's function in Bloch basis: $G^{\sigma}_{\vb*{k},\vb*{k}'} = \langle\langle c_{\vb*{k}\sigma} | c^\dagger_{\vb*{k},-\sigma} \rangle\rangle = \delta_{\vb*{k},\vb*{k}'} G^\sigma_{\vb*{k}}(\omega).$
• Bloch basis to Wannier basis: $G^{\sigma}_{ij}(\omega) = \frac{1}{N} \sum_{\vb*{k}} e^{i\vb*{k}\cdot (\vb*{R}_i - \vb*{R}_j)} G^\sigma_{\vb*{k}}(\omega).$
• Solution (approximate): \begin{align*} \color{darkcyan} G^{\sigma}_{\vb*{k}}(\omega) &= \frac{\omega - T_0 - U(1 - \langle n_{-\sigma} \rangle)}{(\omega - E_{\vb*{k}})(\omega - T_0 - U) + \langle n_{-\sigma} \rangle U(T_0 - E_{\vb*{k}})} \\ &\color{darkcyan}= \frac{A^{(1)}_{\vb*{k}\sigma}}{\omega - E^{(1)}_{\vb*{k}\sigma}} + \frac{A^{(2)}_{\vb*{k}\sigma}}{\omega - E^{(2)}_{\vb*{k}\sigma}}. \end{align*}
• $\left.\begin{array}{l} E^{(1)}_{\vb*{k}\sigma} \\ E^{(2)}_{\vb*{k}\sigma} \end{array}\right\} = \frac{1}{2}\qty{E_{\vb*{k}} + U + T_0 \mp \sqrt{(E_{\vb*{k}} - U - T_0)^2 + 4U\langle n_{-\sigma}\rangle (E_{\vb*{k}} - T_0)}}.$
• $\left.\begin{array}{l} A^{(1)}_{\vb*{k}\sigma} \\ A^{(2)}_{\vb*{k}\sigma} \end{array}\right\} = \frac{1}{2} \qty{1 \mp \frac{E_{\vb*{k}} - U - T_0 + 2U\langle n_{-\sigma} \rangle}{\sqrt{(E_{\vb*{k}} - U - T_0)^2 + 4U\langle n_{-\sigma} \rangle (E_{\vb*{k}} - T_0)}}}.$
• Approximation: $[n_{i,-\sigma}c_{i,\sigma}] = \underbrace{\cdots}_{\text{linear in } c_{i\sigma}} + \cancelto{0}{\sum_{\vb*{l}} T_{i\vb*{l}} (c_{i,-\sigma}^\dagger c_{l,-\sigma} - c^\dagger_{l,-\sigma} c_{i,-\sigma}) c_{i\sigma}}.$
• Local density of states: $\color{darkcyan} \rho_\sigma(\omega) = \frac{1}{N} \sum_{\vb*{k}} \qty{A^{(1)}_{\vb*{k}\sigma} \delta[\omega - E^{(1)}_{\vb*{k}\sigma}] + A^{(2)}_{\vb*{k}\sigma}\delta[\omega - E^{(2)}_{\vb*{k}\sigma}]}.$
• More accurate approximation:
• Split (insulator): $\frac{\Delta}{U} < \frac{2}{\sqrt{3}}.$
• Doesn't split (metal): $\frac{\Delta}{U} > \frac{2}{\sqrt{3}}.$

$$U=0$$, no correlation, $\rho_\sigma(\omega) = D(\omega).$

$$U\ge \Delta$$, $$U \gg \abs{E_{\vb*{k}} - T_0}$$, strong correlation, \begin{align*} G^{\sigma}_{\vb*{k}}(\omega) &\approx \qty{\frac{1-\langle n_{-\sigma}\rangle}{\omega - T_0 - (E_{\vb*{k}} - T_0)(1-\langle n_{-\sigma}\rangle)} + \frac{\langle n_{-\sigma}\rangle}{\omega - T_0 - U - (E_{\vb*{k}} - T_0)\langle n_{-\sigma} \rangle}}. \\ \rho_\sigma(\omega) &= \frac{1}{N} \sum_{\vb*{k}} \begin{cases} (1-\langle n_{-\sigma}\rangle) \delta[\omega - T_0 - (E_{\vb*{k}} - T_0)(1 - \langle n_{-\sigma}\rangle)] \\ \mbox{} + \langle n_{-\sigma} \rangle \delta[\omega - T_0 - U - (E_{\vb*{k}} - T_0)\langle n_{-\sigma} \rangle]. \end{cases} \end{align*}{}

#### Local Magnetic Moments in Metals

##### Phenomenology of Local Magnetic Moments
• Notation:
• $$d$$ denotes an impurity state,
• $$\vb*{k}$$ denote a band state,
• $$i$$ and $$j$$ denote nearest neighbour lattice sites.
• Anderson Hamiltonian: $\color{orange} H^{\mathrm{A}} = \sum_{\vb*{k}\sigma} E_{\vb*{k}} n_{\vb*{k}\sigma} + \sum_\sigma E_d n_{d\sigma} + U n_{d\uparrow} n_{d\downarrow} + \sum_{\vb*{k}\sigma} V_{\vb*{k}d}(a^\dagger_{\vb*{k}\sigma} a_{d\sigma} + a^\dagger_{d\sigma}a_{\vb*{k}\sigma}).$
• Electrons in the pure metal: $\sum_{\vb*{k}\sigma} E_{\vb*{k}} n_{\vb*{k}\sigma}.$
• In the presence of a magnetic field: $\color{orange}E_{\vb*{k}\sigma} = E_{\vb*{k}} + \sigma \mu_{\mathrm{B}} H.$
• Unperturbed impurity atom: correlation not included, $\sum_\sigma E_d n_{d\sigma}.$
• In the presence of a magnetic field: $\color{orange}E_{d\sigma} = E_{d} + \sigma \mu_{\mathrm{B}} H.$
• Correlation between electrons in the impurity atom: $U n_{d\uparrow} n_{d\downarrow}.$
• Coulomb energy between electrons: $U = \bra{dd} V_{\mathrm{ee}} \ket{dd} = \int \dd{\vb*{r}_1} \dd{\vb*{r}_2} \abs{\phi_d(\vb*{r}_1)}^2 \frac{e^2}{\abs{\vb*{r}_1 - \vb*{r}_2}} \abs{\phi_d(\vb*{r}_2)}^2.$
• s-d mixture: $\sum_{\vb*{k}\sigma} V_{\vb*{k}d}(a^\dagger_{\vb*{k}\sigma} a_{d\sigma} + a^\dagger_{d\sigma}a_{\vb*{k}\sigma}).$
• $$V_{\vb*{k}d}$$ describes the magnitude of mixing between s and d states: $V_{\vb*{k}d} = \bra{\phi_d(\vb*{r})} H_0 \ket{\phi_{\vb*{k}}(\vb*{r})},$ where $$H_0$$ is the single-electron Hamiltonian.
• Remarks:
• The interaction of localized electrons with eletrons on other sites are not included. This term may be described by integrals of the form $\bra{i}\bra{j} V_{\mathrm{ee}} \ket{k}\ket{l},$ which is one order of magnitude smaller than $$U$$.
• The Coulomb repulsion $$U$$ favors the formation of local magnetic moments because it inhibits double occupation.
• Phillips: $$E$$ is the energy of the impurity site relative to the Fermi level.
• The Anderson model is governed by serveral parameters: $$\epsilon_{\vb*{k}}$$, $$U$$, and $$\Gamma$$.
• $$\Gamma$$ characterize the transition rate of $$\vb*{k} \rightarrow d$$ and is refered to as the hybridization energy: $\frac{1}{\tau} = 2\pi \frac{\abs{V_{\vb*{k}d}}^2 N(\epsilon_d)}{\hbar} = \frac{2\Gamma}{\hbar}.$
• If $$U\gg \epsilon_d \gg \Gamma$$, then the system supports local moment formation.
• If $$U\gg \Gamma \gg \epsilon_d$$, then the occupation state undergoes rapid flucutations and the system is not magnetic.
• If $$\Gamma \gg U$$, then the impurity level is broadened and is occupied with spin up and spin down electrons with equal probability, resulting in a non-magnetic state. This state is termed as localized spin fluctuation.
##### Density of States of Impurity
• For local moment to form we demand $\langle n_{d\sigma} \rangle \neq \langle n_{d,-\sigma} \rangle.$
• Green's function:
• $\color{darkcyan} \langle\langle a_{\vb*{k}\sigma} | a^\dagger_{\vb*{k}'\sigma}\rangle\rangle = \frac{\delta_{\vb*{k}\vb*{k}'}}{\omega - E_{\vb*{k}\sigma}} + \frac{V_{\vb*{k}d}V_{\vb*{k}'d}}{(\omega - E_{\vb*{k}\sigma})(\omega - E_{\vb*{k}'\sigma})}\langle\langle a_{d \sigma} | a^\dagger_{\sigma}\rangle\rangle.$
• $\color{darkcyan} \langle\langle a_{d\sigma} | a_{d\sigma}^\dagger \rangle\rangle(\omega \pm i\epsilon) = \frac{1}{\omega - E_{d\sigma} - U\langle n_{d,-\sigma}\rangle \pm i\Gamma}.$
• $i\Gamma = \sum_{\vb*{k}} \frac{\abs{V_{\vb*{k}\sigma}}^2}{\omega - E_{\vb*{k}\sigma} + i\epsilon} \approx -i\pi \abs{V_{\vb*{k}d}}^2 \rho^{(0)}(\omega).$
• Approximation: Hatree-Fock approximation, \begin{gather*} \langle\langle n_{d,-\sigma} a_{d\sigma} | a^\dagger_{d\sigma}\rangle\rangle(\omega) \approx \langle n_{d,-\sigma}\rangle \langle\langle a_{d\sigma} | a^\dagger_{d\sigma}\rangle\rangle(\omega) \\ \Updownarrow \\ \frac{U}{2}\sum_\sigma n_{d\sigma}n_{d,-\sigma} \approx U\sum_\sigma \langle n_{d,-\sigma}\rangle n_{d\sigma} - U\langle n_{d\uparrow} \rangle\langle n_{d\downarrow} \rangle. \end{gather*}
• The Coulomb interaction shift the energy level by a real number.
• The mixing between $$d$$ and $$\vb*{k}$$ moves the poles of the Green's function off the real axis and therefore broadens the energy level.
• The d states do decay. $G_{\mathrm{r}}(t) = -i\Theta(t) e^{-i(E_{d\sigma} + U\langle n_{d,-\sigma} \rangle)} e^{-\Gamma t}.$
• The impurity density of states: $\color{darkcyan} \rho_{d\sigma}(\omega) \sim \frac{1}{\pi} \frac{\Gamma}{(\omega - E_{d\sigma} - U\langle n_{d,-\sigma} \rangle)^2 + \Gamma^2}.$
• $$\Gamma \sim \mathrm{FWHM}$$.
• Susceptibility: $\chi = \chi_{\mathrm{P}} + \underbrace{\lim_{H\rightarrow 0} \frac{\mu\sigma_{\mathrm{B}}}{H} \langle a^\dagger_{d\sigma} a_{d\sigma}\rangle}_{\chi_{\mathrm{I}}}.$
• $$\chi_{\mathrm{P}}$$ is the Pauli susceptibility.
• Population of d state: $\langle a^\dagger_{d\sigma} a_{d\sigma}\rangle = \int_{-\infty}^\infty \dd{\omega} f(\omega) \rho_{d\omega}(\omega).$
• $$T=0$$: $\color{darkcyan} \langle n_{d\sigma} \rangle = \frac{1}{\pi} \arccot \qty[\frac{E_d - E_{\mathrm{F}} + U\langle n_{d,-\sigma} \rangle + (\operatorname{sign} \sigma)\mu_{\mathrm{B}}H}{\Gamma}].$
• For $$U\rho_{d\sigma}(E_{\mathrm{F}}) < 1$$: $\chi_{\mathrm{I}} = 2\mu_{\mathrm{B}}^2 \frac{1}{\displaystyle \frac{\pi \Gamma}{\sin^2 n_0 \pi} - U}.$
• Assumption: nonmagnetic solution under $$H=0$$ $\langle n_{d\uparrow} \rangle = \langle n_{d\downarrow} \rangle.$
• $$\Gamma$$ to DoS: $\rho_{d\sigma}(E_{\mathrm{F}}) = \frac{1}{\pi} \frac{\sin^2 \pi n_0}{\Gamma}.$
• For $$U\rho_{d\sigma}(E_{\mathrm{F}}) > 1$$: nonmagnetic solution unstable.
• Magnetic ground state exists only if
• $\color{red}U\rho_{d\sigma}(E_{\mathrm{F}}) > 1.$
• $\color{red} 0 < \frac{E_{\mathrm{F}} - E_{\mathrm{d}}}{U} < 1.$

#### Linear Response

##### Monochromatic Perturbation
• Perturbation: $H_{\mathrm{e}}(t) = B e^{-i\omega t + \eta t}.$
• Heisenberg picture: $H'_{\mathrm{e}}(t) = B(t) e^{-i\omega t + \eta t},$ where $B(t) = e^{iHt/\hbar} B e^{-iHt/\hbar}.$
• Green's function: $G_{\mathrm{r}} = -\frac{i}{\hbar}\Theta(t-t')\langle [A(t), B(t')] \rangle.$
• Spatial formulation: $\Delta A = \int_{-\infty}^{\infty} G_{\mathrm{r}}(t-t') e^{-i\omega t + \eta t'} \dd{t'}.$
• Frequency domain formulation: $\Delta A = G_{\mathrm{r}}(\omega) e^{-i\omega t + \eta t}.$ where $G_{\mathrm{r}}(\omega) = \int_{-\infty}^{\infty} G_{\mathrm{r}}(t) e^{i\omega t - \eta t} \dd{t}.$
##### Example: Conductivity
• Current density operator: $j_\alpha(\vb*{r}) = \frac{1}{2m} \sum_i e_i\qty{\vb*{p}_{i\alpha}\delta(\vb*{r} - \vb*{r}_i) + \delta(\vb*{r} - \vb*{r}_i) p_{i\alpha}},$ or $j_\alpha(\vb*{q}) = \frac{1}{2m} \sum_i e_i\qty[\vb*{p}_{i\alpha} e^{i\vb*{q}\cdot \vb*{r}_i} + e^{i\vb*{q}\cdot \vb*{r}_i}\vb*{p}_{i\alpha}].$
###### DC Conductivity
• DC conductivity: $\Re(\sigma_{\alpha\beta}) = \frac{\pi \beta}{v} \sum_{nm} e^{-\beta E_n} \bra{n} j_\beta \ket{m} \bra{m} j_\alpha \ket{n} \delta(E_n - E_m).$

Expand
 Quantum Field Theory (I) Mathematical Prerequisites

## Quantum Field Theory (II)

Quantization of Fields

#### Foundations of Field Theory

##### Second-Quantization
• The following integration is Lorentz invariant: $\color{red} \int \frac{\dd{^3 p}}{2E_{\vb*{p}}} = \int \frac{\dd{^3 \tilde{p}}}{2E_{\tilde{\vb*{p}}}}.$
• With the convention $\ket{\vb*{p},s} = \sqrt{2E_{\vb*{p}}} a_{\vb*{p}}^{s\dagger}\ket{0},$ we have the Lorentz invariant normalization $\bra{\vb*{p},r}\ket{\vb*{q},s} = 2E_{\vb*{p}} (2\pi)^3 \delta^{(3)} (\vb*{p} - \vb*{q})\delta^{rs}.$
• The annihilation operator transforms like $\color{red} U(\Lambda) a_{\vb*{p}}^s U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda \vb*{p}}}{E_{\vb*{p}}}} a^s_{\Lambda \vb*{p}},$ where we assumed that the spin axis is parallel to the boost or rotation axis. This is equivalent to that $$\sqrt{2E_{\vb*{p}}} a^s_{\vb*{p}}$$ is Lorentz invariant, $\color{red} \sqrt{2E_{\vb*{p}}} a^s_{\vb*{p}} = \sqrt{2E_{\vb*{\tilde{p}}}} a^s_{\vb*{\tilde{p}}}.$
##### Noether Current

If under $\phi(x) \rightarrow \phi'(x) = \phi(x) + \alpha\Delta\phi(x),$ the Lagrangian undergoes $\mathcal{L}(x) \rightarrow \mathcal{L}(x) + \alpha\partial_\mu \mathcal{J}^\mu(x),$ then $\partial_\mu j^\mu(x) = 0$ for $\color{orange} j^\mu(x) = \pdv{\mathcal{L}}{(\partial_\mu \phi)}\Delta \phi - \mathcal{J}^\mu.$

##### Free Particle Solution (Dirac Equation)
• Normalization of spinor: $\color{darkcyan}\xi^\dagger \xi = 1.$
• Example basis of spinors: $\xi^1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix},\quad \xi^2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$
• Superscript $$s$$ always denote the direction of spin.
• Positive frequency: $\psi(x) = u(p) e^{-ip\cdot x},$ where $\color{orange} u^s(p) = \begin{pmatrix} \sqrt{p\cdot \sigma} \xi^s \\ \sqrt{p\cdot \overline{\sigma}} \xi^s \end{pmatrix}.$
• Normalization of solution:
• $\color{darkcyan}\overline{u}^r(p) u^s(p) = 2m\delta^{rs}.$
• $\color{darkcyan}{u}^{r\dagger}(p) u^s(p) = 2E_{\vb*{p}} \delta^{rs}.$
• $\color{darkcyan}\overline{u}(p) = u^\dagger(p) \gamma^0.$
• Negative frequency: $\psi(x) = v(p) e^{+ip\cdot x},$ where $\color{orange}v^s(p) = \begin{pmatrix} \sqrt{p\cdot \sigma} \eta^s \\ -\sqrt{p\cdot \overline{\sigma}} \eta^s \end{pmatrix}.$
• Normalization of solution:
• $\color{darkcyan}\overline{v}^r(p) v^s(p) = -2m\delta^{rs}.$
• $\color{darkcyan}{v}^{r\dagger}(p) v^s(p) = 2E_{\vb*{p}} \delta^{rs}.$
• $\color{darkcyan}\overline{v}(p) = v^\dagger(p) \gamma^0.$
• Orthogonality:
• $\color{darkcyan}\overline{u}^r(p) v^s(p) = \overline{v}^r(p) u^s(p) = 0.$
• $\color{darkcyan}u^{r\dagger}(\vb*{p}) v^s(-\vb*{p}) = v^{r\dagger}(-\vb*{p}) u^s(\vb*{p}) = 0.$
• $\color{darkcyan}u^{r\dagger}(p) v^s(p) \neq 0,\quad v^{r\dagger}(p) u^s(p) \neq 0.$
• Spinor sums:
• $\color{darkcyan}\sum_s u^s(p) \overline{u}^s(p) = \gamma \cdot p + m.$
• $\color{darkcyan}\sum_s v^s(p) \overline{v}^s(p) = \gamma \cdot p - m.$
• Helicity operator: $h = \hat{p} \cdot \vb*{S} = \frac{1}{2}\hat{p}_i \begin{pmatrix} \sigma^i & 0 \\ 0 & \sigma^i \end{pmatrix}.$
• Right-handed: $$h=+1/2$$.
• Left-handed: $$h=-1/2$$.

#### Functional Tools

##### Path Integral
• Path integral in quantum mechanics: $\int \mathcal{D}q\,\mathcal{D}p\, F(q,p).$
• Implicitly from $$t'$$ to $$t''$$.
• Summing all paths that $q(t') = q',\quad q(t'') = q''.$
• Each path given a weight $\dd{q}_1 \cdots \dd{q}_N \cdot \dd{p}_1 \cdot \dd{p}_N.$
• Mathematically doable only before taking $$N\rightarrow \infty$$.
• Path integral in field theory: $\int \mathcal{D}\varphi\, F[\varphi].$
• Summing all possible field configurations.
• Each path (configuration) given a weight $\mathcal{D}\varphi \propto \prod_x \dd{\varphi(x)}.$
• Transition amplitude $\bra{\phi_b(\vb*{x})} e^{-iHT} \ket{\phi_a(\vb*{x})} = \int \mathcal{D}\phi\, \exp\qty[i\int_0^T \dd{^4 x} \mathcal{L}].$
##### Functional Derivative
• Functional derivative: $\frac{\delta}{\delta f(t_1)} f(t_2) = \delta(t_2 - t_1).$
• Rules of ordinary derivatives, including the chain rule, do apply.
##### Conventions
• Fourier: Peskin, Srednicki $\tilde{\varphi}(k) = \int \dd{^4x} e^{-ik\cdot x} \varphi(x),\quad \varphi(x) = \int \frac{\dd{^4 k}}{(2\pi)^4} e^{ik\cdot x} \tilde{\varphi(k)}.$
##### Functional Determinant
• $\zeta_S(z) = \tr S^{-z}.$
• $\det S = e^{-\zeta'_S(0)}.$
• $\frac{1}{\sqrt{\det S}} \propto \int_V \mathcal{D}\phi\, e^{-\bra{\phi} S \ket{\phi}}.$

#### Framework: Quantization of Fields

• Thanks to the LSZ reduction formula, we could focus on correlation functions only.
• We are interested in $\color{orange} \bra{\Omega} T\phi_{H}(x_1) \phi_{H}(x_2) \cdots \phi_{H}(x_n) \ket{\Omega} = ?.$
• Subscript $$S$$ denotes Schrödinger operator and $$H$$ denote Hamiltonian operator.
##### Canonical Quantization
• Classical (free) field $$\phi(x)$$ promoted to operators.
• $$\phi(x)$$ should be expanded as $\phi(x) \propto \int \dd{^3k} a_{\vb*{k}}^\dagger e^{-ik\cdot x} + \cdots,$
• $$a_{\vb*{p}}^\dagger$$ creates a free particle.
• $$a_{\vb*{p}}^\dagger$$ satisfies the canonical commutation relations, e.g. $[a_{\vb*{p}}^\dagger, a_{\vb*{q}}] \propto \delta(\vb*{p} - \vb*{q}).$
• Feynman propagator be given by $D_F(x-y) = T\phi(x)\phi(y) - N\phi(x)\phi(y).$
• $$N$$ denotes the normal ordering.
##### Path Integral Quantization
• Correlation functions given by $\color{red} \bra{\Omega} T\phi_H(x_1) \phi_H(x_2) \ket{\Omega} = \dfrac{\int \mathcal{D}\phi\, \phi(x_1) \phi(x_2) \exp\qty[i\int_{-T}^T \dd{^4 x}\mathcal{L}]}{\int \mathcal{D}\phi\, \exp\qty[i\int_{-T}^T \dd{^4 x}\mathcal{L}]}.$
• We don't care if the field is an interacting field or a free field.
• Denominator given by $$Z_0[0]$$, equivalent to the sum of vacuum diagrams.
• Functional integral: $\color{orange} Z_0[J] = \int \mathcal{D}\varphi\, e^{i\int \dd{^4 x} \qty[\mathcal{L}_0 + J\varphi]}.$
• We want a more explicit form, i.e. $Z[J] = Z_0\qty(\text{path integral concerning }\phi\text{ only}) \cdot \qty(J\text{-dependent term without } \phi).$
• We do this by a careful shift of Jacobian $$1$$, $\phi' = \phi + \qty(\text{something}).$
• The integrand is rewritten as $\exp\qty{i\int \dd{^4 x} \qty[\mathcal{L}_0(\varphi) + J\varphi]} = \exp\qty{i\int \dd{^4 x} \qty[\mathcal{L}_0(\varphi') + f\qty[J]]}.$
• Therefore, $Z_0[J] = Z_0[0] F[J].$
• Functional integral as correlation amplitude: $\color{darkcyan} \bra{0}\ket{0}_J = \frac{Z_0[J]}{Z_0}.$
• Correlation functions: $\color{red} \bra{0} T\varphi(x_1)\cdots \varphi(x_n)\ket{0} = \frac{1}{Z_0}\left.{\frac{\delta}{\delta J(x_1)}\cdots \frac{\delta}{\delta J(x_n)} Z_0[J]}\right\vert_{J=0}.$
##### ϵ-Trick
• $$\color{orange} H\ket{0} = 0$$.
• $$H \rightarrow H(1-i\epsilon)$$.
• Convergence factor:
• $k^0 \rightarrow k^0(1+i\epsilon).$
• $t \rightarrow t(1-i\epsilon).$

For $$H_0 \propto m^2\varphi^2$$, we turn $$m^2$$ into $m^2 \rightarrow m^2 - i\epsilon.$

• Path integral applied to a gauge field: $\int \mathcal{D}A\, e^{iS[A]}.$
• Unit factor: $1 = \int \mathcal{D}\alpha(x)\, \delta(G(A^\alpha)) \det\qty(\frac{\delta G(A^\alpha)}{\delta \alpha}).$
• $$G(A^\alpha)$$: the gauge we choose such that $G(A^\alpha) = 0.$
• $$A^\alpha$$ denotes the gauge-transformed field, e.g. $A^\alpha_\mu(x) = A_\mu(x) + \frac{1}{e} \partial_\mu \alpha(x).$
• Path integral rewritten as $\int \mathcal{D}\alpha \int \mathcal{D}A \, e^{iS[A]} \delta(G(A^\alpha)) \det\qty(\frac{\delta G(A^\alpha)}{\delta \alpha}).$

#### Scalar Field

##### Classical Field (Scalar Field)
• Lagrangian $\mathcal{L}_0 = -\frac{1}{2}\partial^\mu\varphi \partial_\mu\varphi - \frac{1}{2}m^2\varphi^2.$
##### Canonical Quantization (Scalar Field)

$\color{orange} \varphi(x) = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}}\qty(a_{\vb*{p}}e^{-ip\cdot x} + a^\dagger_{\vb*{p}} e^{ip\cdot x}).$

##### Path Integral (Scalar Field)
• Functional integral: $\color{orange} Z_0[J] = Z_0 \exp\qty[-\frac{1}{2} \iint \dd{^4 x}\dd{^4 x'} J(x) D_F(x-x') J(x')].$
• We made the substitution of Jacobian $$1$$: $\phi \rightarrow \phi + \frac{1}{-\partial^2 - m^2 + i\epsilon} J.$
• Feynman propagator: $\color{orange} D_F(x-x') = \int \frac{\dd{^4 k}}{(2\pi)^4} \frac{i}{k^2 - m^2 + i\epsilon}e^{-ik\cdot (x-x')}.$
• $$D_F$$ is the Green's function of the Klein-Gordon equation: $(\partial^2_x + m^2) D_F(x-x') = -i\delta^4(x-x').$
• $\color{darkcyan} D_F(x-x_0) = \bra{0}T\varphi(x_1)\varphi(x_2)\ket{0}.$
• $$D_F$$ is denoted by $$\Delta$$ in Srednicki. $$D_F$$ in Peskin is $$i$$ times $$\Delta$$ in Srednicki.
• Wick's Theorem: $\color{darkcyan}\bra{0}T\varphi(x_1)\cdots \varphi(x_n)\ket{0} = \sum_{\text{pairings}} D_F(x_{i_1} - x_{i_2}) \cdots D_F(x_{i_{2n-1}} - x_{i_{2n}}).$
• The propagator vanishes if there are odd number of $$\varphi$$'s.

#### Dirac Field

##### Lagranian, Hamiltonian, Conservation Laws, etc.
• Dirac equation:
• $$\color{orange}(i\gamma^\mu \partial_\mu - m)\psi(x) = 0.$$
• $$\color{orange}-i\partial_\mu \overline{\psi} \gamma^\mu - m\overline{\psi} = 0.$$
• Lagrangian of Dirac field: $\color{orange} \mathcal{L} = \overline{\psi}(i\gamma^\mu \partial_\mu - m)\psi.$
• Hamiltonian: $\color{orange} H = \int \dd{^3 x} \psi^\dagger [-i\gamma^0 \vb*{\gamma}\cdot \grad + m\gamma^0] \psi.$
• Vector current: $j^\mu(x) = \overline{\psi}(x) \gamma^\mu \psi(x).$
• Divergence: $\partial_\mu j^\mu = 0.$
• $$j^\mu$$ is the Noether current of $\psi(x) \rightarrow e^{i\alpha}\psi(x).$
• The charge associated to $$j^\mu$$ is (up to a constant) $Q = j^0 = \int \frac{\dd{^3 p}}{(2\pi)^3} \sum_s \qty(a_{\vb*{p}}^{s\dagger} a_{\vb*{p}}^s - b_{\vb*{p}}^{s\dagger} b_{\vb*{p}}^s).$
• Axial vector current: $j^{\mu 5}(x) = \overline{\psi}(x) \gamma^\mu \gamma^5 \psi(x).$
• Divergence: $\partial_\mu j^{\mu 5} = 2im\overline{\psi} \gamma^5 \psi.$
• $$j^{\mu 5}$$ is the Noether current of $\psi(x) \rightarrow e^{i\alpha\gamma^5} \psi(x),$ under which the mass term is not invariant.
• Angular momentum:
• Angular momentum operator: $\vb*{J} = \int \dd{^3 x} \psi^\dagger \qty(\vb*{x}\times (-i \grad) + \frac{1}{2}\vb*{\Sigma}) \psi.$
• Angular momentum of zero-momentum fermion:
• $J_z a^{s\dagger}_{\vb*{p} = 0} \ket{0} = \pm \frac{1}{2} a_0^{s\dagger}\ket{0},$
• $J_z b^{s\dagger}_{\vb*{p} = 0}\ket{0} = \mp \frac{1}{2} b^{s\dagger}_{\vb*{p} = 0}\ket{0}.$
• The upper sign is for $$\displaystyle \xi^s = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$ and the lower sign for $$\displaystyle \xi^s = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$.
##### The Quantized Dirac Field
• Expansion using creation and annihilation operators:
• Field operators: \begin{align*} \color{orange} \psi(x) &\color{orange} = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_s \qty(a^s_{\vb*{p}} u^s(p) e^{-ip\cdot x} + b^{s\dagger}_{\vb*{p}} v^s(p) e^{ip\cdot x}), \\ \color{orange} \overline{\psi}(x) &\color{orange} = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_s \qty(b^s_{\vb*{p}} \overline{v}^s(p) e^{-ip\cdot x} + a^{s\dagger}_{\vb*{p}} \overline{u}^s(p) e^{ip\cdot x}). \end{align*}
• Hamiltonian: $\color{orange}H = \int \frac{\dd{^3 p}}{(2\pi)^3} \sum_s E_{\vb*{p}}\qty(a^{s\dagger}_{\vb*{p}} a^s_{\vb*{p}} + b^{s\dagger}_{\vb*{p}} b^s_{\vb*{p}}).$
• Momentum: $\color{orange}\vb*{P} = \int \frac{\dd{^3 p}}{(2\pi)^3} \sum_s \vb*{p} \qty(a^{s\dagger}_{\vb*{p}} a^s_{\vb*{p}} + b^{s\dagger}_{\vb*{p}} b^s_{\vb*{p}}).$
• Anticommutation relations:
• Creation and annihilation operators: $\color{darkcyan}\qty{a_{\vb*{p}}^r,a_{\vb*{q}}^{s\dagger}} = \qty{b_{\vb*{p}}^r, b_{\vb*{q}}^{s\dagger}} = (2\pi)^3 \delta^{(3)}(\vb*{p} - \vb*{q})\delta^{rs},$ with all other commutators equal to zero.
• Field operators: \begin{align*} \color{darkcyan}\qty{\psi_a(\vb*{x}), \psi^\dagger_b(\vb*{y})} &\color{darkcyan}= \delta^{(3)}(\vb*{x} - \vb*{y}) \delta_{ab}, \\ \color{darkcyan}\qty{\psi_a(\vb*{x}), \psi_b(\vb*{y})} &\color{darkcyan}= \qty{\psi_a^\dagger(\vb*{x}), \psi_b^\dagger(\vb*{y})} = 0. \end{align*}

$$\psi_\alpha(x)\ket{0}$$ contains a position as position $$x$$, while $$\overline{\psi}_\alpha\ket{0}$$ contains one electron.

##### The Dirac Propagator

The $$\operatorname{T}$$ operator has a minus sign under reversed order: $\color{red} T\psi(x) \overline{\psi}(y) = \begin{cases} \psi(x) \overline{\psi}(y), & \text{if } x^0 > y^0, \\ -\overline{\psi}(y) {\psi}(x), & \text{if } x^0 < y^0. \\ \end{cases}$

• The Green's functions solve the equation $\color{darkcyan} (i\unicode{x2215}\kern-.5em {\partial_x} - m) S(x-y) = i\delta^{(4)}(x-y) \cdot \mathbb{1}_{4\times 4}.$
• Retarded Green's function: \begin{align*} \color{orange} S^{ab}_R(x-y) &\color{orange} = \Theta(x^0 - y^0) \bra{0} \qty{\psi_a(x), \overline{\psi}(y)}\ket{0} \\ &\color{orange} = (i\unicode{x2215}\kern-.5em {\partial_x} + m)D_R(x-y). \end{align*}
• Feynman Propagator: $\color{orange} S_F(x-y) = \bra{0} T\psi(x) \overline{\psi}(y) \ket{0}.$
##### Classical Field (Dirac Field)
• Grassman field $\psi(x) = \sum_i \psi_i \phi_i(x).$
• $$\phi_i(x)$$: c-number functions.
• $$\psi_i$$: Grassman numbers.
##### Canonical Quantization (Dirac Field)

$\color{orange} \psi(x) \color{orange} = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_s \qty(a^s_{\vb*{p}} u^s(p) e^{-ip\cdot x} + b^{s\dagger}_{\vb*{p}} v^s(p) e^{ip\cdot x}),$

##### Path Integral (Dirac Field)

To obtain the correct Feynman propagator, we should follow the rule of proximity when taking derivatives.

• Functional integral: \begin{align*} \color{orange} Z[\overline{\eta},\eta] &\color{orange}= \int \mathcal{D}\overline{\psi} \mathcal{D}\psi \, \exp\qty[i \int \dd{^4 x}\qty[\overline{\psi}(i\unicode{x2215}\kern-.5em \partial - m)\psi + \overline{\eta}\psi + \overline{\psi}\eta]] \\ &\color{orange}= Z_0 \exp\qty[-\int \dd{^4 x} \dd{^4 y} \overline{\eta}(x) S_F(x-y) \eta(y)]. \end{align*}{}
• Correlation function: $$\mathcal{D}\overline{\psi}$$ identified with $$\mathcal{D}\psi$$, $\color{red} \bra{0} T\psi(x_1) \overline{\psi}(x_2)\ket{0} = \dfrac{\int \mathcal{D}\overline{\psi} \mathcal{D}\psi\, \exp\qty[i\int \dd{^4 x} \overline{\psi}(i\unicode{x2215}\kern-.5em \partial - m)\psi]\psi(x_1)\overline{\psi}(x_2)}{\int \mathcal{D}\overline{\psi}\mathcal{D}\psi\, \exp[i\int\dd{^4 x} \overline{\psi}(i\unicode{x2215}\kern-.5em \partial - m)\psi]}.$
• Obtained via $\bra{0} T\psi(x_1) \overline{\psi}(x_2) \ket{0} = \left.Z_0^{-1} \qty(-i \frac{\delta}{\delta \overline{\eta}(x_1)})\qty(+i \frac{\delta}{\delta \eta(x_2)}) Z[\overline{\eta},\eta]\right\vert_{\overline{\eta},\eta = 0}.$
• Feynman propagator: $\color{orange} S_F(x_1 - x_2) = \int \frac{\dd{^4 k}}{(2\pi)^4} \frac{ie^{-ik\cdot(x_1 - x_2)}}{\unicode{x2215}\kern-.5em k - m + i\epsilon}.$

#### Electromagnetic Field

##### Classical Field (Electromagnetic Field)
• Action: $S = \int \dd{^4 x}\qty[-\frac{1}{4}\qty(F_{\mu\nu})^2] = \frac{1}{2} \int \frac{\dd{^4 k}}{(2\pi)^4} \tilde{A}_\mu(k) (-k^2 g^{\mu\nu} + k^\mu k^\nu) \tilde{A}_\nu(-k).$
##### Canonical Quantization (Electromagnetic Field)

$\color{orange} A_\mu(x) = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_{r=0}^3 \qty{a_{\vb*{p}}^r \epsilon^r_\mu(p) e^{-ip\cdot x} + a^{r\dagger}_{\vb*{p}} \epsilon^{r*}_\mu(p) e^{ip\cdot x}}.$

• $\epsilon^\mu = (0, \vb{\epsilon}).$
• Transversality condition: $\vb*{p}\cdot \vb{\epsilon} = 0.$
##### Path Integral (Electromagnetic Field)
• Faddeev-Popov trick applied: $\int \mathcal{D}A\, e^{iS[A]} = \det\qty(\frac{1}{e}\partial^2)\qty(\int \mathcal{D}A)\int \mathcal{D}A\, e^{iS[A]} \delta(\partial^\mu A_\mu - \omega(x)).$
• We have used gauge invariance to change $S[A] \rightarrow S[A^\alpha].$
• Instead of setting $$\omega(x) = 0$$, we integrate over all $$\omega(x)$$ with a Gaussian weighting function: \begin{align*} & N(\xi) \int \mathcal{D}\omega\, \exp\qty[-i \int \dd{^4 x} \frac{\omega^2}{2\xi}] \det\qty(\frac{1}{e}\partial^2) \qty(\int \mathcal{D}\alpha) \int \mathcal{D}A\, e^{iS[A]}\delta(\partial^\mu A_\mu - \omega(x)) \\ &= N(\xi) \det\qty(\frac{1}{e}\partial^2)\qty(\int \mathcal{D}\alpha) \int \mathcal{D}A\, e^{iS[A]} \exp\qty[-i \int \dd{^4 x} \frac{1}{2\xi} (\partial^\mu A_\mu)^2]. \end{align*}{}
• Correlation function: $$\mathcal{O}(A)$$ be gauge invariant, $\color{red} \bra{\Omega} T\mathcal{O}(A) \ket{\Omega} = \dfrac{\int \mathcal{D}A\, \mathcal{O}(A) \exp\qty[i \int_{-T}^T \dd{^4 x} \qty[\mathcal{L} - \frac{1}{2\xi}(\partial^\mu A_\mu)^2]]}{\int \mathcal{D}A\, \exp\qty[i \int_{-T}^T \dd{^4 x} \qty[\mathcal{L} - \frac{1}{2\xi}(\partial^\mu A_\mu)^2]]}.$
• Feynman propagator: $\tilde{D}_F^{\mu\nu}(k) = \frac{-i}{k^2 + i\epsilon}\qty(g^{\mu\nu} - (1-\xi) \frac{k^\mu k^\nu}{k^2}).$
• Landau gauge: $$\xi = 0$$.
• Feynman gauge: $$\xi = 1$$.

#### (Weak) Gravitational Field

##### Classical Field (Gravitational Field)
• Action: $S = \int \dd{^4 x}\qty(\frac{1}{32\pi G} \mathcal{J} - \frac{1}{2}h_{\mu\nu} T^{\mu\nu}).$
• $\mathcal{J} = \frac{1}{2}\partial_\lambda h^{\mu\nu}\partial^\lambda h_{\lambda\mu} - \frac{1}{2}\partial_\lambda h^\mu_\mu \partial^\lambda h^\nu_\nu - \partial_\lambda h^{\lambda\nu} \partial^\mu h_{\mu\nu} + \partial^\nu h^\lambda_\lambda \partial^\mu h_{\mu\nu}.$
##### Canonical Quantization (Gravitational Field)

$\color{orange} h_{\rho\sigma}(x) = \frac{1}{(2\pi)^{3/2}} \int_{X_0^+} h_{\rho\sigma}(p)e^{ip\cdot x} \dd{\alpha_0^+(p)} + \frac{1}{(2\pi)^{3/2}} \int_{X_0^+} h^\dagger_{\rho\sigma}(p) e^{-ip\cdot x} \dd{\alpha_0^+(p)}.$

##### Path Integral (Gravitational Field)
• Faddeev-Popov trick applied: $S = \frac{1}{32\pi G} \int \dd{^4 x} \qty[h^{\mu\nu} K_{\mu\nu\lambda\sigma} (-\partial^2) h^{\lambda\sigma} + O(h^3)].$
• Feynman propagator: $\tilde{D}_{\mu\nu\lambda\sigma}(k) = \frac{1}{2} \frac{\eta_{\mu\lambda}\eta_{\nu\sigma} + \eta_{\mu\sigma}\eta_{\nu\lambda} - \eta_{\mu\nu}\eta_{\lambda\sigma}}{k^2 + i\epsilon}.$

#### Discrete Symmetries

• Fields under a transformation $$O$$: $U(O)^{-1} \varphi(x) U(O) = \varphi(Ox).$
• $$O$$ is conserved if $U(O)^{-1} \mathcal{L}(x) U(O) = \mathcal{L}(Ox).$
• Transformations acting on creation and annihilation operators: $U(O)^{-1} a^{s\dagger}(\vb*{p}) U(O) = \eta_a a'^{s'\dagger}(\vb*{p}').$
• $$\eta$$ is a phase factor.
• The prime should be isomorphic to $$O$$.
##### Discrete Symmetries of Scalar Field

$$\mathcal{L}$$ is invariant under:

• Parity: $P^{-1}\varphi(x) P = \varphi(\mathcal{P} x).$
• Time reversal: $T^{-1}\varphi(x) T = \varphi(\mathcal{T} x).$
• Charge conjugation: $C^{-1}\varphi(x) C = \varphi^\dagger(x).$
• For $\varphi(x) = \varphi_1(x) + i\varphi_2(x).$
##### Discrete Symmetries of Dirac Field

$$\mathcal{L}$$ is invariant under:

• Parity: $\color{darkcyan}P^{-1} \psi(x) P = -\eta \gamma^0 \psi(\mathcal{P} x).$
• $\color{darkcyan}P^{-1} a^{s\dagger}(\vb*{p})P = \eta a^{s\dagger}(-\vb*{p}).$
• $\color{darkcyan}P^{-1} b^{r\dagger}(\vb*{p})P = \eta b^{r\dagger}(-\vb*{p}).$
• $$\eta = \pm i$$.
• Time reversal: $\color{darkcyan}T^{-1} \psi(x) T = \gamma^1 \gamma^3 \psi(\mathcal{T} x).$
• $\color{darkcyan}T^{-1} a^{s\dagger}(\vb*{p}) T = s a^{-s\dagger}(-\vb*{p}).$
• $\color{darkcyan}T^{-1} b^{s\dagger}(\vb*{p}) T = s b^{-s\dagger}(-\vb*{p}).$
• $$s = \pm.$$
• Charge conjugation: $\color{darkcyan}C^{-1} \psi(x) C = -i(\overline{\psi}(x)\gamma^0 \gamma^2)^T.$
• $\color{darkcyan}C^{-1} a^{s\dagger}(\vb*{p}) C = b^{s\dagger}.$
• $\color{darkcyan}C^{-1} b^{s\dagger}(\vb*{p}) C = a^{s\dagger}.$
$$C$$ $$P$$ $$T$$ $$CPT$$
$$\overline{\psi}\psi$$ $$+1$$ $$+1$$ $$+1$$ $$+1$$
$$\overline{\psi}\gamma^5\psi$$ $$-1$$ $$-1$$ $$+1$$ $$+1$$
$$\overline{\psi}\gamma^\mu\psi$$ $$(-1)^\mu$$ $$(-1)^\mu$$ $$-1$$ $$-1$$
$$\overline{\psi}\gamma^\mu \gamma^5\psi$$ $$-(-1)^\mu$$ $$(-1)^\mu$$ $$+1$$ $$-1$$
$$\overline{\psi}\sigma^{\mu\nu}\psi$$ $$(-1)^\mu(-1)^\nu$$ $$-(-1)^\mu(-1)^\nu$$ $$-1$$ $$+1$$
$$\partial_\mu$$ $$(-1)^\mu$$ $$-(-1)^\mu$$ $$+1$$ $$-1$$

#### Examples

##### Bound States
• A general two-body system with equal constituent: \begin{align*} \vb*{R} &= \frac{1}{2}(\vb*{r}_1 + \vb*{r}_2) & \leftrightarrow \vb*{K} = \vb*{k}_1 + \vb*{k}_2, \\ \vb*{r} &= \vb*{r}_1 - \vb*{r}_2 & \leftrightarrow \vb*{k} = \frac{1}{2}(\vb*{k}_1 - \vb*{k}_2). \end{align*}
• In COMF: $$S=1$$, $$M=1$$, nonrelativistic limit, $\ket{B} = \sqrt{2M} \int \frac{\dd{^3 k}}{(2\pi)^3} \tilde{\psi}(\vb*{k}) \frac{1}{\sqrt{2m}} \frac{1}{\sqrt{2m}} \ket{\vb*{k}\uparrow, -\vb*{k}\uparrow}.$
• $\tilde{\psi}(\vb*{k}) = \int \dd{^3 x} e^{i\vb*{k}\cdot \vb*{r}} \psi(\vb*{r}).$
 Quantum Field Theory (III) Feynman Diagrams

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## Advanced Solid State Physics (I)

Applications of Second Quantization

#### Solid State Physics

##### Heat Capacity

The heat capacity may be given by $C_V = T\qty(\pdv{S}{T})_V.$ The contribution from electrons may be given by $\delta S = \frac{V}{T} \int \dd{\epsilon_{\vb*{p}}} \delta f_{\vb*{p}} N(\epsilon_{\vb*{p}})(\epsilon_{\vb*{p}} - \mu),$ where the DoS of electrons is given by $N(\epsilon_{\vb*{p}}) = \frac{p^2}{\pi^2 \hbar^3} \dv{p}{\epsilon_{\vb*{p}}}.$ The integral of $$\delta S$$ may be evaluated using the Sommerfeld expansion.

#### Second Quantization

• The basis functions of our expansion doesn't have to cooperate with the exact eigenfunctions of the Hamiltonian.
• Field operator: \begin{align*} \hat{\psi}(\vb*{x}) &= \sum_{\vb*{k}} \psi_{\vb*{k}}(\vb*{x}) c_{\vb*{k}}, \\ \hat{\psi}{\vb*{x}} &= \sum_{\vb*{k}} \psi_{\vb*{k}}(\vb*{x})^\dagger c^\dagger_{\vb*{k}}. \end{align*}
• Second-quantized operator:
• Before second-quantization: $J = \sum_{i=1}^N J(\vb*{x}_i).$
• After second-quantization: \begin{align*} \color{orange} \hat{J} &= \sum_{rs} \bra{r}J\ket{s} c_{r}^\dagger c_s \\ &\color{orange}= \int \dd{^3 \vb*{x}} \hat{\psi}^\dagger(\vb*{x}) J(\vb*{x}) \hat{\psi}(\vb*{x}). \end{align*}
• Hamiltonian:
• $\color{orange} \hat H = \sum_{rs} a_{r}^\dagger \bra{r}T\ket{s} a_s + \frac{1}{2} \sum_{rstu} a_r^\dagger a_s^\dagger \bra{rs}V\ket{tu} a_u a_t.$
• $\color{orange} \hat{H} = \int \dd{^3 \vb*{x}} \hat{\psi}^\dagger(\vb*{x}) T \hat{\psi}(\vb*{x}) + \frac{1}{2} \iint \dd{^3 \vb*{x}} \dd{^3 \vb*{x}'} \hat{\psi}^\dagger(\vb*{x}) \hat{\psi}^\dagger(\vb*{x}') V(\vb*{x},\vb*{x}') \hat{\psi}(\vb*{x}')\hat{\psi}(\vb*{x}).$
• Number-density operator: $\color{orange} n(\vb*{x}) = \hat{\psi}^\dagger(\vb*{x}) \hat{\psi}(\vb*{x}).$
• Total-number operator: $\color{orange}\hat{N} = \int \dd{^3 \vb*{x}} \hat{\psi}^\dagger(\vb*{x}) \hat{\psi}(\vb*{x}).$
• The problem in the abstract Hilbert space separates into a sequence of problems in the subspaces corresponding to a fixed total number of particles. Nevertheless, the abstract Hilbert space contains states with any number of particles.

#### Hatree-Fock Approximation

The Hamiltonian is given by $\hat{H}_e = \sum_{\nu\lambda} \bra{\nu} \hat{h}(1) \ket{\lambda} a_\nu^\dagger a_\lambda + \frac{1}{2} \sum_{\nu\lambda\alpha\beta} \bra{\nu\lambda} \frac{e^2}{\abs{\vb*{r}_1 - \vb*{r}_2}}\ket{\alpha\beta} a_\nu^\dagger a_\lambda^\dagger a_\beta a_\alpha,$ where $\hat{h}(1) = \frac{\hat{\vb*{p}}_1^2}{2m} + \hat{V}_{\mathrm{ion}}(\vb*{r}_1).$

##### Non-Interacting Limit

The ket of non-interacting electron gas is given by $\ket{\psi_0} = \ket{\vb*{p}_0 \uparrow, \vb*{p}_0\downarrow, \cdots, \vb*{p}_{\mathrm{F}} \uparrow, \vb*{p}_{\mathrm{F}} \downarrow} = a^\dagger_{\vb*{p}_0\uparrow} a^\dagger_{\vb*{p}_0\downarrow} \cdots a^\dagger_{\vb*{p}_{\mathrm{F}} \uparrow} a^\dagger_{\vb*{p}_{\mathrm{F}} \downarrow} \ket{0}.$

Now we evaluate the expectation value of $$h_1$$.

• The expectation value of kinetic energy is given by $\langle \hat{T} \rangle = 2 \sum_{p<p_{\mathrm{F}}} \frac{p^2}{2m} = \frac{3}{5} \frac{p_{\mathrm{F}}^2 }{2m} N.$
• The expectation value of ion potential energy is given by $\langle \hat V_{\mathrm{ion}} \rangle = 2\sum_{p<p_{\mathrm{F}}}V_{\mathrm{ion}}(0).$ where $V_{\mathrm{ion}}(0) = \frac{1}{V} \int \dd{\vb*{r}} V_{\mathrm{ion}}(\vb*{r}).$

The Hamiltonian may be written in second-quantized form as \begin{align*} \hat{H}_1 &= \sum_\sigma \int \dd{\vb*{r}} \qty[-\frac{\hbar^2}{2m} \psi^\dagger_\sigma (\vb*{r}) \laplacian \psi_\sigma(\vb*{r}) + \psi^\dagger_\sigma(\vb*{r}) V_{\mathrm{ion}}(\vb*{r}) \psi_\sigma(\vb*{r})] \\ &= \sum_\sigma \int \dd{\vb*{r}} \qty[ \frac{\hbar^2}{2m} \abs{\grad \psi_\sigma}^2 + \hat{n}_\sigma(\vb*{r}) \psi_{\mathrm{ion}}(\vb*{r}) ]. \end{align*} This could also be written in the orbital basis as \begin{align*} \langle \hat{H}_1 \rangle &= \sum_\nu \bra{\nu}\hat{h}(1)\ket{\nu} n_\nu \\ &= \sum_{\nu \mathrm{\ (occ.)}} \int \qty[\frac{\hbar^2}{2m}\abs{\grad \phi_\nu}^2 + \phi_\nu^*(\vb*{r}) V_{\mathrm{ion}}(\vb*{r})\phi_\nu(\vb*{r})] \dd{\vb*{r}}. \end{align*}

##### Coulomb Interaction

The two-body part of $$H$$ is given by \begin{align*} \langle \hat{H}_2 \rangle &= \bra{\psi_0} \hat{V}_{\mathrm{ee}}\ket{\psi_0} \\ &= \frac{1}{2} \sum_{\alpha,\beta,\nu,\lambda} \bra{\nu}\bra{\lambda} \frac{e^2}{\abs{\vb*{r}_1 - \vb*{r}_2}} \ket{\alpha}\ket{\beta} \langle a^\dagger_\nu a^\dagger_\lambda a_\beta a_\alpha \rangle \\ &= \frac{}{2} \sum_{\nu,\lambda} \bra{\nu}\bra{\lambda} \frac{e^2}{\abs{\vb*{r}_1 - \vb*{r}_2}} \ket{\nu}\ket{\lambda} n_\lambda n_\nu - \frac{1}{2} \sum_{\nu,\lambda} \bra{\nu}\bra{\lambda} \frac{e^2}{\abs{\vb*{r}_1 - \vb*{r}_2}} \ket{\lambda}\ket{\nu} n_\lambda n_\nu \\ &= \frac{1}{2} \sum_{\nu,\lambda \mathrm{\ (occ.)}} (U_{\nu\lambda} - \delta_{\sigma_\nu \sigma_\lambda}J_{\nu\lambda}). \end{align*} Where we have used Wick's rules and found $\langle a^\dagger_\nu a^\dagger_\lambda a_\beta a_\alpha \rangle = (\delta_{\nu\alpha}\delta_{\lambda\beta} - \delta_{\nu\beta}\delta_{\lambda\alpha}) \langle \hat{n}_\alpha \rangle \langle \hat{n}_\beta \rangle.$

• $$U_{\nu\lambda}$$ is the direct Coulomb interaction, $U_{\nu\lambda} = \int \dd{\vb*{r}_1} \dd{\vb*{r}_2} \abs{\phi_\nu(\vb*{r}_1)}^2 \frac{e^2}{\abs{\vb*{r}_1 - \vb*{r}_2}} \abs{\phi_\lambda(\vb*{r}_2)}^2.$
• $$J_{\nu\lambda}$$ is the exchange interaction, $J_{\nu\lambda} = \int \dd{\vb*{r}_1} \dd{\vb*{r}_2} \phi^*_\nu(\vb*{r}_1) \phi^*_\lambda(\vb*{r}_2) \frac{e^2}{\abs{\vb*{r}_1 - \vb*{r}_2}} \phi_\nu(\vb*{r}_2) \phi_\lambda(\vb*{r}_1).$

The total Hatree-Fock energy is given by $E_{\mathrm{HF}} = \langle \hat{H}_1 \rangle + \frac{1}{2} \sum_{\lambda,\nu \mathrm{\ (occ.)}} [U_{\nu\lambda} - U_{\nu\lambda}].$ To obtain the orbitals, we minimize the ground state energy with respect to the $$\phi$$'s, under the condition that $$\phi$$'s are normalized. We introduce a Lagrange multiplier $$\epsilon_\nu$$ and write down the Euler-Lagrange equation $\frac{\delta E_{\mathrm{HF}}}{\delta \phi_\nu^*(\vb*{r})} = \epsilon_\nu \frac{\delta}{\delta \phi^*_\nu(\vb*{r})} \int \dd{\vb*{r}'} \abs{\phi_\nu(\vb*{r})}^2,$ which yields the Hatree-Fock equations \begin{align*} \qty[-\frac{\hbar^2}{2m}\grad^2 + \hat{V}_{\mathrm{ion}}(\vb*{r}) + \sum_\lambda \int \dd{\vb*{r}'} n_\lambda(\vb*{r}')\frac{e^2}{\abs{\vb*{r} - \vb*{r}'}}]\phi_\nu(\vb*{r}) & \\ -\sum_\lambda \int \dd{\vb*{r}'} \phi_\lambda^*(\vb*{r}') \phi_\nu(\vb*{r}') \frac{e^2}{\abs{\vb*{r} - \vb*{r}'}} \phi_\lambda(\vb*{r}) &= \epsilon_\nu \phi_\nu(\vb*{r}). \end{align*} Therefore we find $\epsilon_\nu = \bra{\nu} h_1 \ket{\nu} + \sum_{\lambda \mathrm{\ occ.}} (U_{\nu\lambda} - J_{\nu\lambda}),$ and $E_{\mathrm{HF}} = \sum_{\nu \mathrm{\ (occ.)}} \epsilon_\nu - \frac{1}{2} \sum_{\lambda,\nu \mathrm{\ (occ.)}} (U_{\nu\lambda} - J_{\nu\lambda}).$

The Hatree-Fock energy $$\epsilon_\nu$$ is the energy required to add a particle in the previously unoccupied orbital $$\nu$$.

#### Interacting Electron Gas

##### Uniform Electron Gas

We evaluated the energy of uniform electron gas under the following prescription:

• The basis functions are given by $\phi_{\vb*{p}}(\vb*{r}) = \frac{e^{i\vb*{p}}\cdot \vb*{r}}{\sqrt{V}}.$
• The charge density is uniform, i.e. $\sum_\lambda n_{\lambda}(\vb*{r}') = n_{\mathrm{e}}(\vb*{r}') = n_{\mathrm{e}}.$
• The ion potential is given by $V_{\mathrm{ion}}(\vb*{r}) = -\sum_i \frac{Ze^2}{\abs{\vb*{r} - \vb*{R}_i}} \rightarrow -e^2 n_{\mathrm{e}}\int \frac{\dd{\vb*{R}}}{\abs{\vb*{r} - \vb*{R}}},$ which cancels the direct Coulomb interaction.
• The exchange interaction is given by \begin{align*} -e^2 \sum_\lambda \int \dd{\vb*{r}'} \phi_\nu(\vb*{r}') \phi^*_\lambda(\vb*{r}') \phi_\lambda(\vb*{r})\frac{1}{\abs{\vb*{r} - \vb*{r}'}} &\rightarrow -e^2 \int \frac{\dd{\vb*{p}'}\dd{\vb*{x}}}{(2\pi\hbar)^3} \frac{e^{i(\vb*{p}-\vb*{p}')\cdot \vb*{x}/\hbar}}{\abs{\vb*{x}}} \phi_{\vb*{p}}(\vb*{r}). \end{align*}

Therefore, the energy of a particle in a momentum state $$\vb*{p}$$ is given by $\epsilon(\vb*{p}) = \frac{p^2}{2m} + \epsilon_{\mathrm{exch}}(\vb*{p})$ where \begin{align*} \epsilon_{\mathrm{exch}}(\vb*{p}) &= - e^2 \int_0^{p_{\mathrm{F}}} \frac{\dd{\vb*{p}'}}{(2\pi\hbar)^3} \int \dd{\vb*{x}} \frac{e^{i(\vb*{p} - \vb*{p}')\cdot \vb*{x}/\hbar}}{\abs{\vb*{x}}} \\ &= -\frac{e^2p_{\mathrm{F}}}{\pi\hbar} \qty(1 + \frac{(p_{\mathrm{F}}^2 - p^2)}{2pp_F}\ln \abs{\frac{p+p_{\mathrm{F}}}{p - p_{\mathrm{F}}}}). \end{align*} The total Hatree-Fock energy is therefore given by \begin{align*} \color{orange}E_{\mathrm{HF}} &= 2V\int_0^{p_{\mathrm{F}}} \frac{\dd{\vb*{p}'}}{(2\pi\hbar)^3} \qty(\frac{p^2}{2m} + \frac{1}{2}\epsilon_{\mathrm{exch}}\qty(\vb*{p})) \\ &= \frac{Ne^2}{2a_0}\qty[\frac{3}{5}\qty(\frac{p_{\mathrm{F}}a_0}{\hbar})^2 - \frac{3}{2\pi}\qty(\frac{p_{\mathrm{F}}a_0}{a_0})] \\ &\color{orange}= N\qty(\frac{2.21}{r_{\mathrm{s}}^2} - \frac{0.916}{r_{\mathrm{s}}}) \operatorname{Ry}, \end{align*} where $r_{\mathrm{s}} = \qty(\frac{9\pi}{4})^{1/3} \frac{e^2}{\hbar v_{\mathrm{F}}}.$

Introducing $F(x) = \frac{1}{2} + \frac{1-x^2}{4x} \ln \abs{\frac{1+x}{1-x}}$ we find $\frac{\epsilon(\vb*{p})}{\epsilon_{\mathrm{F}}^0} = x^2 - 0.663 r_{\mathrm{s}} F(x),$ where $x = \frac{p}{p_{\mathrm{F}}}.$

The exchange energy is negative because electrons of like spin avoid one another.

##### Hatree-Fock Excitation Spectrum

The DoS evaluates to $N(\epsilon) = \frac{p_{\mathrm{F}}^2}{\pi^2 \hbar^3} \pdv{p}{\epsilon} \approx \frac{mp_{\mathrm{F}}/\pi^2 \hbar^2}{1 - (me^2/\pi\hbar p_{\mathrm{F}})\ln\abs{(p-p_{\mathrm{F}})/2p_{\mathrm{F}}}}.$ With $$p^2/2m - p_{\mathrm{F}}^2/2m \approx T$$ we find $C_V \sim \frac{T}{\ln T},$ which deviates from the (correct) linear dependence, because the screening of Coulomb potential was not taken into account.

Hatree-Fock approximation overestimate the energy bandwidth to $\Delta = \epsilon(x=1) - \epsilon(x=0) = \epsilon_{\mathrm{F}}^0(1+0.331r_{\mathrm{s}}),$ while in the free-electron model $$\Delta = \epsilon_{\mathrm{F}}^0$$ fits well with experimental data.

##### Cohesive Energy of Metals

The cohesive energy is defined by $\epsilon_{\mathrm{coh}} = (\epsilon/\mathrm{atom})_{\mathrm{matal}} - (\epsilon/\mathrm{atom})_{\mathrm{free}}.$ We write the cohesive energy as $\epsilon_{\mathrm{coh}} = \epsilon_0 - \epsilon_{\mathrm{atom}} + \epsilon_{\mathrm{kin}} + \epsilon_{\mathrm{coul}}.$ Using the Wigner-Seitz method for alkali metals, we find the following:

• $$\epsilon_0$$ is given by $\qty{\frac{\hat{p}^2}{2m} + V_{\mathrm{ion}}} \varphi_0(\vb*{r}) = \epsilon_0 \varphi_0(\vb*{r})$ subjected to the boundary condition $\varphi_0'(\vb*{r}) = 0,$ which may be estimated to be $\epsilon_0 \approx -n_e \int_{a_0}^{r_{\mathrm{e}}} \dd{\vb*{r}} \frac{e^2}{r} = -\frac{40.82\, \mathrm{eV}}{r_{\mathrm{s}}}\qty(1-\frac{1}{r_{\mathrm{s}}^2}).$
• The kinetic energy may be given by $\epsilon_{\mathrm{kin}}^0 = \frac{3}{5} \frac{p_{\mathrm{F}}^2}{2m} = \frac{2.21}{r_{\mathrm{s}}^2} \operatorname{Ry}.$
• $$\epsilon_{\mathrm{coul}} = \epsilon_{\mathrm{direct}} + \epsilon_{\mathrm{exch}}$$ where $\epsilon_{\mathrm{direct}} = \int_0^{r_{\mathrm{e}}} \dd{\vb*{r}} \frac{e^2}{r} n_{\mathrm{e}} n(r) = \frac{6}{5r_{\mathrm{s}}}\operatorname{Ry}$ where $$n(r) = 4\pi r^3 n_{\mathrm{e}}/3$$ and $\epsilon_{\mathrm{exch}} = -\frac{0.916}{r_{\mathrm{s}}}\operatorname{Ry}.$

Using perturbation theory we add the correlation energy \begin{align*} \epsilon_{\mathrm{corr}} &\approx \begin{cases} \displaystyle -\frac{0.884}{r_{\mathrm{s}}} \operatorname{Ry},& r_{\mathrm{s}} \rightarrow \infty, \\ \displaystyle 0.062\ln r_{\mathrm{s}} - 0.096, & r_{\mathrm{s}} \rightarrow 0, \end{cases} \end{align*} which yields more accurate results.

##### Wigner Solid
• The Hatree-Fock approximation applies at $$r_{\mathrm{s}}\rightarrow 0$$.
• In the low-density limit $$r_{\mathrm{s}}\rightarrow \infty$$, we should use the Wigner solid model by allowing electrons to crystallize in a Wigner solid.
• Energy $\color{orange} \frac{E}{N} = \frac{e^2}{2a_0}\qty{-\frac{1.79}{r_{\mathrm{s}}} + \frac{2.66}{r_{\mathrm{s}}^{3/2}} + \cdots}.$

Expand

## PyVISA Programming

#### Instrument Controller

The InstrumentController communicates with instruments. It could be initialized with

import instrument_control
instrument_controller = instrument_control.InstrumentController()

##### List Instruments
instrument_controller.handle({"function": "list"})
# ["ASRL1::INSTR", "GPIB0::12::INSTR", "GPIB0::17::INSTR"]

##### Connect to Instrument
{"function": "open", "name": "H", "address": "GPIB0::12::INSTR"}
# {"instruments": "{'H': <'GPIBInstrument'('GPIB0::12::0::INSTR')>}"}

##### Write Command to Instrument
instrument_controller.handle({"function": "write", "name": "H", "command": "*IDN?"})
# {"wrote": "*IDN?"}

instrument_controller.handle({"function": "read", "name": "H"})

##### Query from Instrument
instrument_controller.handle({"function": "query", "name": "H", "command": "*IDN?"})

##### Query Status Byte
instrument_controller.handle({"function": "stb", "name": "H"})
# {"name": "H", "status": 0}

instrument_controller.handle({"function": "dataReady", "name": "E"})


Once dataReady is queried, further query will yield false.

#### Interacting with Keithley-4200

##### Channel Definition Page (DE)

This page is entered via the command DE.

CH command

CHA, 'BBBBBB', 'CCCCCC', D, E

• A: SMU channel number
• BBBBBB: voltage name
• CCCCCC: current name
• D: mode
• 1: voltage source mode
• 2: current source mode
• 3: common
• E: source function
• 1: VAR1 sweep source function
• 2: VAR2 step source function
• 3: constant (fixed) source function
• 4: VAR1' source function

Notes:

• If no parameters are supplied after the channel number, the channel is turned off.
• When the source mode D is set to common 3, the source function E must be set to constant 3.
• The behaviors of source functions are listed below.
• The VAR1 source function performs a linear or logarithmic sweep that is synchronized to the steps of VAR2. The VAR1 sweep is performed whenever VAR2 goes up to a new value.
• The constant source function outputs a fixed source value.
• The VAR1' source function is similar to the VAR1 function, except that each sweep step is scaled by the Ratio value (RT) and an Offset (FS) as $\text{VAR1' sweep step} = (\text{VAR1 sweep step} \times \text{RT}) + \text{FS}.$
• The names BBBBBB and CCCCCC could have up to 6 characters.
CH3, 'V1', 'I1', 1, 3


sets SMU3 to source a fixed voltage (1V). The specified names for voltage and current are V1 and I1, respectively.

CH1; CH2; CH3; CH4


disables channels 1 through 4.

##### Source Setup Page (SS)

This page is entered via the command SS.

VR and IR command sets up the VAR1 source function

AAB, [+-]CCC.CCCC, [+-]DDD.DDDD, [+-]EEE.EEEE, [+-]FFF.FFFF

• AA: the source mode
• VR: voltage source
• IR: current source
• B: the sweep type
• 1: linear sweep
• 2: $$\log_{10}$$ sweep
• 3: $$\log_{25}$$ sweep
• 4: $$\log_{50}$$ sweep
• CCC.CCCC: start value
• DDD.DDDD: stop value
• EEE.EEEE: step value (linear sweep only)
• for a log sweep, do not set a step value
• FFF.FFFF: compliance value (linear sweep only)
• for a log sweep, do not set a compliance value

Notes:

• Start value CCC.CCCC, stop value DDD.DDDD, step value EEE.EEEE and compliance value FFF.FFFF should not exceed
• $$\pm 210.00\, \mathrm{V}$$ for a voltage source,
• $$\pm 0.1050\, \mathrm{A}$$ for a current source on 4200-SMU, and
• $$\pm 1.0500\, \mathrm{A}$$ on 4210-SMU.
• Voltage start or stop value below $$0.001\,\mathrm{V}$$ will be automatically sets to zero.
• The maximum number of points for VAR1 is 1024, where $\text{Number of points} = \lfloor \frac{\abs{\text{Stop value} - \text{Start value}}}{\text{Step Value}} + 1.5 \rfloor.$
• Compliance current is set for a voltage source, while compliance voltage is set for a current source.
• A compliance below the minimum allowable value is set to the minimum allowable value.
• With a logarithmic sweep mode selected, only the start and stop values should be specified.
VR1, 1, 5, 1, 0.01


sets up a VAR1 linear sweep with a start $$1\,\mathrm{V}$$, stop $$5\,\mathrm{V}$$, step $$1\,\mathrm{V}$$ and compliance $$10\,\mathrm{mA}$$.

VP and IP command sets up the VAR2 step sweep

AA [+-]BBB.BBBB, [+-]CCC.CCCC, DD, [+-]EEE.EEEE [, FF]

• AA: the source mode
• VP: voltage source
• IP: current source
• BBB.BBBB: start value
• CCC.CCCC: step value
• DD: number of steps
• EEE.EEEE: compliance value
• FF: VAR2 source stepper index (1 to 4)

Notes:

• Start value BBB.BBBB, stop value CCC.CCCC, and compliance value EEE.EEEE are subjected to the limit stated in the VR command.
• The stepper index FF may be omitted, in which case the default value is 1. If multiple VAR2 sources are defined via the CH command, the first one defined will have index 1, the second one 2, etc.
VP 5, 5, 3, 0.01


sets up a VAR2 voltage sweep with a start $$5\,\mathrm{V}$$, step $$5\,\mathrm{V}$$, in $$3$$ steps, and a compliance $$10\,\mathrm{mA}$$.

FS command sets the offset value when VAR1' is a selected source function

FS [+-]AAA.A [,B]

• AAA.A: offset value
• B: SMU channel; offset applies to all channels configured to VAR1' if B is not applied

Notes:

• The offset value AAA.A should not exceed 210.

RT command sets the ratio value when VAR1' is a selected source function

RT [+-]AAA.A [, B]

• AAA.A ratio value
• B: SMU channel; ratio applies to all channels configured to VAR1' if B is not applied
RT +3,2
FS +2,2


sets up the VAR1' sweep with ratio 3 and offset 2 for Channel 2.

VC and IC command sets up the SMU to output a fixed (constant) voltage or current level

AAB, [+-]CCC.CCCC, [+-]DDD.DDDD

• AA: the source mode
• VC: voltage source
• IC: current source
• B: SMU channel number
• CCC.CCCC: output value
• DDD.DDDD compliance value

Notes:

• Output value CCC.CCCC and compliance value DDD.DDDD are subjected to the limit stated in the VR command.

HT command sets a hold time that delays the start of a sweep

HT AAA.A

• AAA.A: hold time in seconds (0 to 655.3)

DT command sets the time duration spent on each step of the sweep for a VAR1 sweep

DT A.AAA

• A.AAA: delay time in seconds (0 to 6.553)
##### Measurement Setup Page (SM)

This page is entered via the command SM.

WT command delays the start of a test sequence for time domain measurements.

WT AAA.AAA

• AAA.AAA: wait time in seconds (0 to 100)

Notes:

• For time domain measurements, the test sequence could be delayed by setting a wait time. Then the test sequence starts after the wait time period expires.
WT 0.1


sets the wait time to 100 ms.

IN command sets the time interval between sample measurements

IN AA.AA

• AA.AA: interval in seconds (0.01 to 10)

Notes:

• After a sample measurement is made, the next measurement starts after the time interval expires.
IN 0.1


sets the interval to 100 ms.

NR command sets the number of readings that can be made for time domain measurements.

NR AAAA

• AAAA: number of measurements to make

Notes:

• For 4200 command set, the number of measurements AAAA ranges from 1 to 4096.
NR 200


sets up the 4200A-SCS to make 200 sample measurements.

DM command selects the Keysight 4145B display mode

DMA

• A: display modes
• 1: graphics display mode
• 2: list display mode
DM1


prepares the 4200A-SCS to receive graphics commands.

LI command enables voltage and current functions to be measured when the 4200A-SCS is in list display mode

LI 'AAAAAA'
LI 'AAAAAA', 'AAAAAA'
LI 'AAAAAA', 'AAAAAA', 'AAAAAA'
LI 'AAAAAA', 'AAAAAA', 'AAAAAA', 'AAAAAA'
LI 'AAAAAA', 'AAAAAA', 'AAAAAA', 'AAAAAA', 'AAAAAA'
LI 'AAAAAA', 'AAAAAA', 'AAAAAA', 'AAAAAA', 'AAAAAA', 'AAAAAA'

• AAAAAA: the name assigned for CH, VS, or VM
LI 'V1', 'I1', 'VS1'


enables V1, I1 and VS1 to be measured.

##### Measurement Control Page (MD)

This page is entered via the command MD.

ME command controls measurements

MEA

• A: measurement action
• 1: SingleTrigger test, store readings in cleared buffer
• 2: RepeatTrigger test, store readings in cleared buffer
• 3: AppendTrigger test, append readings to buffer
• 4: StopAbort test
ME1


triggers the start of the test and stores the readings in the cleared buffer.

##### General Commands

The following commands are valid in any pages.

BC command clears all readings from the buffer

BC


Notes:

• BC also clears bit B0 (Data Ready) of the status byte.

DR command enables or disables service request for Data Ready.

DRA

• A: set service request for Data Ready
• 0: disable service request for data ready
• 1: enable service request for data ready

DO command requests readings

DO 'AAAAAA'

• AAAAAA: user-specified name of the channel that made the measurement

Notes:

• The prefix in the readings are listed below.
• N: normal
• I: interval too short
• V: overflow reading
• X: oscillation
• C: this channel in compliance
• T: other channel in compliance
DO 'Volt'


requests the reading string for a SMU channel that is named Volt.

IT command sets the integration time

ITA
IT4, X, Y, Z

• A: integration time
• 1: short (0.1 PLC)
• 2: medium (1.0 PLC)
• 3: long (10 PLC)
• 4: custom (4200A command set only)
• X: delay factor for custom setting (0.0 to 100)
• Y: filter factor for custom setting (0.0 to 100)
• Z: A/D converter integration time in number of PLCs for custom setting (0.01 to 10.0)
IT2


sets integration time to 1.0 PLC.

IT4, 2.5, 0.6, 1.3


sets the delay factor to 2.5, the filter factor to 0.6, and the A/D converted integration time to 1.3 PLCs.

*RST command resets the instrument settings to default settings

*RST


Notes:

• This command returns the instrument to default settings, cancels all pending commands, etc.

RG command sets the lowest current range to be used when measuring.

RG A,B

• A: SMU channel number
• B: the lowest autoranged range

Notes:

• The lowest autoranged range is subjected to the following limits:
• 4200-SMU without a preamplifier: -100e-9 to 100e-3 A,
• 4200-SMU with a preamplifier: -1e-12 to 100e-3 A,
• 4210-SMU without a preamplifier: -100e-9 to 1 A,
• 4210-SMU with a preamplifier: -1e-12 to 1 A.
RG 2, 10E-12


sets the lowest range of SMU2 with a preamplifier to $$10\,\mathrm{pA}$$.

Expand

## Mathematica Tricks

A collection of operators may be found here.

#### Pattern

(* _ matches any expression *)
f[x_] := x^2;
f[3] (* 9 *)

(* _h matches expression with head h *)
g[r_Rational | n_Integer] := 0;
g[x_Real] := 1;
g[0.5] (* 1 *)
g[1] (* 0 *)
g[1/2] (* 0 *)

(* x_?test matches expression that passes the test *)
h[x_Integer?(#>=0&)] := x;
h[x_Integer?(#<0&)] := -x;
h[3] (* 3 *)
h[-3] (* -3 *)

(* x_/;test also matches expression that passes the test *)
h[x_Integer?x>=0] := x;
h[x_Integer?x<0] := -x;
h[3] (* 3 *)
h[-3] (* -3 *)

(* f[x_:default] assigns default value to x *)
f[a_, b_:1] := a^b;
f[2, 3] (* 8 *)
f[2]    (* 2 *)

(* another way to assign default value *)
Default[f] = 0;
f[x_., y_.] = {x, y};
f[a] (* {a, 0} *)
f[]  (* {0, 0} *)

(* -> for replacement *)
rule1 = x_Symbol?(ContainsNone[{Plus, Times, Derivative}, {#}] &) ->
x';
rule2 = x_ + y_ -> x ** y;
rule3 = x_ y_ -> x + y;
rule4 = x_ ** y_ -> x y;
rule5 = x_'' -> x;

(x (a + b + c (d + e)) + y (z' + u' v' w')) /. rule1 //. rule2 //.
rule3 //. rule4 //. rule5
(* (z (u + v + w) + y') (a' b' (c' + d' e') + x') *)

(* x_ to match One argument *)
f[x_] := x;
f[]     (* f *)
f[1]    (* 1 *)
f[1, 2] (* f[1,2] *)

(* x__ to match One or More arguments *)
g[x__] := {x};
g[]     (* g[] *)
g[1]    (* {1} *)
g[1, 2] (* {1, 2} *)

(* x___ to match Zero, One or More arguments *)
h[x___] := {x}
h[]     (* {} *)
h[1]    (* {1} *)
h[1, 2] (* {1, 2} *)


#### Simplification

Assumptions = {a > 0, r > 0, (x | y | z) ∈ Reals}; Sqrt[a^2] // Simplify (* a *) Sqrt[x^2] // Simplify (* Abs[x] *) Abs'[x] // Simplify (* Sign[x] *) Grad[s[Norm[{x, y, z}]], {x, y, z}] /. {x_^2 + y_^2 + z_^2 -> r^2} // FullSimplify (* { x s'[r] / r, y s'[r] / r, z s'[r] / r } *)  ComplexExpand[(x + I y)^2] (* x^2 - y^2 *)  #### Integral To type $\int_a^b f(x) \dd{x},$ we have the following shortcuts: • <Esc> int <Esc> for $$\int$$, • <Ctrl+_> for the lower bound, • <Ctrl+%> for the upper bound, and • <Esc> dd <Esc> for $$\dd{}$$. #### Map (* f@x is equivalent to f[x] *) Sqrt@4 (* 2 *) Total@{1,2,3} (* 6 *) (* f@@x is equivalent to Apply[f, x] *) (* i.e. f@@{a, b, c} is equivalent to f[a, b, c] *) Solve@@{x + 1 == 0, x} (* {{x -> -1}} *) (* f@@@{ ... } applies on the first level *) f@@@{{a, b}, {c, d}} (* {f[a, b], f[c, d]} *) (* f/@{a, b, c, d, e} is equivalent to {f[a], f[b], f[c], f[d], f[e]} *) #^2&/@{1, 2, 3} (* {1, 4, 9} *) f /@ <|"a" -> x, "b" -> y|> (* <|"a" -> f[x], "b" -> f[y]|> *) (* f@*g composites functions *) f@*g@*h@x (* f[g[h[x]]] *) (* f/*g composites functions in reversed order) f/*g/*h@x (* h[g[f[x]]] *)  #### Association (Dictionary) dict = <|"a" -> x, "b" -> y|>; dict["a"] (* x *) {#b, 1 + #b} & [<|"a"->x, "b" -> y |>] (* {y, 1 + y} *) <|"a" -> x, "b" -> {5, 6}|>[["b", 1]] (* 5 *)  #### Matrix of Symbols element[A_String, i_Integer, j_Integer] := Symbol[A <> ToString@i <> "x" <> ToString@j]; matrix[A_String, d_Integer] := Table[element[A, i, j], {i, d}, {j, d}]; sol = Solve[{matrix["A", 2].matrix["A", 2] == PauliMatrix[1]}, Flatten[matrix["A", 2]]] // FullSimplify; matrix["A", 2] /. sol[[1]] // MatrixForm  Expand ## Special Relativity #### Principle of Relativity ##### Newtonian Mechanics Symmetry has two meanings in physics. • Symmetry of a solution: • the solution $$X(t) = 0$$ is invariant under $$t \rightarrow t+c$$; • the solution $$X(t) = A\cos\omega t$$ is invariant under $$t \rightarrow t + 2\pi n/\omega$$. • Symmetry of action: • the action $$L = \dot{x}^2/2$$ is invariant under $$x \rightarrow x+c$$. • the action $$L = \dot{x}^2/2 - \omega x^2/2$$ is invariant under $$x \rightarrow -x$$. Galileo's Relativity Principle states that the Lagranian is invariant under Galileo transformations. ##### Special Relativity The Poincaré transformations includes • $$t \rightarrow t+c$$, • $$\mathbb{r} \rightarrow \vb*{r} + \vb*{a}$$, • $$\vb*{r}' = O\vb*{r}$$, and • Lorentz boost \begin{align*} \vb*{r}' &= \vb*{r} + \gamma \vb*{v} t + \frac{\gamma-1}{v^2} (\vb*{r}\cdot \vb*{v})\cdot \vb*{v}, \\ t' &= \gamma\qty(t + \frac{\vb*{r}\cdot \vb*{v}}{c^2}). \end{align*} Under such transformations, \begin{align*} \delta S &= \int \delta L(u^2)\dd{t} + \int L(u^2) \delta \dd{t} \\ &= \int \dd{t} \qty(2 \dv{L(u^2)}{(u^2)} \qty(1-\frac{u^2}{c^2}) + L\qty(u^2)) \dot{\vb*{r}} \cdot \vb*{\epsilon}, \end{align*} which demands $L = A + B\sqrt{1-\frac{u^2}{c^2}}.$ Finally we obtain $S = -mc^2 \int \sqrt{1-\frac{u^2}{c^2}}\dd{t}.$ For many particles, $S = \int \dd{t} \qty{-\sum_i \qty[mc^2 \sqrt{1-\frac{u_i^2}{c^2}} + V\qty(\vb*{r}_i, \vb*{u}_i, t)]}.$ For a particle in a electromagnetic field, we have $L = -mc^2 \sqrt{1-\frac{v^2}{c^2}} - q\phi + q\dot{\vb*{r}}\cdot \vb*{A}.$ We obtain from the Euler-Lagrange equation that $\dv{}{t}\qty(m\gamma \dot{\vb*{r}}) = q\vb*{E} + q \dot{\vb*{r}}\times \vb*{B}.$ In curvilinear coordinates we have $L = \sqrt{1-g_{ij} \dv{y_i}{t} \dv{y_j}{t}}.$ In covariant form we write $S = -m \int \dd{\tau} = -m \int \sqrt{-\eta_{\mu\nu} \dd{x^\mu} \dd{x^\nu}}.$ The invariance of $$\dd{s^2}$$ may be written as $\eta_{\mu\nu} {\Lambda}{^\mu}{_\rho} {\Lambda}{^\nu}{_\sigma} = \eta_{\rho\sigma},$ or $\Lambda^T \eta \Lambda = \eta,$ which demands that $$\abs{\Lambda} = \pm 1$$. Poincarétransformations are given by ${X}{^{\mu'}} = {\Lambda}{^\mu}{_\nu} X^\nu + a^\mu.$ For a massless particle, we have $\dd{s^2} = -\dd{t}^2 + \dd{x}^2$ in any frame. Therefore, the particle travels at the speed of light in any frame. It could be proven that the invariance of $$\dd{s^2}$$ demands that the transformation matrix be constant (independent on $$x$$). Since $\eta_{\mu\nu}{\Lambda}{^\mu}{_0} {\Lambda}{^\nu}{_0} = \eta_0,$ we find that $(\Lambda{^0}{_0})^2 = 1 + \sum_i (\Lambda{^i}_0)^2.$ Therefore, the Lorentz transformations may be classified as follows:  $$\abs{\Lambda} = 1$$ (Proper, $$\operatorname{SO}{(1,3)}$$) $$\abs{\Lambda} = -1$$ $$\Lambda{^0}{_0} \ge 1$$ (Orthochronous) Restricted, $$\operatorname{SO}{(1,3)}^\uparrow$$ Parity $$\Lambda{^0}{_0} \le -1$$ Time Reversal A Lorentz boost may be regarded as a $$t$$-$$x$$ rotation, i.e. ${\Lambda}{^\mu}{_\nu} = \begin{pmatrix} \cosh \phi & \sinh \phi & 0 & 0 \\ \sinh \phi & \cosh \phi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix},$ where $$v = \tanh \phi$$ and $$\gamma = \cosh\phi$$. A more general boost may be written as \begin{align*} \Lambda{^0}{_0} &= \gamma, \\ \Lambda{^i}{_0} &= \gamma v^i, \\ \Lambda{^0}{_j} &= \gamma v^j, \\ \Lambda{^i}{_j} &= \delta_{ij} + v^i v^j \frac{\gamma-1}{v^2}. \end{align*} #### Lorentz Covariant Formulation ##### Energy and Momentum Definitions and properties of the 4-velocity and 4-momentum are listed below. • For a massive particle: $U^\mu = \dv{x^\mu}{\tau}.$ • $$U_\mu U^\mu = -1$$. • In the rest frame, $U^\mu = (1,0,0,0).$ • In the observer frame, $U^\mu = (\gamma, \gamma\vb*{u}).$ • $$p^\mu = mU^\mu$$. • In the observer frame, $p^\mu = (E,\vb*{p}) = (m\gamma, m\gamma\vb*{u}).$ • In particular, $\vb*{p} = E\vb*{u}.$ • Dispersion relation: $E^2 = \vb*{p}^2 + m^2.$ For massless particles, we write the action $S = \frac{1}{2} \int \dd{\lambda} \eta_{\mu\nu} \dv{x^\mu}{\lambda} \dv{x^\nu}{\lambda}$ and $L = \frac{1}{2} \eta_{\mu\nu} \dv{x^\mu}{\lambda} \dv{x^\nu}{\lambda}.$ The conjugate momentum is $p_\mu = \pdv{L}{\displaystyle\qty(\pdv{x^\mu}{\lambda})} = \eta_{\mu\nu} \dv{x^\mu}{\lambda}.$ A few properties of massless particles are listed below. • $$p^\mu p_\mu = 0$$. • $$p^\mu = (E, \vb*{p}) = (\abs{\vb*{p}}, \vb*{p})$$. • $$\vb*{u} = 1$$, $$\vb*{p} = E\vb*{u}$$. ##### Particle Dynamics Properties of the 4-acceleration and 4-force are listed below. • Definition of 4-acceleration: $a^\mu = \dd{U^\mu}{\tau} = \dd[2]{x^\mu}{\tau^2}.$ • In the rest frame, $a^\mu = (0, \vb*{a}).$ • Definition of 4-force: $f^\mu = m a^\mu.$ • In the rest frame $F^0 = 0,\quad \vb*{F} = \dv{\vb*{p}}{t}.$ • In the observer frame $f^0 = \gamma \dv{m\gamma}{t},\quad \vb*{f} = \gamma \dv{\vb*{p}}{t}.$ • The 4-forces in the rest frame and the observer frame are related by \begin{align*} f^0 &= \gamma \vb*{v}\cdot \vb*{f} = \vb*{v}\cdot \vb*{f},\\ \vb*{f} &= \vb*{F} + \frac{\gamma-1}{\gamma} (\vb*{v}\cdot \vb*{F}) \vb*{v}. \end{align*} Expand ## General Relativity (II) Riemannian Manifold #### Riemann Manifolds ##### Metric Tensors Let $$M$$ be a differentiable manifold. A Riemannian metric $$g$$ on $$M$$ is a type $$(0,2)$$ tensor field on $$M$$ which satisfies the following axioms at each point $$p\in M$$: • $$g_p(U,V) = g_p(V,U)$$; • $$g_p(U,U) \ge 0$$, where the equality holds only when $$U=0$$. A tensor field $$g$$ of type $$(0,2)$$ is a pseudo-Riemiann metric if • $$g_p(U,V) = g_p(V,U)$$; • if $$g_p(U,V) = 0$$ for any $$U\in T_p M$$ then $$V=0$$. $$g_p(U,)$$ gives rise to a one-form. Thus $$g_p$$ gives rise to an isomorphism between $$T_p M$$ and $$T_p^* M$$. We adopt the following notations: • $$g^{\mu\nu}$$ is the inverse of $$g_{\mu\nu}$$; • $$g$$ denotes $$\det(g^{\mu\nu})$$. If there are $$i$$ positive eigenvalues and $$j$$ negative eigenvalues, the pair $$(i,j)$$ is called the index of the metric. • The Euclidean metric is defined by $$\delta = \operatorname{diag}{(1,\cdots,1)}$$. • The Minkowski metric is defined by $$\eta = \operatorname{diag}{(-1,1,\cdots,1)}$$. If $$(M,g)$$ is Lorentzian, the elements of $$T_p M$$ are divided into three classes as follows. 1. spacelike if $$g(U,U)>0$$; 2. lightlike or null if $$g(U,U) = 0$$; 3. timelike if $$g(U,U) < 0$$. $$(M,g)$$ is a • Riemannian manifold if $$g$$ is Riemannian; • pesudo-Riemannian manifold if $$g$$ is pseudo-Riemannian; • Lorentz if $$g$$ is Lorentzian. ##### Induced Metric If $$f: M \rightarrow N$$ is the embedding which induces the submanifold structure of $$M$$, the pullback map $$f^*$$ induces the natural metric $$g_M = f^* g_N$$ on $$M$$. The components are given by $g_{M\mu\nu} (x) = g_{N \alpha \beta}(f(x)) \pdv{f^\alpha}{x^\mu} \pdv{f^\beta}{x^\nu}.$ #### Parallel Transport, Connection and Covariant Derivative ##### Affine Connections An affine connection $$\grad$$ is a map $$\grad: \mathcal{X}\times \mathcal{X} \rightarrow \mathcal{X}$$, or $$(X,Y)\rightarrow \grad_X Y$$ which satisfies the following conditions: • $$\grad_X (Y+Z) = \grad_X Y + \grad_X Z$$; • $$\grad_{X+Y} Z = \grad_X Z + \grad_Y Z$$; • $$\grad_{fX} Y = f\grad_X Y$$; • $$\grad_X (fY) = X[f]Y + f\grad_X Y$$. The connection coefficients are defined by $\grad_\nu e_\mu = e_\lambda {\Gamma}{^\lambda}_{\nu\mu}.$ The covariant derivative of any vectors are given by $\grad_V W = V^\mu \qty(\pdv{W^\lambda}{x^\mu} + W^\nu {\Gamma}{^\lambda}_{\mu\nu}) e_\lambda,$ or $\grad_\mu W^\lambda = \pdv{W^\lambda}{x^\mu} + {\Gamma}{^\lambda}_{\mu\nu} W^\nu.$ Note • $$\grad_\mu W^\lambda$$ is the $$\lambda$$th component of a vector $$\grad_\mu W$$ and should not be confused with the covariant derivative of a component $$W^\lambda$$. • $$\grad_V W$$ is independent of the derivative of $$V$$, unlike the Lie derivative $$\mathcal{L}_V W = [V,W]$$. ##### Parallel Transport and Geodesics If a vector field $$X$$ satisfies the condition $\grad_V X = 0,$ where $$\displaystyle V = \dv{}{t}$$ is the tangent vector to $$c(t)$$, then $$X$$ is said to be parallel transported along $$c(t)$$. The condition is written in terms of the components as $\dv{X^\mu}{t} + {\Gamma}{^\mu}_{\nu\lambda} \dv{x^\nu}{t} X^\lambda = 0.$ If the tangent vector of $$c(t)$$ itself is parallel transported along $$c(t)$$, i.e. $\dv[2]{x^\mu}{t} + {\Gamma}{^\mu}_{\nu\lambda} \dv{x^\nu}{t} \dv{x^\lambda}{t} = 0,$ then $$x(t)$$ is called the geodesic. More generally, the equation should be written as $\grad_V V = fV.$ ##### The Covariant Derivative of Tensor Fields We require that $\grad_X (T_1 \otimes T_2) = (\grad_X T_1)\otimes T_2 + T_1 \otimes (\grad_X T_2).$ For one-forms we define $(\grad_X \omega)_\nu = X^\mu \partial_\mu \omega_\nu - X^\mu {\Gamma}^{\lambda}_{\mu\nu} \omega_\lambda,$ or $(\grad_\mu \omega)_\nu = \partial_\mu \omega_\nu {\Gamma}^{\lambda}_{\mu\nu}\omega\lambda,$ especially for the basis we have $\grad \dd{x^\nu} = -{\Gamma}{^\nu}_{\mu\lambda} \dd{x^\lambda}.$ For a general tensor we have \begin{align*} \grad_\nu t^{\lambda_1 \cdots \lambda_p}_{\mu_1 \cdots \mu_q} &= \partial_\nu t^{\lambda_1 \cdots \lambda_p}_{\mu_1 \cdots \mu_q} \\ &{\phantom{{}={}}} + {\Gamma}{^{\lambda_1}}_{\nu\kappa} t^{\kappa\lambda_2 \cdots \lambda_p}_{\mu_1 \cdots \mu_q} + \cdots + {\Gamma}{^{\lambda_p}}_{\nu\kappa} t^{\lambda_1 \cdots \lambda_{p-1}\kappa}_{\mu_1 \cdots \mu_q} \\ &{\phantom{{}={}}} - {\Gamma}{^\kappa}_{\nu\mu_1} t^{\lambda_1 \cdots \lambda_p}_{\kappa\mu_2 \cdots \mu_q} + \cdots - {\Gamma}{^\kappa}_{\nu\mu_q} t^{\lambda_1 \cdots \lambda_p}_{\mu_1 \cdots \mu_{q-1}\kappa}. \end{align*} ##### The Transformation Properties of Connection Coefficients To make $$\grad_V W$$ transform like a vector, the connection coefficients should satisfy $\tilde{\Gamma}{^\gamma}_{\alpha\beta} = \pdv{x^\lambda}{y^\alpha} \pdv{x^\mu}{y^\beta} \pdv{y^\gamma}{x_\nu} {\Gamma}{^\nu}_{\lambda\mu} + \frac{\partial^2 x^\nu}{\partial y^\alpha \partial y^\beta} \pdv{y^\gamma}{x^\nu}.$ ##### The Metric Connection We demand that if $$X$$ and $$Y$$ are parallel transported along any curve, then $$g(X,Y)$$ is constant along the curve, i.e. $(\grad_\kappa g)_{\mu\nu} = 0.$ We make a few definitions here: • the torsion tensor by ${T}{^\kappa}_{\lambda\mu} = {\Gamma}{^\kappa}_{\lambda\mu} - {\Gamma}{^\kappa}_{\mu\lambda};$ • and we have $${T}{^\kappa}_{\lambda\mu} = -{T}{^\kappa}_{\mu\lambda}$$; • the Christoffel symbols by $\binom{\kappa}{\mu\nu} = {\Gamma}{^\kappa}_{(\mu\nu)} = \frac{1}{2} g^{\kappa\lambda}\qty(\partial_\mu g_{\nu\lambda} + \partial_\nu g_{\mu\lambda} - \partial_\lambda g_{\mu\nu});$ • the cotorsion tensor by ${K}{^\kappa}_{\mu\nu} = \frac{1}{2} ({T}{^\kappa}_{\mu\nu} + {T}{_\nu}{^\kappa}{_\nu} + {T}{^\kappa}{_\mu}{_\nu});$ • and we have $$\displaystyle {K}{^\kappa}_{[\mu\nu]} = \frac{1}{2} {T}{^\kappa}_{\mu\nu}{}$$ • and $$K_{\kappa\mu\nu} = -K_{\nu\mu\kappa}$$. #### Curvature and Torsion ##### Definition of Curvature and Torsion The torsion tensor $$T: \mathcal{X}(M) \otimes \mathcal{X}(M) \rightarrow \mathcal{X}(M)$$ is defined by $T(X,Y) = \grad_X Y - \grad_Y X - [X,Y]$ and the Riemann curvature tensor $R: \mathcal{X}(M) \otimes \mathcal{X}(M) \otimes \mathcal{X}(M) \rightarrow \mathcal{X}(M)$ by $R(X,Y,Z) = \grad_X \grad_Y Z - \grad_Y \grad_X Z - \grad_{[X,Y]}Z.$ It can be shown that $$T$$ and $$R$$ are multilinear objects that satisfies $T(X,Y) = -T(Y,X)$ and $R(X,Y)Z = -R(Y,X)Z,$ or, in component forms, that ${T}{^\lambda}_{\mu\nu} = -{T}{^\lambda}_{\nu\mu}$ and ${R}{^\kappa}_{\lambda\mu\nu} = -{R}{^\kappa}_{\lambda\nu\mu}.$ In component form we have ${T}{^\lambda}_{\mu\nu} = \langle \dd{x^\lambda},T(e_\mu,e_\nu)\rangle = {\Gamma}{^\lambda}_{\mu\nu} - {\Gamma}{^\lambda}_{\nu\mu},$ and \begin{align*} {R}{^\kappa}_{\lambda\mu\nu} &= \langle \dd{x^\kappa}, R(e_\mu,e_\nu)e_\lambda \rangle \\ &= \partial_\mu {\Gamma}{^\kappa}_{\nu\lambda} - \partial_\nu {\Gamma}{^\kappa}_{\mu\lambda} + {\Gamma}{^\eta}_{\nu\lambda} {\Gamma}{^\kappa}_{\mu\eta} - {\Gamma}{^\eta}_{\mu\lambda} {\Gamma}{^\kappa}_{\nu\eta} \\ &= \partial_{[\mu} {\Gamma}{^\kappa}_{\nu]\lambda} + {\Gamma}{^\kappa}_{[\mu|\eta}{\Gamma}{^\eta}_{|\nu]\lambda}. \end{align*} ##### The Ricci Tensor and the Scalar Curvature The Ricci tensor $$\operatorname{Ric}$$ is defined by $\operatorname{Ric}{(X,Y)} = \langle \dd{x^\mu}, R(e_\mu,Y) X\rangle,$ or, in component form, by $\operatorname{Ric}_{\mu\nu} = {R}{^\lambda}_{\mu\lambda\nu}.$ The scalar curvature $$\mathcal{R}$$ is defined by $\mathcal{R} = g^{\mu\nu} \operatorname{Ric}{(e_\mu,e_\nu)} = \operatorname{Ric}{^\mu}{_\nu}.$ #### Levi-Civita Connections ##### The Fundamental Theorem A connection $$\grad$$ is called a symmetric connection if the torsion tensor vanishes, i.e. ${\Gamma}{^\lambda}_{\mu\nu} = {\Gamma}{^\lambda}_{\nu\mu}.$ The Fundamental Theorem of (pseudo-)Riemannian Geometry On a pseudo-Riemannian manifold $$(M,g)$$, there exists a unique symmetric connection which is compatible with the metric $$g$$, called the Levi-Civita connection. Since the torsion is zero, we have $\grad_\mu \grad_\nu f = \grad_\nu \grad_\mu f.$ The Levi-Civita connection of a one-form $$\omega$$ is the exterior differentiation, $\dd{\omega} = (\grad_\mu \omega)_\nu \dd{x^\mu} \wedge \dd{x^\nu}.$ ##### Examples of Levi-Civita Connections In polar coordinates on $$\mathbb{R}^2$$, $g = \dd{r}\otimes \dd{r} + r^2 \dd{\phi} \otimes \dd{\phi}$ and we have \begin{align*} {\Gamma}{^\phi}_{r\phi} &= {\Gamma}{^\phi}_{\phi r} = r^{-1}, \\ {\Gamma}{^r}_{\phi\phi} &= -r. \end{align*} On $$S^2$$ we have $g = \dd{\theta}\otimes \dd{\theta} + \sin^2\theta \dd{\phi}\otimes \dd{\phi}$ and \begin{align*} {\Gamma}{^\theta}_{\phi\phi} &= -\cos\theta \sin\theta, \\ {\Gamma}{^\phi}_{\theta\phi} &= {\Gamma}{^\phi}_{\phi\theta} = \cot\theta. \end{align*} ##### Geodesics The Euler-Lagrange equation of $F = \frac{1}{2} g_{\mu\nu} x'^\mu x'^\nu$ is given by $\dv{}{s} \qty(\pdv{F}{\dot{x}^\lambda}) - \pdv{F}{x^\lambda} = 0,$ which could be utilized to compute the Christoffel symbols. On $$S^2$$ we have $F = \frac{1}{2}(\theta'^2 + \phi'^2 \sin^2\theta ),$ which yields the Euler-Lagrange equations \begin{gather*} \dv[2]{\theta}{s} - \sin\theta \cos\theta \qty(\dv{\phi}{s})^2 = 0, \\ \dv[2]{\phi}{s} + 2\cot\theta \dv{\phi}{s} \dv{\theta}{s} = 0. \end{gather*} We could read off \begin{gather*} {\Gamma}{^\theta}_{\phi\phi} = -\sin\theta\cos\theta, \\ {\Gamma}{^\phi}_{\phi\theta} = {\Gamma}{^\phi}_{\theta\phi} = \cot\theta. \end{gather*} ##### The Normal Coordinate System The normal coordinate system based on $$p$$ with basis $$\qty{e_\mu}$$ is defined such that the coordinate of $$q$$ in the neighbourhood of $$q$$ is given by the tangent vector $$X_q$$ such that the geodesic $$c_q(t)$$ starting from $$p$$ with initial velocity $$X_q$$ satisfies $$c_q(1) = q$$. We define $$\operatorname{EXP}: T_p M \rightarrow M$$ by $\operatorname{EXP}: X_q \mapsto q.$ By definition we have $\varphi(\operatorname{EXP}X_q^\mu e_\mu) = X^\mu_q.$ If $$c$$ is a geodesic, then we have $\varphi(c(t)) = X^\mu = X^\mu_q t.$ In the normal coordinate system at $$p$$, we have ${\Gamma}{^\mu}_{\nu\lambda}(p) = 0.$ The covariant derivative of any tensor $$t$$ at $$p$$ in this coordinate system could be written as $\grad_X t = X[t].$ ##### Riemann Curvature Tensor with Levi-Civita Connection The following symmetries holds for Levi-Civita connection: • Swapping the latter two indices: $R_{\kappa\lambda\mu\nu} = -R_{\kappa\lambda\nu\mu}.$ • Swapping the first two indices: $R_{\kappa\lambda\mu\nu} = -R_{\lambda\kappa\mu\nu}.$ • Swapping the first pair with the second pair: $R_{\kappa\lambda\mu\nu} = R_{\mu\nu\kappa\lambda}.$ • The Ricci tensor is symmetric: $R_{\mu\nu} = R_{\nu\mu}.$ • The first Bianchi identity: \begin{gather*} R(X,Y)Z + R(Z,X)Y + R(Y,Z)X = 0, \\ \Updownarrow \\ {R}{^\kappa}_{\lambda\mu\nu} + {R}{^\kappa}_{\mu\nu\lambda} + {R}{^\kappa}_{\nu\lambda\mu} = 0. \end{gather*} • The second Bianchi identity: \begin{gather*} (\grad_X R)(Y,Z)V + (\grad_Z R)(X,Y)V + (\grad_Y R)(Z,X)V = 0, \\ \Updownarrow \\ (\grad_\kappa R){^\xi}_{\lambda\mu\nu} + (\grad_\mu R){^\xi}_{\lambda\nu\kappa} + (\grad_\nu R){^\xi}_{\lambda\kappa\mu} = 0. \end{gather*} • Contracted Bianchi identity: $\grad_\rho R{^\rho}_\mu = \frac{1}{2}\grad_\mu R.$ The number of independent components of the Riemann tensor is $\frac{1}{12} n^2 (n^2 - 1).$ On a two-dimensional manifold we have $R_{\kappa\lambda\mu\nu} = \frac{\mathcal{R}}{2}(g_{\kappa\mu}g_{\lambda\nu} - g_{\kappa\nu}g_{\lambda\mu}).$ #### Non-Coordinate Bases ##### Vierbeins and Vielbeins • We denote by $$e{^\alpha}_\mu$$ the inverse of $$e{_\alpha}^\mu$$, i.e. \begin{align*} e{^\alpha}_\mu e{_\alpha}^\nu &= \delta{_\mu}^\nu, \\ e{^\alpha}{_\mu} e{_\beta}{^\mu} &= \delta{^\alpha}_\beta. \end{align*} • We define a new set of bases $\hat{e}_\alpha = e{_\alpha}^\mu \pdv{}{x^\mu}$ and $\hat{\theta}^\alpha = e{^\alpha}_\mu \dd{x^\mu}$ such that • $g(\hat{e}_\alpha, \hat{e}_\beta) = e{_\alpha}^\mu e{_\beta}^\nu g_{\mu\nu} = \delta_{\alpha\beta},$ • $g = g_{\mu\nu} \dd{x^\mu}\otimes \dd{x^\nu} = \delta_{\alpha\beta} \hat{\theta}^\alpha \otimes \hat{\theta}^\beta.$ • The bases $$\qty{\hat{e}_\alpha}$$ and $$\qty{\hat{\theta}^\alpha}$$ are called non-coordinate bases. • The coefficients $$e{_\alpha}^\mu$$ are called the vierbeins if the space is four dimensional and vielbeins if it is many-dimensional. • The Lie brackets of non-coordinate bases are given by $[\hat{e}_\alpha, \hat{e}_\beta]\vert_p = c_{\alpha\beta}{^\gamma}(p) \hat{e}_\gamma\vert_p$ where $c_{\alpha\beta}{^\gamma}(p) = e{^\gamma}{_\nu}[e{_\alpha}{^\mu} \partial_\mu e{_\beta}^\nu - e{_\beta}^\mu \partial_\mu e{_\alpha}^\nu](p).$ The zweibeins on $$S^2$$ are given by $\begin{array}{cc} e{^1}_\theta = 1, & e{^1}_\phi = 0, \\ e{^2}_\theta = 0, & e{^2}_\phi = \sin\theta, \end{array}$ which give rise to $\hat{\theta}^1 = \dd{\theta},\quad \hat{\theta}^2 = \sin\theta\dd{\phi}$ and $g = \hat{\theta}^1 \otimes \hat{\theta}^1 + \hat{\theta}^2 \otimes \hat{\theta}^2.$ The non-vanishing components of $$c_{\alpha\beta}^\gamma$$ are $c_{12}{^2} = -c_{21}{^1} = -\cot\theta.$ We use sub-indices $$\kappa,\lambda,\mu\nu,\cdots$$ for coordinate basis while $$\alpha,\beta,\gamma,\delta,\cdots$$ for non-coordinate basis. The transformation laws are given by $V^\mu = V^\alpha e{_\alpha}^\mu,\quad V^\alpha = e{^\alpha}_\mu V^\mu.$ ##### Cartan's Structure Equations We define $\grad_{\hat{e}_\alpha} \hat{e}_\beta = \Gamma{^\gamma}_{\alpha\beta} \hat{e}_\gamma$ and find that • ${\Gamma}{^\gamma}_{\alpha\beta} = e{^\gamma}_\nu e{_\alpha}^\mu \grad_\mu e{_\beta}{^\nu}.$ • $T{^\alpha}_{\beta\gamma} = \Gamma{^\alpha}_{\beta\gamma} - \Gamma{^\alpha}_{\gamma\beta} - c_{\beta\gamma}{^\alpha}.$ • $R{^\alpha}_{\beta\gamma\delta} = \hat{e}_\gamma [{\Gamma}{^\alpha}_{\delta\beta}] - \hat{e}_\delta [{\Gamma}{^\alpha}_{\gamma\beta}] + \Gamma{^\epsilon}_{\delta\beta} \Gamma{^\alpha}_{\gamma\epsilon} - {\Gamma}{^\epsilon}_{\gamma\beta} \Gamma{^\alpha}_{\delta\epsilon} - c_{\gamma\delta}{^\epsilon} \Gamma{^\alpha}_{\epsilon\beta}.$ • We define the connection one-form by $\omega{^\alpha}{_\beta} = \Gamma{^\alpha}_{\gamma\beta} \hat{\theta}^\gamma,$ • the torsion two-form by $T^\alpha = \frac{1}{2} T{^\alpha}_{\beta\gamma} \hat{\theta}^\beta \wedge \hat{\theta}^\gamma,$ • and the curvature two-form by $R{^\alpha}_{\beta} = \frac{1}{2} R{^\alpha}_{\beta\gamma\delta} \hat{\theta}^\gamma \wedge \hat{\theta}^\delta.$ • We have the following identities: • Cartan's first structure equation: $\dd{\hat{\theta}}^\alpha + {\omega}{^\alpha}{_\beta}\hat{\theta}^\beta = T^\alpha.$ • Cartan's second structure equations: $\dd{\omega{^\alpha}{_\beta}} + \omega{^\alpha}{_\gamma} \wedge \omega{^\gamma}{_\beta} = R{^\alpha}{_\beta}.$ • First Bianchi identity: $\dd{T^\alpha} + \omega{^\alpha}{_\beta} \wedge T^\beta = R{^\alpha}{_\beta} \wedge \hat{\theta}^\beta.$ • Second Bianchi identity: $\dd{R{^\alpha}{_\beta}} + \omega{^\alpha}{_\gamma} \wedge R{^\gamma}{_\beta} - R{^\alpha}{_\gamma} \wedge \omega{^\gamma}{_\beta} = 0.$ ##### The Local Frame Since in the non-coordinate basis the metric is given by the Minkowski metric, the transformations that keeps the metric invariant are elements of the Lorentz group. The transformation laws of tensors under these rotations are preserved. • The curvature two-form transforms homogeneously as $R{^\alpha}{_\beta} \rightarrow R'{^\alpha}{_\beta} = {\Lambda}{^\alpha}{_\gamma} R{^\gamma}{_\delta} (\Lambda^{-1}){^\delta}{_\beta}.$ • However, the transformation laws of the torsion two-form and the connection two-form are quite different. ##### The Levi-Civita Connection in a Non-Coordinate Basis We define the Ricci rotation coefficient $$\Gamma_{\alpha\beta\gamma}$$ $\Gamma_{\alpha\beta\gamma} = \eta_{\alpha\delta}\Gamma{^\delta}_{\beta\gamma}$ and express the metric compatibility as \begin{gather*} \Gamma_{\alpha\beta\gamma} = -\Gamma_{\gamma\beta\alpha} \\ \Updownarrow \\ \omega_{\alpha\beta} = -\omega_{\beta\alpha}. \end{gather*} The torsion-free condition is $\dd{\hat{\theta}^\alpha} + \omega{^\alpha}{_\beta} \wedge \hat{\theta}^\beta = 0.$ Under this condition we have $c_{\alpha\beta}{^\gamma} = \Gamma{^\gamma}_{\alpha\beta} - \Gamma{^\gamma}_{\beta\alpha}.$ We could express the Riemann curvature tensor in terms of $$\Gamma$$ only. $R{^\alpha}_{\beta\gamma\delta} = \hat{e}_\gamma [{\Gamma}{^\alpha}_{\delta\beta}] - \hat{e}_\delta [{\Gamma}{^\alpha}_{\gamma\beta}] + \Gamma{^\epsilon}_{\delta\beta} \Gamma{^\alpha}_{\gamma\epsilon} - {\Gamma}{^\epsilon}_{\gamma\beta} \Gamma{^\alpha}_{\delta\epsilon} - (\Gamma{^\epsilon}_{\gamma\delta} - \Gamma{^\epsilon}_{\delta\gamma}) \Gamma{^\alpha}_{\epsilon\beta}.$ On $$S^2$$ we have $e{^1}{_\theta} = 1,\quad e{^1}{_\phi} = 0,\quad e{^2}{_\theta} = 0,\quad e{^2}{_\phi} = \sin\theta.$ • We note first that $$\omega_{11} = \omega_{22} = 0$$. • We obtain the conenction one-forms from the torsion-free conditions: \begin{align*} \dd{(\dd{\theta})} + \omega{^1}{_2} \wedge (\sin\theta\dd{\phi}) &= 0, \\ \dd{(\sin\theta \dd{\phi})} + \omega{^2}{_1}\wedge \dd{\theta} &= 0, \end{align*} that $\omega{^2}{_1} = -\omega{^1}{_2} = \cos\theta \dd{\phi}.$ • From Cartan's structure equations we obtain the curvature tensor. \begin{align*} \omega{^1}{_2} \wedge \omega{^2}{_1} &= \frac{1}{2} R{^1}_{1\alpha\beta} \hat{\theta}^\alpha \wedge \hat{\theta}^\beta, \\ \dd{\omega{^1}{_2}} &= \frac{1}{2} R{^1}_{2\alpha\beta} \hat{\theta}^\alpha \wedge \hat{\theta}^\beta, \\ \dd{\omega{^2}{_1}} &= \frac{1}{2} R{^2}_{1\alpha\beta} \hat{\theta}^\alpha \wedge \hat{\theta}^\beta, \\ \omega{^2}{_1} \wedge \omega{^1}{_2} &= \frac{1}{2} R{^2}_{2\alpha\beta} \hat{\theta}^\alpha \wedge \hat{\theta}^\beta. \end{align*} The non-zero components are obtained to be \begin{align*} R{^1}_{212} = -R{^1}_{221} &= \sin\theta, \\ R{^2}_{112} = -R{^2}_{121} &= -\sin\theta. \end{align*} • The curvature in the original coordinate is obtained by a transformation back, e.g. $R{^\theta}_{\phi\theta\phi} = e{_\alpha}{^\theta} e{^\beta}{_\phi} e{^\gamma}{_\theta} e{^\delta}{_\phi} R{^\alpha}_{\beta\gamma\delta} = \frac{1}{\sin^2 \theta} R{^1}_{212} = \frac{1}{\sin\theta}.$ The Schwarzschild metric is given by $\dd{s^2} = -\hat{\theta}^0 \otimes \hat{\theta}^0 + \hat{\theta}^1 \otimes \hat{\theta}^1 + \hat{\theta}^2 \otimes \hat{\theta}^2 + \hat{\theta}^3 \otimes \hat{\theta}^3.$ where \begin{align*} \hat{\theta}^0 &= \qty(1-\frac{2M}{r})^{1/2} \dd{t}, \\ \hat{\theta}^1 &= \qty(1-\frac{2M}{r})^{-1/2} \dd{r}, \\ \hat{\theta}^2 &= r\dd{\theta}, \\ \hat{\theta}^3 &= r\sin\theta \dd{\phi}. \end{align*} • We first note that $\omega{^0}{_0} = \omega{^1}{_1} = \omega{^2}{_2} = \omega{^3}{_3} = 0.$ • The torsion-free condition yields \begin{align*} \dd{\qty[\qty(1-\frac{2M}{r})^{1/2}\dd{t}]} + \omega{^0}{_\beta}\wedge \hat{\theta}^\beta &= 0, \\ \dd{\qty[\qty(1-\frac{2M}{r})^{-1/2}\dd{t}]} + \omega{^0}{_\beta}\wedge \hat{\theta}^\beta &= 0, \\ \dd{(r\dd{\theta})} + \omega{^2}{_\beta}\wedge\hat{\theta}^\beta &= 0, \\ \dd{(r\sin\theta\dd{\theta})} + \omega{^3}{_\beta} \wedge \hat{\theta}^\beta &0. \end{align*} • We obtain the connection one-forms \begin{align*} \omega{^0}{_1} = \omega{^1}{_0} &= \frac{M}{r^2} \dd{t}, \\ \omega{^2}{_1} = -\omega{^1}{_2} &= \qty(1-\frac{2M}{r})^{1/2}\dd{\theta}, \\ \omega{^3}{_1} = -\omega{^1}{_3} &= \qty(1-\frac{2M}{r})^{1/2}\sin\theta \dd{\phi}, \\ \omega{^3}{_2} = -\omega{^2}{_3} &= \cos\theta\dd\phi. \end{align*} • The curvature two-forms are found to be \begin{align*} R{^0}{_1} = R{^1}{_0} &= \frac{2M}{r^3} \hat{\theta}^0 \wedge \hat{\theta}^1, \\ R{^0}{_2} = R{^2}{_0} &= -\frac{M}{r^3} \hat{\theta}^0 \wedge \hat{\theta}^2, \\ R{^0}{_3} = R{^3}{_0} &= -\frac{M}{r^3} \hat{\theta}^0 \wedge \hat{\theta}^3, \\ R{^1}{_2} = -R{^2}{_1} &= -\frac{M}{r^2} \hat{\theta}^1 \wedge \hat{\theta}^2, \\ R{^1}{_3} = -R{^3}{_1} &= -\frac{M}{r^2} \hat{\theta}^1 \wedge \hat{\theta}^3, \\ R{^2}{_3} = -R{^3}{_2} &= \frac{2M}{r^2} \hat{\theta}^2 \wedge \hat{\theta}^3. \end{align*} #### Moving Frame %%% Cartan's Moving Frame Abstract IndexTorsionless ##### Structure Equations Cartan's First Structure Equation. $\dd{\vb*{e}^\nu} = -\vb*{e}^\mu \wedge {\omega}{_\mu}{^\nu}.$ Cartan's Second Structure Equation. ${\vb*{R}}{_\mu}{^\nu} = \dd{\omega{_\mu}{^\nu}} + {\omega}{_\mu}{^\lambda} \wedge {\omega}{_\lambda}{^\nu}.$ To obtain the Riemann tensor, we use ${\vb*{R}}{_\mu}{^\nu} = \frac{1}{2} R_{\rho\sigma\mu}{^\nu} \vb*{e}^\rho \wedge \vb*{e}^\sigma.$ To obtain the connection 1-form, we let $\Lambda_{\mu\nu\rho} = [(e_\nu)_{\lambda,\tau} - (e_\nu)_{\tau,\lambda}](e_\mu)^\lambda (e_\rho)^\tau$ and apply $\omega_{\mu\nu\rho} = \frac{1}{2}(\Lambda_{\mu\nu\rho} + \Lambda_{\rho\mu\nu} - \Lambda_{\nu\rho\mu}).$ The number of independent elements of $$\Lambda$$ is $\frac{1}{2}n^2 (n-1),$ which becomes $$2,9,24$$ in $$d=2,3,4$$, respectively. ##### Obtaining the Curvature Tensor We obtain the curvature tensor for $\dd{s}^2 = -e^{2A(r)}\dd{t}^2 + e^{2B(r)}\dd{r}^2 + r^2(\dd{\theta}^2 + \sin^2\theta \dd{\varphi}^2).$ 1. We take \begin{align*} (e^0)_a &= e^{A(r)} (\dd{t})_a, \\ (e^1)_a &= e^{B(r)} (\dd{r})_a, \\ (e^2)_a &= r (\dd{\theta})_a, \\ (e^3)_a &= r\sin\theta (\dd{\varphi})_a, \end{align*} such that $$\dd{s}^2$$ under this basis becomes $$\operatorname{diag}{(-1,1,1,1)}$$. We have the vector version \begin{align*} (e_0)^a &= e^{-A(r)}\qty(\pdv{}{t})^a, \\ (e_1)^a &= e^{-B(r)}\qty(\pdv{}{r})^a, \\ (e_2)^a &= r^{-1}\qty(\pdv{}{\theta})^a, \\ (e_3)^a &= \frac{1}{r\sin\theta}\qty(\pdv{}{\varphi})^a, \end{align*} and the lower index version \begin{align*} (e_0)_a &= -e^{A(r)} (\dd{t})_a, \\ (e_1)_a &= e^{B(r)} (\dd{r})_a, \\ (e_2)_a &= r (\dd{\theta})_a, \\ (e_3)_a &= r\sin\theta (\dd{\varphi})_a. \end{align*} 2. Obtain $$\Lambda_{\mu\nu\rho}$$. First $$\Lambda_{\mu 0 \rho}$$, and then $$\Lambda_{\mu 1 \rho}$$, etc. • Utilize the anti-symmetry $\Lambda_{\mu\nu\rho} = -\Lambda_{\rho\nu\mu}.$ In particular, $$\Lambda_{\mu\nu\rho} = 0$$ if $$\mu = \rho$$. • In $\Lambda_{\mu\nu\rho} = [(e_\nu)_{\lambda,\tau} - (e_\nu)_{\tau,\lambda}](e_\mu)^\lambda (e_\rho)^\tau,$ we use the lower-index one-form basis for $[(e_\nu)_{\lambda,\tau} - (e_\nu)_{\tau,\lambda}]$ and the vector basis for $(e_\mu)^\lambda (e_\rho)^\tau.$ • We obtain \begin{align*} \Lambda_{001} & = -\Lambda_{100} = -A'(r) e^{-B(r)}, \\ \Lambda_{122} &= -\Lambda_{221} = -\frac{e^{-B(r)}}{r}, \\ \Lambda_{133} &= -\Lambda_{331} = -\frac{e^{-B(r)}}{r}, \\ \Lambda_{233} &= -\Lambda_{332} = -\frac{\cot \theta}{r}. \end{align*} • Then we use $\omega_{\mu\nu\rho} = \frac{1}{2}(\Lambda_{\mu\nu\rho} + \Lambda_{\rho\mu\nu} - \Lambda_{\nu\rho\mu})$ to obtain $$\omega_{\mu\nu\rho}$$. Note that $\omega_{\mu\nu\rho} = -\omega_{\nu\mu\rho}.$ In particular, $$\omega_{\mu\nu\rho} = 0$$ if $$\mu = \nu$$. • The nonzero $$\omega_{\mu\nu\rho}$$'s are found to be \begin{align*} \omega_{010} &= -\omega_{100} = -A'(r) e^{-B(r)}, \\ \omega_{122} &= -\omega_{212} = -\frac{e^{-B(r)}}{r}, \\ \omega_{133} &= -\omega_{313} = -\frac{e^{-B(r)}}{r}, \\ \omega_{233} &= -\omega_{323} = -\frac{\cot\theta}{r}. \end{align*} • Therefore, the six independent one-forms $$\vb*{\omega}_{\mu\nu}$$ are \begin{align*} \vb*{\omega}_{01} &= -A'(r) e^{-B(r)} \vb*{e}^0, \\ \vb*{\omega}_{02} &= 0, \\ \vb*{\omega}_{03} &= 0, \\ \vb*{\omega}_{12} &= -\frac{e^{-B(r)}}{r} \vb*{e}^2, \\ \vb*{\omega}_{13} &= -\frac{e^{-B(r)}}{r} \vb*{e}^3, \\ \vb*{\omega}_{23} &= -\frac{\cot\theta}{r} \vb*{e}^3. \end{align*} 3. Use Cartan's second structure equation to obtain $$R$$. • When calculating the exterior differentiations, we use the original basis vectors and write \begin{align*} \omega_{01} &= -A'(r) e^{A(r) - B(r)} \dd{t}, \\ \omega_{12} &= -e^{-B(r)} \dd{\theta}, \\ \omega_{13} &= -e^{-B(r)} \sin \theta \dd{\varphi}, \\ \omega_{23} &= -\cos\theta \dd{\varphi}. \end{align*} • Since $$\omega_{ij}$$'s are associated with Minkowski metric, we have $\omega{_\mu}{^i} = \omega_{\mu i}$ for $$i=1,2,3$$. • With Cartan's second structure equation we find \begin{align*} \vb*{R}{_0}{^1} &= e^{-2B(r)}(A''(r) - A'(r)B'(r) + A'(r)^2)\, \vb*{e}^0 \wedge \vb*{e}^1, \\ \vb*{R}{_0}{^2} &= \frac{A'(r) e^{-2B(r)}}{r}\, \vb*{e}^0 \wedge \vb*{e}^2, \\ \vb*{R}{_0}{^3} &= \frac{A'(r) e^{-2B(r)}}{r}\, \vb*{e}^0 \wedge \vb*{e}^3, \\ \vb*{R}{_1}{^2} &= \frac{B'(r) e^{-2B(r)}}{r}\, \vb*{e}^1 \wedge \vb*{e}^2, \\ \vb*{R}{_1}{^3} &= \frac{B'(r) e^{-2B(r)}}{r}\, \vb*{e}^1 \wedge \vb*{e}^3, \\ \vb*{R}{_2}{^3} &= \frac{1-e^{-2B(r)}}{r^2}\, \vb*{e}^2 \wedge \vb*{e}^3. \\ \end{align*} 4. From $\vb*{R}{_\mu}{^\nu} = \frac{1}{2} R_{\rho\sigma\mu}{^\nu} \vb*{e}^\rho \wedge \vb*{e}^\sigma$ we could read off the Riemann tensor. Note that we should expand $$\vb*{e}^\rho$$ and $$\vb*{e}^\sigma$$ to get their components in the original frame. For example, from $\vb*{R}{_0}{^1} = e^{-2B(r)}(A''(r) - A'(r)B'(r) + A'(r)^2)\, \vb*{e}^0 \wedge \vb*{e}^1$ we read that \begin{align*} R_{010}{^1} &= e^{-2B(r)}(A''(r) - A'(r)B'(r) + A'(r)^2) \times e^{A(r)} e^{B(r)} e^{A(r)} e^{-B(r)} \\ &= e^{2(A(r)-2B(r))}(A''(r) - A'(r)B'(r) + A'(r)^2). \end{align*} Note that we should transform (see Comparison of Conventions). $R_{010}{^1} \leftrightarrow R{^1}_{010}.$ #### Holonomy Definition Let $$p$$ be a point in $$(M,g)$$ with connection $$\grad$$ and consider the set of closed loops at $$p$$, $\Set{c(t) | 0\le t\le 1, c(0) = c(1) = p}.$ Take a vector $$X\in T_p M$$ and parallel transport $$X$$ along $$c(t)$$, which results in a new vector $$X_c \in T_p M$$. Therefore, the parallel transport induces a linear transformation $P_c: T_p M \rightarrow T_p M.$ The set of these transformations is denoted by $$H(p)$$ and is called the holonomy group at $$p$$. The product of elements in $$H(p)$$ is the linear transformation induced by the product of the $$c(t)$$'s. If $$(M,g)$$ is parallelizable, $$H(p)$$ may be made trivial. We assume that $$H(p)$$ acts on $$T_p M$$ from the right, i.e. $P_c X = Xh \Leftrightarrow P_cX = X^\mu h{_\mu}{^\nu} e_\nu.$ If $$M$$ is (arcwise-)connected, and the curve $$a$$ connects $$p$$ and $$q$$. Then the parallel transport along $$a$$ defines a linear transform $$\tau_a: T_p M \rightarrow T_q M$$. The holonomy groups $$H(p)$$ and $$H(q)$$ are related by $H(q) = \tau_a^{-1} H(p)\tau_a.$ If $$\grad$$ is a metric connection, $$\grad$$ preserves the length of vector and therefore $$H(p)\subset \operatorname{SO}{(m)}$$ if it's orientable and Riemannian, and $$\operatorname{SO}{(m-1,1)}$$ if it's orientable and Lorentzian. Parallel transport along $$\theta = \theta_0$$ on $$S^2$$ is given by \begin{align*} \partial_\phi X^\theta - \sin\theta_0 \cos\theta_0 X^\phi &= 0, \\ \partial_\phi X^\phi + \cot\theta_0 X^\theta = 0, \end{align*} which has the solution for $$X^\theta=1$$ at $$\phi_0$$ $\begin{cases} \displaystyle X^\theta = \cos(2\pi \cdot C_0), \\ \displaystyle X^\phi = -\frac{\sin(2\pi C_0)}{\sin\theta_0}. \end{cases}$ In general, $$S^m$$ for $$m>2$$ admits the holonomy group $$\operatorname{SO}{(m)}$$. #### Isometries and Conformal Transformations ##### Isometries A diffeomorphism $$f:M\rightarrow M$$ is an isometry if it preserves the metric $f^* g_{f(p)} = g_p,$ i.e. if $g_{f(p)}(f_* X, f_* y) = g_p(X,Y).$ All isometries form a group. ##### Conformal Transformations A diffeomorphism $$f: M\rightarrow M$$ is called a conformal transformation if it preserves the metric up to a scale, $f^* g_{f(p)} = e^{2\sigma} g_p,$ i.e. $g_{f(p)} (f_* X, f_* Y) = e^{2\sigma} g_p (X,Y),$ where $$\sigma \in \mathcal{F}(M)$$. The map $$f:(x,y) \mapsto (u,v)$$ defined by a analytic function is conformal. Let $$g$$ and $$\overline{g}$$ be metrics on a manifold $$M$$. $$\overline{g}$$ is said to be conformally related to $$g$$ if $\overline{g}_p = e^{2\sigma(p)} g(p).$ The equivalence class is called the conformal structure. The transformation $$g\rightarrow e^{2\sigma} g$$ is called a Weyl rescaling. ##### Riemann Tensor under Conformal Transformation Let $K(X,Y) = \overline{\grad}_X Y - \grad_X Y,$ where $$\overline{\grad}$$ denotes the connection with respect to $$\overline{g} = e^{2\sigma} g$$. Proposition Let $$U$$ be a vector field which corresponds to the one-form $$\dd{\sigma}$$. Then $K(X,Y) = X[\sigma]Y + Y[\sigma X] - g(X,Y)U.$ We rewrite $$\overline{R}(X,Y)Z$$ using $$\grad$$ and $$K$$, and expand $$K$$ using the above equation. Then we arrive at $\overline{R}{^\kappa}_{\lambda\mu\nu} = R{^\kappa}_{\lambda\mu\nu} - g_{\nu\lambda} B{_\mu}{^\kappa} + g_{\xi\lambda}B{_\mu}{^\xi} \delta{^\kappa}{_\nu} - g_{\xi\lambda} B{_\nu}{^\xi} \delta{^\kappa}{_\mu} + g_{\mu\lambda}B{_\nu}{^\kappa},$ where $B{_\mu}{^\kappa} = -\partial_\mu \sigma U^\kappa + (\grad_\mu U)^\kappa + \frac{1}{2} U[\sigma] \delta{_\mu}{^\kappa},$ which satisfies $B_{\mu\nu} = B_{\nu\mu}.$ Let $$m$$ denote the dimension of $$M$$. By contraction we find • $\overline{\operatorname{Ric}}_{\mu\nu} = \operatorname{Ric}_{\mu\nu} - g_{\mu\nu} B{_\lambda}{^\lambda} - (m-2) B_{\nu\mu},$ • $e^{2\sigma} \overline{\mathcal{R}} = \overline{\mathcal{R}} - 2(m-1) B{_\lambda}{^\lambda}.$ We may eliminate $$B{_\lambda}{^\lambda}$$ and $$B_{\mu\nu}$$ in $$\overline{R}{^\kappa}_{\lambda\mu\nu}$$ using the contracted equations and find that the Weyl tensor (defined for $$m\ge 4$$) \begin{align*} C{^\kappa}_{\lambda\mu\nu} &= R{^\kappa}_{\lambda\mu\nu} \\ &{\phantom{{}={}}} + \frac{1}{m-2} (\operatorname{Ric}_{\kappa\mu} g_{\lambda\nu} - \operatorname{Ric}_{\lambda\mu}g_{\kappa\nu} + \operatorname{Ric}_{\lambda\nu} g_{\kappa\mu} - \operatorname{Ric}_{\kappa\nu} g_{\lambda\mu}) \\ &{\phantom{{}={}}} + \frac{\mathcal{R}}{(m-2)(m-1)}(g_{\kappa\mu}g_{\lambda\nu} - g_{\kappa\nu} g_{\lambda\mu}) \end{align*} is invariant under conformal transformations, i.e. $\overline{C}{^\kappa}_{\lambda\mu\nu} = C{^\kappa}_{\lambda\mu\nu}.$ If every point $$p$$ of a (pseudo-)Riemann manifold $$(M,g)$$ has a chart $$(U,\varphi)$$ containing $$p$$ such that $$g_{\mu\nu} = e^{2\sigma} \delta_{\mu\nu}$$, then $$(M,g)$$ is said to be conformally flat. • If $$\operatorname{dim} M\ge 4$$, then $$C=0$$ is equivalent to conformal flatness. • If $$\operatorname{dim}{M} = 3$$, then $$C{^\kappa}_{\lambda\mu\nu}$$ vanishes identically. • If $$\operatorname{dim}{M} = 2$$, then $$M$$ is always conformally flat. Let $$\dd{s^2} = \dd{\theta^2} + \sin^2\theta \dd{\phi^2}$$ be the metric on $$S^2$$. Then $f: (\theta,\phi) \rightarrow \qty{u = \log \abs{\tan \frac{\theta}{2}}, v=\phi}$ yields a conformally flat metric: $\dd{s^2} = \sin^2\theta \qty(\frac{\dd{\theta}^2}{\sin^2\theta} + \dd{\phi^2}) = \sin^2\theta (\dd{u^2} + \dd{v^2}).$ #### Killing Vector Fields and Conformal Killing Vector Fields ##### Killing Vector Fields Let $$X$$ be a vector field on $$(M,g)$$. The following condition are equivalent: • $$X$$ is a Killing vector field; • the displacement $$\epsilon X$$ generates an isometry; • $\pdv{(x^\kappa + \epsilon X^\kappa)}{x^\mu} \pdv{(x^\lambda + \epsilon x^\lambda)}{x^\nu} g_{\kappa\lambda}(x+\epsilon X) = g_{\mu\nu}(x);$ • the Killing equation $X^\xi \partial_\xi g_{\mu\nu} + \partial_\mu X^\kappa g_{\kappa\nu} + \partial_\nu X^\lambda g_{\mu\lambda} = 0;$ • $(\mathcal{L}_X g)_{\mu\nu} = 0;$ • or (for Levi-Civita connection) $(\grad_\mu X)_\nu + (\grad_\nu X)_\mu = \partial_\mu X_\nu + \partial_\nu X_\mu - 2{\Gamma}{^\lambda}_{\mu\nu}X_\lambda = 0.$ Let $$X$$ and $$Y$$ be two Killing vector fields. • a linear combination $$aX+bY$$ is also a Killing vector field; and • the Lie bracket $$[X,Y]$$ is a Killing vector field. Therefore, all the Killing vector fields form a Lie algebra of the symmetric operations on the manifold $$M$$. A set of Killing vector fields are defined to be dependent if one of them is expressed as a linear combination of others with constant coefficients. For the Minkowski spacetime $$(\mathbb{R}^4,\eta)$$, the Killing equations becomes $\partial_\mu X_\nu + \partial_\nu X_\mu = 0.$ $$X_\mu$$ could be atmost linear in $$x$$. The constant solutions generate translations. Let $X_\mu = a_{\mu\nu} x^\mu.$ We then find that $$a_{\mu\nu}$$ should be anti-symmetric. This form admits $$6$$ independent solutions, $$3$$ of which space rotation and $$3$$ boosts. In $$m$$-dimensional Minkowski spacetime ($$m\ge 2$$), there are $$m(m+1)/2$$ Killing vector fields. Those spacetime which admit $$m(m+1)/2$$ Killing vector fields are called maximally symmetric spaces. The Killing equations on $$S^2$$ is written as \begin{align*} \partial_\theta X_\theta + \partial_\theta X_\theta &= 0, \\ \partial_\phi X_\phi + \partial_\phi X_\phi + 2\sin\theta \cos\theta X_\theta &= 0, \\ \partial_\theta X_\phi + \partial_\phi X_\theta - 2\cot\theta X_\phi &= 0. \end{align*} Therefore, $$X_\theta$$ is independent of $$\theta$$. The solution is given by \begin{align*} X_\theta &= A\sin\phi + B\cos\phi, \\ X_\phi &= (A\cos\phi - B\sin\phi) \sin\theta\cos\theta + C_1 \sin^2\theta. \end{align*} These give rise to vector fields generated by \begin{align*} L_x &= -\cos\phi \pdv{}{\theta} + \cot\theta\sin\phi \pdv{}{\phi}, \\ L_y &= \sin\phi \pdv{}{\theta} + \cot\theta\cos\phi \pdv{}{\phi}, \\ L_z &= \pdv{}{\phi}. \end{align*} These basis vectors generate the Lie algrabra $$\mathfrak{so}(3)$$. In general, $$S^n = \mathrm{SO}(n+1)/\mathrm{SO}(n)$$ admits $\dim \mathrm{SO}(n+1) = \frac{n(n+1)}{2}$ Killing vector fields. They are maximally symmetric spaces. ##### Conformal Killing Vector Fields Let $$X$$ be a vector field on $$(M,g)$$. The following condition are equivalent: • $$X$$ is a conformal Killing vector field (CKV); • the displacement $$\epsilon X$$ generates an conformal transformation; • $\pdv{(x^\kappa + \epsilon X^\kappa)}{x^\mu} \pdv{(x^\lambda + \epsilon x^\lambda)}{x^\nu} g_{\kappa\lambda}(x+\epsilon X) = e^{2\sigma} g_{\mu\nu}(x);$ • the conformal Killing equation $(\mathcal{L}_X g)_{\mu\nu} = X^\xi \partial_\xi g_{\mu\nu} + \partial_\mu X^\kappa g_{\kappa\nu} + \partial_\nu X^\lambda g_{\mu\lambda} = \psi g_{\mu\nu}.$ Let $$X$$ and $$Y$$ be two CKVs. • a linear combination $$aX+bY$$ is also a CKV; and • the Lie bracket $$[X,Y]$$ is a CKV. Let $$x^\mu$$ be the coordinate of $$(\mathbb{R}^m,\delta)$$. Then $D = x^\mu \pdv{}{x^\mu}$ is a CKV since $\mathcal{L}_D \delta_{\mu\nu} = \partial_\mu x^\kappa \delta_{\kappa\nu} + \partial_\nu x^\lambda \delta_{\mu\lambda} = 2\delta_{\mu\nu}.$ #### Differential Forms and Hodge Theory ##### Invariant Volume Elements The invariant volume element is defined by $\Omega_M = \sqrt{\abs{g}} \dd{x^1}\wedge \dd{x^2} \wedge\cdots \wedge \dd{x^m}.$ An integration of $$f\in \mathcal{F}(M)$$ over $$M$$ may be defined by $\int_M f\Omega_M = \int_M f\sqrt{\abs{g}} \dd{x^1} \cdots \dd{x^m}.$ ##### Duality Transformations We define the totally anti-symmetric tensor by $\epsilon_{\mu_1 \cdots \mu_m} = \operatorname{sign}{(\mu_1 \cdots \mu_m)}.$ By raising the indices we get $\epsilon^{\mu_1 \cdots \mu_m} = g^{-1} \epsilon_{\mu_1 \cdots \mu_m}.$ The Hodge * operation is a linear map $*: \Omega^r(M) \rightarrow \Omega^{m-r}(M)$ defined by $*(\dd{x^{\mu_1}} \wedge \cdots \wedge \dd{x^{\mu_r}}) = \frac{\sqrt{\abs{g}}}{(m-r)!} \epsilon{^{\mu_1 \cdots \mu_r}}{_{\nu_{r+1} \cdots \nu_m}} \dd{x^{\nu_r + 1}} \wedge \cdots \wedge \dd{x^{\nu_m}}.$ • $$*1$$ is the invariant volume element: $*1 = \sqrt{\abs{g}} \dd{x^1}\wedge \cdots \wedge \dd{x^m}.$ • More generally for $\omega = \frac{1}{r!} \omega_{\mu_1 \cdots \mu_r} \dd{x^{\mu_1}} \wedge \cdots \wedge dd{x^{\mu_r}}, \\$ we have $*\omega = \frac{\sqrt{\abs{g}}}{r!(m-r)!} \omega_{\mu_1 \cdots \mu_r} \epsilon{^{\mu_1 \cdots \mu_r}}{_{\nu_{r+1} \cdots \nu_m}} \dd{x^{\nu_{r+1}}} \wedge \dd{x^{\nu_m}}.$ • Using the non-coordinate basis we find $*(\hat{\theta}^{\alpha_1} \wedge \cdots \wedge \hat{\theta}^{\alpha_r}) = \frac{1}{(m-r)!} \epsilon{^{\alpha_1 \cdots \alpha_r}}{_{\beta_{r+1}\cdots \beta_m}} \hat{\theta}^{\beta_{r+1}} \wedge \cdots \wedge \hat{\theta}^{\beta_m},$ where the indices are raised by $$\eta^{\alpha\beta}$$. Applying the Hodge * twice does not yield the original form. • If $$(M,g)$$ is Riemannian, $**\omega = (-1)^{r(m-r)} \omega \Leftrightarrow *^{-1} = (-1)^{r(m-r)}.$ • If $$(M,g)$$ is Lorentzian, $**\omega = -(-1)^{r(m-r)} \omega \Leftrightarrow *^{-1} = (-1)^{1+r(m-r)}.$ ##### Inner Products of Forms The inner product or two $$r$$-forms is defined by \begin{align*} (\omega,\eta) &= \int \omega \wedge *\eta \\ &= \frac{1}{r!} \int_M \omega_{\mu_1 \cdots \mu_r} \eta^{\mu_1 \cdots \mu_r} \sqrt{\abs{g}} \dd{x^1} \cdots \dd{x^m}. \end{align*} The inner product is • symmetric: $(\omega,\eta) = (\eta,\omega);$ • positive-definite if $$(M,g)$$ is Riemannian: $(\alpha,\alpha) > 0.$ ##### Adjoints of Exterior Derivatives We define the adjoint exterior derivative $\dd{^\dagger}: \Omega^r(M) \rightarrow \Omega^{r-1}(M)$ by • if $$(M,g)$$ is Riemannian: $\dd{^\dagger} = (-1)^{mr+m+1} *\dd{*};$ • if $$(M,g)$$ is Lorentzian: $\dd{^\dagger} = (-1)^{mr+m} *\dd{*}.$ We have that • $$\dd{^\dagger}$$ is nilpotent; • let $$(M,g)$$ be a compact orientable manifold without a boundary, then $(\dd{\beta},\alpha) = (\beta,\dd{^\dagger}\alpha)$ where $$\alpha \in \Omega^r(M)$$ and $$\beta \in \Omega^{r-1}(M)$$. ##### The Laplacian, Harmonic Forms, and the Hodge Decomposition Theorem The Laplacian $\bigtriangleup: \Omega^r(M) \rightarrow \Omega^r(M)$ is defined by $\bigtriangleup = (\dd{} + \dd{^\dagger})^2 \dd{\dd{^\dagger}} + \dd{^\dagger\dd{}}.$ Let $$f \in \mathcal{F}(M)$$, we could easily verify that $\bigtriangleup f = -\frac{1}{\sqrt{\abs{g}}} \partial_\nu [\sqrt{\abs{g}}g^{\nu\mu}\partial_\mu f].$ The Maxwell's equations may be written as \begin{align*} \dd{F} = \dd{^2 A} &= 0, \\ \dd{^\dagger F} = \dd{^\dagger \dd{A}} = j, \end{align*} where $$A = A_\mu \dd{x^\mu}$$. With the Lorentz gauge $\dd{^\dagger A} = 0,$ the inhomogeneous equation may be written as $\bigtriangleup A = j.$ If $$(M,g)$$ is a compact Riemannian manifold without boundary, then $$\bigtriangleup$$ is a positive operator on $$M$$: $(\omega, \bigtriangleup \omega) = (\dd{\omega},\dd{\omega}) + (\dd{^\dagger \omega}, \dd{^\dagger \omega}) \ge 0.$ • An $$r$$-form is called harmonic if $$\bigtriangleup\omega = 0$$; the set of harmonic $$r$$-forms on $$M$$ is denoted by $$\operatorname{Harm}{^r(M)}$$. • closed if $$\dd{\omega} = 0$$; • coclosed if $$\dd{^\dagger \omega} = 0$$; • exact if $$\omega = \dd{\beta}$$; the set of exact $$r$$-forms on $$M$$ is denoted by $$\dd{\Omega^{r-1}M}$$; and • coexact if $$\omega = \dd{^\dagger \beta}$$; the set of coexact $$r$$-forms on $$M$$ is denoted by $$\dd{^\dagger \Omega^{r+1}(M)}$$. Theorem. An $$r$$-form $$\omega$$ is harmonic if and only if $$\omega$$ is closed and coclosed. Hodge Decomposition Theorem. Let $$(M,g)$$ be a compact orientable Riemannian manifold without a boundary. Then $\Omega^r(M) = \dd{\Omega^{r-1}}(M) \oplus \dd{^\dagger \Omega^{r+1}(M)} \oplus \mathrm{Harm}^r{(M)}.$ If $$r=0$$, we define $$\Omega^{-1}(M) = \qty{0}$$. Theorem. If $$\eta_r$$ is orthogonal to $$\mathrm{Harm}^r{(M)}$$, then $\eta_r = \bigtriangleup \psi_r$ for some $$\psi_r \in \Omega^r(M)$$. In particular, the Hodge decomposition may be given by $\omega_r = \dd{\dd{^\dagger \psi_r}} + \dd{^\dagger \dd{\psi_r}} + \Pi_{\operatorname{Harm}} \omega_r.$ ##### Harmonic Forms and de Rham Cohomology Groups Hodge's Theorem. On a compact orientable Riemannian manifold $$(M,g)$$, $H^r(M) \cong \mathrm{Harm}^r{(M)}.$ The isomorphism is provided by identifying $$[\omega]$$ with $$\Pi_{\operatorname{Harm}} \omega$$. In particular, we have $\operatorname{dim}\mathrm{Harm}^r{(M)} = \operatorname{H}{^r (M)} = b^r,$ and $\chi(M) = \sum (-1)^r b^r = \sum (-1)^r \operatorname{dim}\mathrm{Harm}^r{(M)}.$ #### Applications to General Relativity ##### The Einstein Field Equation The Einstein tensor is defined by $G^{\mu\nu} = \operatorname{Ric}^{\mu\nu} - \frac{1}{2}g^{\mu\nu}\mathcal{R}.$ The contracted Bianchi identity guarantees the conservation law $\grad_\mu G^{\mu\nu} = 0.$ The Einstein field equation is given by $G_{\mu\nu} = 8\pi GT_{\mu\nu}.$ ##### Einstein-Hilbert Action The Einstein-Hilbert action is defined by $\color{orange} S_{\mathrm{EH}} = \frac{1}{16\pi G} \int \mathcal{R} \sqrt{-g}\dd{^4 x}.$ The action of matter is given by $S_{\mathrm{M}} = \int \mathcal{L}(\phi) \sqrt{-g} \dd{^4 x}.$ Under the variation $g_{\mu\nu} \rightarrow g_{\mu\nu} + \delta g_{\mu\nu},$ we have • $\delta g^{\mu\nu} = -g^{\mu\kappa}g^{\lambda\nu}\delta g_{\kappa\lambda},$ • $\delta g = gg^{\mu\nu}\delta g_{\mu\nu},$ • $\delta \sqrt{\abs{g}} = \frac{1}{2}\sqrt{\abs{g}}g^{\mu\nu}\delta g_{\mu\nu},$ • Palatini identity. $\delta \operatorname{Ric}_{\mu\nu} = \grad_\kappa \delta \Gamma{^\kappa}_{\nu\mu} - \grad_{\nu} \delta \Gamma{^\kappa}_{\kappa\mu}.$ $\displaystyle \delta g^{\mu\nu} R_{\mu\nu} \neq \delta g_{\mu\nu} R^{\mu\nu}.$ • $\color{darkcyan}\delta S_{\mathrm{EH}} = -\frac{1}{16\pi G} \int G^{\mu\nu} \delta g_{\mu\nu} \sqrt{-g}\dd{^4 x}.$ • We have used $\qty(\sqrt{-g}V^\mu)_{;\mu} = \qty(\sqrt{-g}V^\mu)_{,\mu},$ which may be proved by the Jacobi's formula, to eliminate total divergence terms. • $\color{orange} \delta S_{\mathrm{M}} = \frac{1}{2} \int T^{\mu\nu} \delta g_{\mu\nu} \sqrt{-g} \dd{^4 x}.$ • $$T^{\mu\nu}$$ is the energy-momentum tensor. • Action principle $\delta(S_{\mathrm{EH}} + S_{\mathrm{M}}) = 0.$ • Einstein field equations: $\color{red} G_{\mu\nu} = 8\pi GT_{\mu\nu}.$ ##### String Theory • String: • One-dimension object. • Parametrized by $$(\sigma,\tau)$$. • Position given by $$X^\mu(\sigma,\tau)$$. • We let $$\alpha,\beta,\cdots$$ take value $$0$$ and $$1$$, representing $$\sigma$$ and $$\tau$$ respectively. • $$\mu,\nu,\cdots$$ take value up to $$D$$. • Nambu action: $S = -\frac{1}{2\pi\alpha'} \int_0^\pi \dd{\sigma} \int_{\tau_{\mathrm{i}}}^{\tau_{\mathrm{f}}} \dd{\tau} [-\det(\partial_\alpha X^\mu \partial_\beta X_\mu)]^{1/2}.$ • $$\partial_0$$ for $$\partial/\partial\tau$$ and $$\partial_1$$ for $$\partial/\partial\sigma$$. • Polyakov action: $S = -\frac{1}{4\pi\alpha'} \int_0^\pi \dd{\sigma} \int_{\tau_{\mathrm{i}}}^{\tau_{\mathrm{f}}} \dd{\tau} \sqrt{-g} g^{\alpha\beta} \partial_\alpha X^\mu \partial_\beta X_\mu.$ • Agrees with the Nambu action after quantization only for $$D=26$$. • Invariant under • local reparametrization of the world sheet $\begin{cases} \tau \rightarrow \tau'(\tau,\sigma), \\ \sigma \rightarrow \sigma'(\tau,\sigma), \end{cases}$ • Weyl rescaling $g_{\alpha\beta} \rightarrow g'_{\alpha\beta} = e^{\phi(\sigma,\tau)} g_{\alpha\beta}.$ • global poincare transformation $X^\mu \rightarrow X^{\mu'} = {\Lambda}{^\mu}{_\nu} X^\nu + a^\nu.$ • $$\Lambda \in \mathrm{SO}(D-1,1)$$. • Equations of motion: • Variation with respect to $$X^\mu$$: $\partial_\alpha(\sqrt{-g} g^{\alpha\beta}\partial_\beta X_\mu) = 0.$ • Variation with respect to $$g_{\alpha\beta}$$: $g_{\alpha\beta} = \partial_\alpha X^\mu \partial_\beta X^\nu \eta_{\mu\nu}.$ #### Miscellany ##### Geometrical Meaning of the Riemann Tensor and the Torsion Tensor An $$m$$-dimensional manifold $$M$$ is said to be parallelizable if $$M$$ admits $$m$$ vector fields which are linearly independent everywhere. $$S^m$$ is parallelizable only if $$m=1,3$$ or $$7$$. ##### Comparison of Conventions Source Definition Conversion Nakahara, Frankel, Schutz, Carroll, TensoriaCalc $$\displaystyle [\grad_\mu, \grad_\nu] X^\kappa = {R}{^\kappa}_{\lambda\mu\nu} X^\lambda$$ $${R}{^\kappa}_{\lambda\mu\nu}={R}{^\kappa}_{\lambda\mu\nu}{}$$ C.B. Liang, Wald Abstract Index $$\displaystyle [\grad_a,\grad_b]X^c = -R_{abd}{^c} X^d$$ $$\displaystyle R_{abd}{^c}\leftrightarrow R{^c}_{dba}$$ ##### Curvature Tensor Using Computer Algebra System The TensoriaCalc package could calculate $$R{^\kappa}_{\lambda\mu\nu}$$ from the metric given. << "Documents/Wolfram Mathematica/TensoriaCalc.m"; g["Example"] = Metric[ SubMinus[i], SubMinus[j], DiagonalMatrix[{-Exp[2 A[r]], Exp[2 B[r]], r^2, r^2 Sin[$Theta]]^2}], CoordinateSystem -> {t, r, \[Theta], \[Phi]}, TensorName -> "h", StartIndex -> 0, ChristoffelOperator -> FullSimplify, RiemannOperator -> FullSimplify, RicciOperator -> FullSimplify, RicciScalarOperator -> FullSimplify ]; Rie[i_, j_, k_, l_] := Riemann[SuperMinus[i], SubMinus[j], SubMinus[k], SubMinus[l], g["Example"] ]; Module[{i, j, k, l}, For[i = 0, i < 4, i++, For[j = 0, j < 4, j++, For[k = 0, k < 4, k++, For[l = 0, l < 4, l++, If[! Rie[i, j, k, l] === 0, Print[Subscript[Superscript[R, i], j, k, l], "=", Rie[i, j, k, l]] ] ] ] ] ] ];  We could see \[ R{^1}_{010} = e^{2(A(r) - B(r))}(A'(r)^2 - A'(r)B'(r) + A''(r)),$ which coincides with our result in Obtaining the Curvature Tensor. ##### Energy-Momentum Tensor of Various Fields • Scalar field: \begin{align*} \mathcal{L} &= -g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi + m^2\phi^2, \\ T_{\mu\nu} &= 2\partial_\mu \phi \partial_\nu \phi - g_{\mu\nu}(g^{\kappa\lambda}\partial_\kappa \phi \partial_\lambda \phi + m^2 \phi^2). \end{align*}{} ##### Spinors in Curved Spacetime $\color{orange} S_\psi = \int \dd{^4 x} \sqrt{-g} \overline{\psi}\qty[i\gamma^\alpha e{_\alpha}{^\mu} (\partial_\mu + A_\mu + \frac{1}{2} i\Gamma{^\beta}{_\mu}{^\gamma} \Sigma_{\beta\gamma}) + m]\psi.$ • $$e_{\alpha}{^\mu}$$ is the vierbein. • $\Sigma_{\alpha\beta} = \frac{i}{4}[\gamma_\alpha,\gamma_\beta]$ is the representation of the generators of the Lorentz transformation. Expand ## Thermodynamics and Statistical Physics (I) #### Thermodynamics ##### The Zeroth Law Let consider the following three systems: 1. a wire of length $$L$$ with tension $$F$$, 2. a paramagnetic of magnetization $$M$$ in a magnetic field $$B$$, and 3. a gas of volume $$V$$ at pressure $$P$$. The equilibrium of the first and the third ststem implies $\qty(P+\frac{a}{V^2})(V-b)(L-L_0) - c\qty[F-K(L-L_0)] = 0$ while of the second and the thrid implies $\qty(P+\frac{a}{V^2})(V-b)M - dB = 0.$ These conditions induces the following definition of temperature: $\Theta \propto \qty(P+\frac{a}{V^2})(V-b) = c\qty(\frac{F}{L-L_0}-K) = d\frac{B}{M}.$ The constraints above implies the following equations of states: \begin{gather*} (P+a/V^2)(V-b) = Nk_B T, \\ M = (M\mu_B^2 B)/(3k_B T), \\ F = (K+DT)(L-L_0). \end{gather*} ##### The First Law The first law may be written as $\dd{E} = \mathfrak{d} Q + \mathfrak{d} W.$ A quasi-static transformation is one that is performed sufficiently slowly so that the system is always in equilibrium. Thus in any stage of the process the thermodynamics coordinates of the system exist and can in principle be computed. By slowing stretching a spring, the external force is matched by the internal force $$F$$ from the spring. The change in the potential energy is $$\int F\dd{L}$$. However, if the spring is stretched abruptly, some of the external work is converted into kinetic energy. We may introduce a set of generalized displacements $$\qty{\vb*{x}}$$ and their conjugate generalized forces $$\qty{\vb*{J}}$$, such that for an infinitesimal quasi-static transformation $\mathfrak{d} W = \sum_i J_i \dd{x_i}.$ System Force Displacement Wire tension $$F$$ length $$L$$ Film surface tension $$\mathcal{S}{}$$ area $$A$$ Fliud pressure $$-P$$ volume $$V$$ Magnet magnetic field $$H$$ magnetization $$M$$ Dielectric electric field $$E$$ polarization $$P$$ Chemical reaction chemical potential $$\mu$$ particle number $$N$$ Heat capacities may be expressed as \begin{align*} C_V &= \eval{\pdv{E}{T}}_P, \\ C_P &= \eval{\pdv{E}{T}}_P + P\eval{\pdv{V}{T}}}_P. \end Force constants are the ratio of displacement to force. • Isothermal compressibility of a gas $\kappa_T = -\frac{1}{V}\eval{\pdv{V}{P}}_T.$ • susceptibility of a magnet ##### Ensembles System Exchange Isolated None Closed Energy Open Energy and Particle ##### Conditions of Equilibrium • Mechanical equilibrium: $$\sum \vb*{f}=0$$ every where. • Chemical equilibrium: no net flow of composition. • Phase equilibrium: no net flow of compositions in different phases. • Thermal equilibrium: no net flow of heat. Expand ## Galois Theory (I) #### Review: Field Theory ##### Roots of Polynomials Let $$f$$ be an irreducible polynomial in $$F[x]$$. • $$f$$ has no multiple roots in any extension of $$F$$, unless $$f'=0$$. • If $$F$$ has characteristic $$0$$, then $$f$$ has no multiple roots in any extension of $$F$$. Heuristically, we write a polynomial with multiple roots as $f(x) = g(x)(x-\alpha)^2.$ Then $f'(x) = g'(x)(x-\alpha)^2 + 2g(x)(x-\alpha)$ has a common factor $$(x-\alpha)$$ with $$f(x)$$. ##### Primitive Element Theorem Let $$K$$ be an extension of $$F$$. If the extension $$K/F$$ is generated by a single element $$\alpha$$, then $$\alpha$$ is called the primitive element of $$K/F$$. We have the following theorem: Primitive Element Theorem. If $$F$$ has characteristic $$0$$, then any extension $$K/F$$ has a primitive element. The extension $$K=\mathbb{Q}\qty(\sqrt{2},\sqrt{3})$$ is of degree $$4$$. We could easily prove that $$K=\mathbb{Q}\qty(\sqrt{2}+\sqrt{3})$$, i.e. the extension has a primitive element $$\alpha = \sqrt{2} + \sqrt{3}{}$$. We may verify that $\sqrt{2} = \frac{1}{2}\qty(\alpha^3-9\alpha)$ and that $\sqrt{3} = -\frac{1}{2}\qty(\alpha^3-11\alpha).$ Such decomposition exists because $$\alpha^i$$ could always be written as linear combinations of $$\sqrt{2}{}$$, $$\sqrt{3}{}$$ and $$\sqrt{6}{}$$. By solving a linear equation we retrieve $$\sqrt{2}{}$$ and $$\sqrt{3}{}$$ from powers of $$\alpha$$. #### Symmetries of Polynomials ##### Symmetric Polynomials $$S_n$$ acts on $$R[u]$$ by $f=f(u_1,\cdots,u_n)\mapsto f(u_{\sigma 1},\cdots,u_{\sigma n}) = \sigma f.$ This action is straightforward for someone who has experience with quantum mechanics. By regarding $$f$$ as the wavefunction of a many-body system, $$\sigma f$$ becomes the wavefunction after permutation of particles. A polynomial $$g$$ is symmetric if $$g$$ is invariant under any permutation, i.e. if all monomials in the same orbit have the same coefficient. Theorem. Any symmetric polynomial could be written (uniquely) as a linear combination of elementary symmetric polynomials. We define the elementary symmetric polynomials by \begin{align*} s_1 &= u_1 + u_2 + \cdots + u_n, \\ s_2 &= u_1u_2 + u_1u_3 + \cdots, \\ s_3 &= u_1u_2u_3 + \cdots, \\ s_n &= u1 u_2 \cdots u_n. \end{align*} We have $u_1^2 + \cdots + u_n^2 = s_1^2 - 2s_2.$ Corollary. If $$f(x) = x^n - a_1x^{n-1} + \cdots + a_n$$ has coefficients in $$F$$ and splits over $$K$$ with roots $$\alpha_1$$, $$\cdots$$, $$\alpha_n$$. Let $$g\qty(u_1,\cdots,u_n)$$ be a symmetric polynomial with coefficients in $$F$$, then $$g(\alpha_1,\cdots,\alpha_n)$$ is in $$F$$. From the Vieta's formulae we know that the roots $$x_1$$ and $$x_2$$ of $x^2 - bx + c = 0$ satisfies $x_1 + x_2 = b,\quad x_1x_2 = c$ even if the roots are in $$\mathbb{R}\backslash \mathbb{Q}$$ while the coefficients are in $$\mathbb{Q}$$. Moreover, symmetric polynomials like $x_1^2 + x_2^2 = (x_1+x_2)^2 - 2x_1x_2$ evaluate to values in $$\mathbb{Q}$$ as well. Proposition. Let $$h$$ be a symmetric polynomial and $$\qty{p_1,\cdots,p_k}$$ be the orbit of $$p_1$$ under $$S_n$$, then $h(p_1,\cdots,p_k)$ is a symmetric polynomial. For the simplest case we take $h = p_1 + \cdots + p_k,$ i.e. the sum of every element in the orbit. ##### Discriminant Let the roots of $P(x) = x^n - s_1 x^{n-1} + s_2 x^{n-2} - \cdots \pm s_n$ be $$\qty{u_1,\cdots,u_n}$$. The discriminant of $$P$$ is defined by $D(u) = (u_1 - u_2)^2 (u_1 - u_3)^2 \cdots (u_{n-1} - u_n)^2 = \prod_{i<j} (u_i - u_j)^2.$ • $$D(u)$$ is a symmetric polynomial with integer coefficients. • In particular, if the coefficients are in $$F$$, then $$D(u)$$ evaluates to an element in $$F$$. • If $$\qty{\alpha_i}$$ are elements in a field, then $$D(u) = 0$$ if and only if $$P$$ has a multiple root. We are able to write $$D$$ as a sum of elementary symmetric polynomials. In the case of degree 2, we have $D = (u_1 - u_2)^2 = s_1^2 - 4s_2.$ #### Fields and Polynomials We assume that $$F$$ has characteristic $$0$$ hereinafter. ##### Splitting Field $$f$$ splits over $$K$$ if $f(x) = (x-\alpha_1) \cdots (x-\alpha_n)$ in $$K$$. Let $$f$$ be a polynomial over $$F$$. The splitting field of $$f$$ is an extension $$K/F$$ such that • $$f$$ spilits in $$K$$; and • $$K$$ is generated by the roots, i.e. $K = F\qty(\alpha_1,\cdots,\alpha_n).$ Lemma. Let $$f$$ be a polynomial over $$F$$. • Let $$F\subset L\subset K$$ be a field, and $$K$$ be the splitting field of $$f$$, then $$K$$ is also the splitting field of $$f$$ as a polynomial over $$L$$. • Every $$f$$ over $$F$$ has a splitting field. • The splitting field is a finite extension of $$F$$. Every finite extension of $$F$$ is contained in a splitting field. Splitting Field Theorem. Let $$K$$ be the splitting field of $$f$$ as a polynomial over $$F$$. Let $$g$$ be an irreducible polynomial over $$F$$. If $$g$$ has a root in $$K$$, then $$g$$ split over $$K$$. Outline of Proof. Let $$K$$ be generated by $$\alpha_1$$,$$\cdots$$,$$\alpha_n$$, and $$\beta_1$$ be a root of $$G$$ in $$K$$. Let $p_1(\alpha) = \beta_1,$ and $$\qty{p_1,\cdots,p_k}$$ be the orbit of $$p_1$$ under $$S_n$$ and $\beta_k = p_k(\alpha).$ We may prove that the coefficients of $h(x) = (x-\beta_1)\cdots (x-\beta_k)$ are in $$F$$. Therefore, $$g$$ divides $$h$$ since $$h$$ has $$\beta_1$$ as a root and $$g$$ splits since $$h$$ splits. ##### Isomorphism of Field Extension • Let $$K$$ and $$K'$$ be extensions of $$F$$. An $$F$$-isomorphism is a field isomorphism between $$K$$ and $$K'$$ that fixes every element in $$F$$. • An $$F$$-automorphism of $$K$$ is an $$F$$-isomorphism from $$K$$ to itself, i.e. the symmetries of the extension $$K/F$$. • The $$F$$-automorphisms of a finite extensions $$K/F$$ form a group $$G(K/F)$$, called the Galois group. • A finite extension $$K/F$$ is called a Galois extension if $\abs{G(K/F)} = \qty[K:F].$ For any quadratic extension $$K/F$$, e.g. $$\mathbb{C}/\mathbb{R}$$ and $$\mathbb{Q}(\sqrt{2})/\mathbb{Q}$$, the Galois group has two elements: the trivial one, and the conjugation, i.e. $a+bi \mapsto a-bi$ or $a+b\sqrt{2} \mapsto a-b\sqrt{2}.$ Let $$F=\mathbb{Q}$$ and $$K=F[\sqrt[3]{2}]$$. Since $$x^3-2=0$$ has two more roots which are complex and do not belong to $$K$$, we have $G(K/F) = \qty{e}$ while $$[K:F]=3$$. Therefore, the extension is not Galois. Lemma. Let $$K$$ and $$K'$$ be extensions of $$F$$. • Let $$f(x)$$ be a polynomial with coefficients in $$F$$, and $$\sigma$$ be an $$F$$-isomorphism from $$K$$ to $$K'$$. If $$\alpha$$ is a root of $$f$$ in $$K$$, then $$\sigma(\alpha)$$ is a root of $$f$$ in $$K'$$. • Let $$K=F(\alpha_1,\cdots,\alpha_n)$$, and $$\sigma$$ and $$\sigma'$$ be $$F$$-isomorphisms from $$K$$ to $$K'$$. If for $$i=1,\cdots,n$$ we have $$\sigma(\alpha_i) = \sigma'(\alpha_i)$$, then $$\sigma = \sigma'$$. If $$\sigma$$ fixes every $$\alpha_i$$, then $$\sigma$$ is the identity mapping. • If $$f$$ is an irreducible polynomial over $$F$$, and $$\alpha$$ and $$\alpha'$$ are roots of $$f$$ in $$K$$ and $$K'$$ respectively, then there exists a unique $$F$$-isomorphism $$\sigma$$ that maps $$\alpha$$ to $$\alpha'$$. If $$F(\alpha) = F(\alpha')$$ then $$F$$ is the identity mapping. Proposition. Let $$f$$ be a polynomial over $$F$$. • An extension $$L/F$$ contains at most one splitting field of $$f$$ over $$F$$. • Splitting fields of $$f$$ over $$F$$ are all isomorphic to each other. ##### Fixed Field Let $$H$$ be an automorphism group of $$K$$. The fixed field of $$H$$ consists of all the elements invariant under every elements of $$H$$, $K^H = \Set{\alpha\in K|\sigma(\alpha)=\alpha,\ \text{for all } \sigma \in H}.$ Theorem. Let $$H$$ be a finite automorphism group of $$K$$, and $$F$$ be the fixed field $$K^H$$. Let $$\beta_1$$ be an element of $$K$$ and $$\qty{\beta_1,\cdots,\beta_r}$$ be the $$H$$-orbit of $$\beta_1$$. • The irreducible polynomial of $$\beta_1$$ over $$F$$ is given by $g(x) = (x-\beta_1)\cdots (x-\beta_r).$ • $$\beta_1$$ is algebraic over $$F$$, and the degree thereof is the cardinality of the $$H$$-orbit thereof. • In particular, degree of $$\beta_1$$ divides the order of $$H$$. Let $$K=\mathbb{C}$$ and $$H = \qty{e, c}$$ where $$c$$ denotes conjugation. Then $$F = \mathbb{R}$$ and the irreducible polynomial of $$\beta \in \mathbb{C}$$ is given by • $$g(x) = x-\beta$$ if $$\beta \in \mathbb{R}$$, or • $$g(x) = (x-\beta)(x-\overline{\beta})$$ if $$\beta \in \mathbb{C}\backslash \mathbb{R}$$. Lemma. Let $$K$$ be an algebraic extension of $$F$$ and not be a finite extension. Then there exists elements in $$K$$ of arbitrary large degrees over $$F$$. Fixed Field Theorem. Let $$H$$ be a finite automorphism group of $$K$$ and $$F=K^H$$ be the fixed field thereof. Then $$K$$ is a finite extension over $$F$$, the degree $$[K:H]$$ of which equals $$\abs{H}$$. $[K:K^H] = \abs{H}.$ Let $$K=\mathbb{C}(t)$$, and $$\sigma$$ and $$\tau$$ be $$\mathbb{C}$$-automorphisms of $$K$$, such that $\sigma(t) = it$ and $\tau(t) = t^{-1}.$ We have $\sigma^4 = 1,\quad \tau^2 = 1,\quad \tau\sigma = \sigma^{-1}\tau.$ Therefore, $$\sigma$$ and $$\tau$$ generate the $$D_4$$ group. We may easily verify that $$u = t^4 + t^{-4}$$ is transcendental over $$\mathbb{C}$$. Now we prove that $$\mathbb{C}(u) = K^H$$. Since $$u$$ is fixed by $$\sigma$$ and $$\tau$$ we have $$\mathbb{C}(u)\in K^H$$. The $$H$$-orbits of $$t$$ is given by $\qty{t,it,-t,-it,t^{-1},-it^{-1},-t^{-1},it^{-1}}.$ From the theorem above we obtain the irreducible polynomial of $$t$$ over $$\mathbb{C}(u)$$: $x^8 - ux^4 + 1.$ • Therefore, $$[K:\mathbb{C}(u)] \le 8$$. • From the fixed field theorem we know $$[K:K^H] = 8$$. • We have already proved that $$\mathbb{C}(u) \subset K^H$$. • Thus we conclude that $$\mathbb{C}(u) = K^H$$. The example above illustrates the Lüroth theorem: Let $$F \subset \mathbb{C}(t)$$ be a field that contains $$\mathbb{C}$$ and not be $$\mathbb{C}$$ itself. Then $$F$$ is isomorphic to $$\mathbb{C}(u)$$. #### Fundamentals of Galois Theory ##### Galois Extension Lemma. • Let $$K/F$$ be a finite extension. Then $$G(K/F)$$ is a fintie group and $\abs{G(K/F)}\mid [K:F].$ • Let $$H$$ be a finite automorphism group of $$K$$. Then $$K$$ is a galois extension of $$K^H$$, and $$H=G(K/K^H)$$. Lemma. Let $$\gamma_1$$ be the primitive element of a finite extension $$K/F$$ and $$f(x)$$ be the irreducible polynomial of $$\gamma_1$$ over $$F$$. Let $$\gamma_1,\cdots,\gamma_r$$ be roots of $$f$$ in $$K$$. Then there exists a unique $$F$$-automorphism $$\sigma_i$$ of $$K$$ such that $$\sigma_i(\gamma_1) = \gamma_i$$. These are all the $$F$$-automorphisms. Therefore, $$\abs{G(K/F)}=r$$. Characteristics of Galois Extension. Let $$K/F$$ be a finite extension and $$G$$ be the Galois group thereof.Then the following statements are equivalent: • $$K/F$$ is a Galois extension, i.e. $$\abs{G} = [K:F]$$. • $$K^G=F$$. • $$K$$ is a splitting field over $$F$$. If $$K$$ is a splitting field of $$f$$ over $$F$$, we may refer to $$G(K/F)$$ as the Galois group of $$f$$. Corollary. • Every finite extension $$K/F$$ is contained in a Galois extension. • If $$K/F$$ is a Galois extension, and $$L$$ is an intermediate field, then $$K/L$$ is also a Galois extension, and $$G(K/L)$$ is a subgroup of $$G(K/F)$$. Theorem. Let $$K/F$$ be a Galois extension and $$G$$ be its Galois group, and $$g$$ be a polynomial over $$F$$ that splits in $$K$$ with roots $$\beta_1,\cdots,\beta_r$$. • $$G$$ acts on $$\qty{\beta_i}$$. • If $$K$$ is the splitting field of $$G$$ over $$F$$, then the permutation of $$G$$ on $$\qty{\beta_i}$$ is faithful and $$G$$ is a subgroup of $$S_r$$. • If $$g$$ is irreducible over $$F$$, then the action of $$G$$ on $$\qty{\beta_i}$$ is transitive. • If $$K$$ is the splitting field of $$g$$ over $$F$$ and that $$g$$ is irreducible over $$F$$, then $$G$$ is a transitive subgroup of $$S_r$$. ##### The Fundamental Theorem The Fundamental Theorem. Let $$K$$ be a Galois extension over $$F$$, and $$G$$ be the Galois group thereof. Then there is a one-to-one correspondence between $\qty{\text{subgroup } H \text{ of } G}$ and $\qty{\text{intermediate field } L \text{ of } K/F}.$ The mapping $H\mapsto K^H$ is the inverse of $L\mapsto G(K/L).$ Corollary. • If $$L\subset L'$$ are intermediate fields, and the corresponding subgroups are $$H$$ and $$H'$$, then $$H\supset H'$$. • The subgroup corresponding to $$F$$ is $$G(K/F)$$, while to $$G$$ is the trivial group $$\qty{e}$$. • If $$L$$ corresponds to $$H$$, then $$[K:L]=H$$ and $$[L:F]=[G:H]$$. Corollary. A finite extension $$K/F$$ has finitely many intermediate fields $$F\subset L\subset K$$. Let $$F=\mathbb{Q}$$, and $$\alpha=\sqrt{2}$$, $$\beta=\sqrt{5}$$, and $$K = F(\alpha,\beta)$$. Then $$G=G(K/\mathbb{Q})$$ is of order $$4$$. Since $F(\alpha),\quad F(\beta),\quad F(\alpha\beta)$ are three distinct intermediate fields, $$G$$ has three nontrivial subgroups and therefore $$G$$ is the Klein group. Therefore, there are not other intermediate fields. Theorem. Let $$K/F$$ is a Galois extension and $$G$$ be the Galois group thereof. Let $$L=K^H$$ where $$H$$ is a subgroup of $$G$$. Then • $$L/F$$ is Galois if and only if $$H$$ is a normal subgroup of $$G$$; and • in this case, $$G(L/F) \cong G/H$$. #### Applications ##### Cubic Equation Let $f(x) = x^3 - a_1 x^2 + a_2 x - a_3$ be a irreducible polynomial over $$F$$ and $$K$$ be the splitting field thereof. Let $$\qty{\alpha_1,\alpha_2,\alpha_3}$$ denotes the roots of $$f$$. We have $F\subset F(\alpha_1) \subset F(\alpha_1,\alpha_2) = F(\alpha_1,\alpha_2,\alpha_3) = K$ since $\alpha_3 = a_1 - \alpha_1 - \alpha_2.$ Let $$L=F(\alpha_1)$$, then $f(x) = (x-\alpha_1) q(x)$ over $$L$$. • If $$q$$ is irreducible over $$L$$, then $$[K:L] = 2$$ and $$[K:F] = 6$$. • If $$q$$ is reducible over $$L$$, then $$L=K$$ and $$[K:F]=3$$. Let $$f(x) = x^3 + 3x + 1$$. $$f(x)$$ is irreducible over $$\mathbb{Q}$$ and has one real root and two complex roots. Therefore, $$[K:F]=6$$. Let $$f(x) = x^3 - 3x + 1$$. $$f(x)$$ is irreducible over $$\mathbb{Q}$$. If $$\alpha_1$$ is one root then $$\alpha_2 = \alpha_1^2 - 2$$ is another. Therefore, $$[K:F]=3$$. Let $$f$$ be a irreducible polynomial (having no multiple root therefore) over $$F$$. Then $$G(K/F)$$ is a transitive subgroup of $$S_3$$. We have either $$G = S_3$$ or $$G = A_3$$, corresponding to $$[K:F] = 6$$ and $$[K:F] = 3$$ respectively. Galois Theory of Cubic Equation. Let $$K$$ be the splitting field of a cubic polynomial $$f$$ over $$F$$, and $$G = G(K/F)$$. • If $$D$$ is a square over $$F$$, then $$[K:F] = 3$$ and $$G = A_3$$. • If $$D$$ is not a square over $$F$$, then $$[K:F] = 6$$ and $$G = S_3$$. The discriminant of $$f=x^2+3x+1$$ is $$-5\cdot 3^2$$, which is not a square. Therefore, $$[K:F] = 6$$. The discriminant of $$f=x^2-3x+1$$ is $$3^4$$, which is a square. Therefore, $$[K:F] = 3$$. ##### Quartic Equation Galois Theory of Quartic Equation. Let $$G$$ be the Galois group of an irreducible quartic polynomial. Then the discriminant $$D$$ of $$f$$ is a square in $$F$$ if and only if $$G$$ does not contain odd permutations. Therefore, • If $$D$$ is a square in $$F$$, then $$G = A_4$$ or $$D_2$$. • If $$D$$ is not a square in $$F$$, then $$G = S_4$$, $$D_4$$ or $$C_4$$. Denote the roots by $$\alpha_i$$ and let $\beta_1 = \alpha_1 \alpha2 + \alpha_3 \alpha_4,\quad \beta_2 = \alpha_1 \alpha_3 + \alpha_2 \alpha_4,\quad \beta_3 = \alpha_1 \alpha_4 + \alpha_2 \alpha_3,$ and $g(x) = (x-\beta_1)(x-\beta_2)(x-\beta_3).$  $$D$$ is a square $$D$$ is not a square $$g$$ is reducible $$G=D_2$$ $$G=D_4$$ or $$C_4$$ $$g$$ is irreducible $$G=A_4$$ $$G=S_4$$ ##### Quintic Equation Proposition. Let $$F$$ be a subfield of $$\mathbb{C}$$. Then $$\alpha\in\mathbb{C}$$ is solvable over $$F$$ if it satisfy either of the following two equivalent conditions: • There is a chain of subfields of $$\mathbb{C}$$ $F = F_0 \subset F_1 \subset \cdots \subset F_r = K$ such that • $$\alpha\in K$$; and • $$F_j = F_{j-1}(\beta_j)$$, where a power of $$\beta_j$$ is inF_{j-1}.
• There is a chain of subfields of $$\mathbb{C}$$ $F = F_0 \subset F_1 \subset \cdots \subset F_r = K$ such that
• $$\alpha\in K$$; and
• $$F_{j+1}/F_j$$ is a Galois extension of prime degree.

Theorem. Let $$f$$ be a irreducible quintic polynomial over $$F$$ and that the Galois group thereof is $$A_5$$ or $$S_5$$. Then the roots of $$F$$ are not solvable over $$F$$.

Proof. We assume that $$G=A_5$$, and let

• $$K$$ be the splitting field of $$f$$,
• $$F'/F$$ be a Galois extension of prime degree $$p$$,
• $$F'$$ be the splitting field of $$g$$ with roots $$\beta_1,\cdots,\beta_p$$,
• $$\alpha_1,\cdots,\alpha_5$$ be the roots of $$f$$, and
• $$K'$$ denote the splitting field of $$fg$$.

The Galois extensions and Galois groups are listed in the graph below.

Now we prove that $$H\cong G = A_5$$.

• $$H\cap H'$$ is the trivial group $$\qty{e}$$ since an element thereof fixes all the roots $$\alpha_i$$ and $$\beta_j$$.
• We restrict the canonical mapping $$\mathcal{G} \mapsto \mathcal{G}/H \cong G'$$ to $$H'$$, i.e. permutation of $$\beta$$ in $$K' = F(\alpha,\beta)$$ to permutation of $$\beta$$ in $$F' = F(\beta)$$.
• The kernel is the trivial group $$\qty{e}$$, i.e. this is an injection.
• Either $$H'$$ is trivial, or $$H'$$ is the cyclic group of order $$p$$.
• $$H'$$ can not be trivial since $$G=A_5$$ has no normal subgroups.
• Therefore, $$H'$$ is the cylic group of order $$p$$, and $$\abs{H} = \abs{G} = \abs{A_5}$$. Now we restrict the canonical mapping $$\mathcal{G} \mapsto \mathcal{G}/H' \cong G$$ to $$H$$, i.e. permutation of $$\alpha$$ in $$K' = F(\alpha,\beta)$$ to permutation of $$\alpha$$ in $$F' = F(\alpha)$$. Again this is an interjection and we found that $$H=A_5$$.

Take $$x^4 - 4x^2 + 2$$ for example. The roots are given by $\alpha = \pm\sqrt{2\pm\sqrt{2}}.$ Taking $$F = \mathbb{Q}$$, $$K'=K = F(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$$ and $$F' = F(\sqrt{2})$$, we find that $K'=K=F(\sqrt{2},\sqrt{2+\sqrt{2}},\sqrt{2-\sqrt{2}}).$ $$K'=K$$ looks in $$F'$$ like $K'=K=F'(\sqrt{2+\sqrt{2}})$ since $(\sqrt{2}-1)\sqrt{2+\sqrt{2}} = \sqrt{2-\sqrt{2}}.$ We find that $G'\cong C_2,\quad H \cong C_2.$

##### Remark

To obtain the Galois group of a certain polynomial, one may go to the Magma online calculator and submit the following code:

P<x> := PolynomialAlgebra(Rationals());
f := x^4 - 2*x^2 - 1;
G := GaloisGroup(f);
print G;


Expand

To make an SSH tunnel, one should

1. On the remote machine, open port 12345 (for example), e.g.
sudo iptables -I INPUT -p tcp --dport 12345 -j ACCEPT

1. On the remote machine, append the following line to /etc/ssh/sshd_config:
GatewayPorts clientspecified

1. On the remote machine, restart the sshd service with
systemctl restart sshd

1. On the local machine, run
ssh -v -R 0.0.0.0:12345:127.0.0.1:80 -N user@remote.site


which redirect port 12345 on the remote machine to port 80 on the local machine.

Expand

## Usage of 連用形

Note from ren'youkei 連用形.

#### As Noun

しばto tie with ropeしばa restraint, tying

かんじる to feel → このかんthis feeling; いいかんgood feeling

##### After の

Replacing the が particle by the の particle refers to the action of the subject.

For する-verbs, the し is not preserved after の.

キノが旅する Kino journeys

キノの旅 the journey of Kino

It's also possible to change the adverb that modified a verb into an adjective that modified the nound.

##### Before に

Adding に after the noun form expresses the objective of another action.

##### Limitations as Noun
1. 連用形 only turns the verb alone into noun.

私わ漫画を読みが好きだ

私は漫画を読むのが好きだ

2. Sometimes the meaning is idiosyncratic.

彼は話すのが早い he talks fast

彼は話が早い he understands things quickly

An example after の:

私の読みが正しければ if my guess is correct

3. Sometimes the noun form is spelled without 送り仮名.

ひかる → ひかり

はなす → はなし

#### Before ながら

The 連用形 coming before ながら means that two events are occuring concurrently.

#### Before つつ

つつ can be synonymous with ながら.

つつ can also mean the continuation of an event (changing), in the pattern つつある.

##### Comparison: つつある and 続く

つつ is used to express how a change keeps progressing, while 続く is used to express that something continues happening.

Expand

## Topological Insulator (I)

#### From the Hall Effect to Quantum Spin Hall Effect

The Hall resistance is given by $R_H = \frac{V_H}{I}.$ In the anomalous Hall effect (AHE), the Hall resistance may have in addition to the contribution from $$B$$ a contribution from $$M$$, i.e. $R_H = R_O B + R_A M.$ The AHE has extrinsic and intrinsic origin.

• Extrinic origin: disorder-related spin-dependent scattering of carriers.
• Intrinic origin: spin-dependent structure of conduction electrons. The spin-orbit effect induces a spin-dependent transverse force and deflects electrons of different spins to different directions.

The spin Hall effect results in spin accumulation on the lateral surfaces of a current-carrying sample, the signs of the spin orientations being opposite on two opposite boundaries.

• Extrinsic: spin-dependent scattering of carriers.
• Intrinsic: occurs even without impurity scattering.
• Resonant spin Hall effect: a small current induces a finite spin current and polarization.

Schematic of the SHE

Schematic of the inverse SHE

The quantum Hall effect is the phenomenon that the in a strong magnetic field, the longitudinal conductance becomes zero while the quantum plateau of Hall conductance appears at $$\nu e^2/h$$, where $$\nu = 1,2,\cdots$$. When one Laudau level is fully filled, the filling factor is $$\nu = 1$$.

In a sample of higher mobility, there may be fractional quantum Hall effect where $$\nu = 1/3,2/3,1/5,2/5,...$$ Such effect in a topological phase of composite fermions, which breaks time reversal symmetry.

In a lattice system of spinless electrons in a periodic magnetic flux, electrons may be driven to form a conducting edge channel by the periodic magnetic flux, leading to the quantum anomalous Hall effect, i.e. the QHE in the absence of an external field or landau levels.

The quantum spin Hall effect is a quantum version of the SHE or a spin version of the QHE, and can be regarded as a combination of two quantum anomalous Hall effects of spin-up and spin-down electrons with opposite chirality.

 Hall Effect$$B\neq 0$$1879 → AHE$$B=0$$, $$M\neq 0$$1880 → SHE$$B=0$$, $$M=0$$2004-2006 ↓ ↓ ↓ QHE$$\sigma_H = n\dfrac{e^2}{h}$$1980/1982 → QAHE$$\sigma_H = \dfrac{e^2}{h}$$not yet confirmed → QSHE$$\sigma_S = \dfrac{e}{4\pi}$$2007

#### Topological Insulators as Generalization of QSHE

A topological insulator is a state of a quantum matter that behaves as an insulator in its interior while as a metal on its boundary.

• Electron spins in the surface states are locked to their momenta.
• In a weak topological insulator, the resultant surface states are not so stable to disorder or impurities.
• In a strong topological insulator the surface states are protected by time reversal symmetry.

Some examples are listed as follows.

• One-dimsional: polyacetylene.
• Two-dimensional: $$\ce{HgTe/CdTe}{}$$ quantum well and $$\ce{InAs/CdTe}{}$$ quantum well.
• Three-dimensional: $$\ce{Bi_{1-x}Sb_x}{}$$, $$\ce{Bi2Te3}{}$$, $$\ce{Bi2Te2Se}{}$$.

#### The Dirac Equation

In the Dirac equation (see my note) we set $H = c\vb*{p}\cdot \alpha + mc^2 \beta.$ Under a transformation of mass $$m\rightarrow -m$$, the equation is invariant if we replace $$\beta \rightarrow -\beta$$. There is no topological distinction between particles with positive and negative masses.

The absence of a negatively charged electron that has a negative mass and kinetic energy would manifest itself as a positively charged particle which has an equal positive mass and positive energy.

• In one dimension we take $\alpha_x = \sigma_x,\quad \beta = \sigma_z.$
• In two dimensions we take $\alpha_x = \sigma_x,\quad \alpha_y = \sigma_y,\quad \beta = \sigma_z.$
• In three dimensions we take $\alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix},\quad \beta = \begin{pmatrix} \mathbb{1}_{2\times 2} & 0 \\ 0 & \mathbb{1}_{2\times 2} \end{pmatrix}.$

#### Solutions of Bound States

##### Jackiw-Rebbi Solution in One Dimension

Let $h(x) = -iv\hbar \partial_x \sigma_x + m(x) v^2 \sigma_z$ where $m(x) = \begin{cases} -m_1, & \text{if } x<0, \\ +m_2, & \text{otherwise}. \end{cases}$

The solution for $$x>0$$ is given by $\begin{pmatrix} \varphi_1(x) \\ \varphi_2(x) \end{pmatrix} = \begin{pmatrix} \varphi_1^+ \\ \varphi_2^+ \end{pmatrix} e^{-\lambda_+ x}$ where $\lambda_+ = \frac{\sqrt{m_2^2 v^4 - E^2}}{\nu \hbar}$ For $$x<0$$ the solution is similar. The continuity condition allows a solution at $$E=0$$.

More generally, for any $$m(x)$$ that changes from negative to positive at two ends, we also have a solution $\varphi_\eta(x) \propto \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ \eta i \end{pmatrix} \exp\qty[-\int^x \eta \frac{m(x')v}{\hbar}\dd{x'}],$ where $$\eta$$ is a sign, taking value $$\pm 1$$.

##### Two Dimensions

With $$p_z=0$$ we find two solutions $\Psi_+(x,k_y) = \sqrt{\frac{v}{h}\frac{m_1m_2}{m_1+m_2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ i \end{pmatrix} e^{-\abs{m(x) vx}/\hbar + ik_y y}$ and $\Psi_-(x,k_y) = \sqrt{\frac{v}{h}\frac{m_1m_2}{m_1+m_2}}\begin{pmatrix} 0 \\ 1 \\ i \\ 0 \end{pmatrix} e^{-\abs{m(x) vx}/\hbar + ik_y y}$ with dispersion relations $\epsilon_{k,+} = v\hbar k_y$ and $\epsilon_{k,-} = -v\hbar k_y,$ respectively. The two states move with velocity $v_\pm = \frac{\partial \epsilon_{k,\pm}}{\hbar \partial k} = \pm v.$

#### Quadratic Correction to the Dirac Equation

We introduce a correction in the Hamiltonian and get $H = v\vb*{p}\cdot \alpha + (mv^2 - B\vb*{p}^2)\beta.$

• $$p = 0$$: the spin orientation is determined by $$mv^2 \beta$$, i.e. the sign of $$m$$.
• $$p \rightarrow \infty$$: determined by $$-B\vb*{p}^2 \beta$$ or sign of $$B$$.
• $$mB>0$$: as $$p\rightarrow \infty$$, the spin rotate eventually to the opposite $$z$$-direction. This consists of two merons in the momentum space, a.k.a. skyrmion.
• $$mB<0$$: as $$p\rightarrow \infty$$, the spin rotate back to the $$z$$-direction.
##### One Dimension

The Dirac equation in this case is given by $h(x) = vp_x \sigma_x + (mv^2 - Bp_x^2) \sigma_z,$ which admits solutions for $$x\ge 0$$ under the Dirichlet condition $$\varphi(0) = 0$$ $\varphi_\eta(x) = \frac{C}{\sqrt{2}} \begin{pmatrix} \operatorname{sign}{B} \\ i \end{pmatrix}\qty(e^{-x/\xi_+} - e^{-x/\xi_-})$ where $\xi_\pm^{-1} = \frac{v}{2\abs{B}\hbar}\qty(1 \pm \sqrt{1-4mB}).$

• For $$B\rightarrow 0$$ we have
• $$\xi_+ \rightarrow \abs{B}\hbar/v$$,
• and $$\xi_- = \hbar/mv$$.
• Further let $$m\rightarrow 0$$ we have
• $$\xi_- \rightarrow \infty$$, where a topological phase transition occurs.

The four-component forms of the solutions are written as $\Psi_1 = \frac{C}{\sqrt{2}} \begin{pmatrix} \operatorname{sign}{B} \\ 0 \\ 0 \\ i \end{pmatrix} \qty(e^{-x/\xi_+} - e^{-x/\xi_-})$ and $\Psi_2 = \frac{C}{\sqrt{2}} \begin{pmatrix} 0 \\ \operatorname{sign}{B} \\ i \\ 0 \end{pmatrix} \qty(e^{-x/\xi_+} - e^{-x/\xi_-}).$

##### Two Dimensions: Helical Edge States

The exact solutions to $h_\pm = vp_x \sigma_x \pm vp_y\sigma_y + (mv^2 - Bp^2)\sigma_z$ is given by \begin{align*} \Psi_1 &= \frac{C}{\sqrt{2}} \begin{pmatrix} \operatorname{sign}{B} \\ 0 \\ 0 \\ i \end{pmatrix} \qty(e^{-x/\xi_+} - e^{-x/\xi_-}) e^{+ip_y y/\hbar}; \\ \Psi_2 &= \frac{C}{\sqrt{2}} \begin{pmatrix} 0 \\ \operatorname{sign}{B} \\ i \\ 0 \end{pmatrix} \qty(e^{-x/\xi_+} - e^{-x/\xi_-}) e^{+ip_y y/\hbar}, \\ \end{align*} where the characteristic length is given by $\xi_\pm^{-1} = \frac{v}{2\abs{B}\hbar}\qty(1\pm \sqrt{1-4mB + 4B^2p_y^2/v^2}).$

In two dimensions, the Chern number for a system with Hamiltonian $H = \vb{d}(p)\cdot \sigma$ is given by $n_c = -\frac{1}{4\pi}\int \dd{\vb*{p}} \frac{\vb{d}\cdot \qty(\partial_{p_x}\vb{d}\times \partial_{p_y}\vb{d})}{d^3}$ where the integral runs over the first BZ. The result is given by $n_\pm = \pm\qty(\operatorname{sign}{m} + \operatorname{sign}{B})/2.$ The system is topologically nontrivial if $$n\neq 0$$, i.e. $$m$$ and $$B$$ have the same sign.

Using the perturbation theory to the first order, the dispersion relation may be obtained to be $\epsilon_{p_y,\pm} = \pm vp_y.$

##### Three Dimensions: Surface States

In three dimensions the Hamiltonian is given by $H = H_{1D}(x) + H_{3D}(x)$ where $H_{3D}(x) = vp_y \alpha_y + vp_z\alpha_z - B(p_y^2 + p_z^2)\beta.$

The exact solutions are given by $\Psi_\pm = C \Psi_\pm^0 \qty(e^{-x/\xi_+} - e^{-x/\xi_-})\exp\qty[+i\qty(p_y y + p_z z)/\hbar]$ where \begin{align*} \Psi_+^0 &= \begin{pmatrix} \cos \frac{\theta}{2} \operatorname{sign}{B} \\ -i \sin \frac{\theta}{2} \operatorname{sign}{B} \\ \sin \frac{\theta}{2} \\ i \cos \frac{\theta}{2} \end{pmatrix}; \Psi_-^0 &= \begin{pmatrix} \sin\frac{\theta}{2}\operatorname{sign}{B} \\ i \cos \frac{\theta}{2} \operatorname{sign}{B} \\ -\cos \frac{\theta}{2} \\ i\sin \frac{\theta}{2} \end{pmatrix}, \tan\theta &= \frac{p_y}{p_z}, \end{align*} and the dispersion is given by $\epsilon_{p\pm} = \pm vp \operatorname{sign}{B}.$ The penetration depth is $\xi_\pm^{-1} = \frac{v}{2\abs{B}\hbar} \qty(1\pm \sqrt{1-4mB + 4B^2 p^2 /\hbar^2}).$

Under the condition $$mB>0$$,

• In one dimension, there exists bound state of zero energe near the ends.
• In two dimensions, there exists helical edge states nea the edge.
• In three dimensions, there exists surface states near the surface.

#### Berry Phase

##### General Formalism
• $$\vb*{R}$$ a set of parameter of the Hamiltonian.
• Schrödinger equation: $H(\vb*{R}) \ket{n(\vb*{R})} = E_n(\vb*{R}) \ket{n(\vb*{R})}.$
• Gauge freedom: an arbitrary phase factor $$e^{-i\theta(\vb*{R})}$$,
• or, in the case of degenerate states, a matrix. $\ket{\psi(t)} = e^{-i\theta(t)} \ket{n(\vb*{R}(t))}.$
• Smooth and singled-valued gauge may not be possible.
• Under slowing moving $$\vb*{R}(t)$$, $\ket{\psi(t)} = \exp\qty({\frac{1}{\hbar} \int_0^t E_n(\vb*{R}(t')) \dd{t'}}) e^{i\gamma_n} \ket{n(\vb*{R}(t))}.$
• Geometric phase: $\gamma_n = \int \dd{\vb*{R}}\cdot \vb*{A}_n(\vb*{R}) = -\Im \int \bra{n(\vb*{R})}\grad_{\vb*{R}} \ket{n(\vb*{R})} \dd{\vb*{R}}.$
• Berry connection: $\vb*{A}_n(\vb*{R}) = i\bra{n(\vb*{R})} \pdv{}{\vb*{R}} \ket{n(\vb*{R})}.$
• Gauge transformation: under $\ket{n(\vb*{R})} \rightarrow e^{i\xi(\vb*{R})} \ket{n(\vb*{R})},$ the Berry connection transforms like $\vb*{A}_n(\vb*{R}) \rightarrow \vb*{A}_n(\vb*{R}) - \pdv{}{\vb*{R}} \xi(\vb*{R}).$
• Under such a transformation, the geometric phase is changed by $\gamma_n \rightarrow \gamma_n + \xi(\vb*{R}(0)) - \xi(\vb*{R}(T)).$
• Berry curvature: $\mathcal{F} = \dd{\mathcal{A}} = \qty(\pdv{\bra{\vb*{R}}}{R^\mu}) \qty(\pdv{\ket{\vb*{R}}}{R^\nu}) \dd{R^\mu} \wedge \dd{R^\nu}.$
• Geometric phase: $\gamma_n = -\Im \int \dd{\vb*{S}}\cdot \qty(\grad \bra{n(\vb*{R})}) \times \qty(\grad \ket{n(\vb*{R})}).$
##### Numerical Calculation of Berry Phase
• Gauge-independent formula: for singly degenerate level, $\gamma_n = -\iint \dd{\vb*{S}} \cdot \vb*{V}_n = -\iint \dd{\vb*{S}} \cdot \Im \sum_{m\neq n} \frac{\bra{n(\vb*{R})}(\grad_{\vb*{R}} H(\vb*{R})) \ket{m(\vb*{R})} \times \bra{m(\vb*{R})}(\grad_{\vb*{R}} H(\vb*{R})) \ket{n(\vb*{R})}}{(E_m(\vb*{R}) - E_n(\vb*{R}))^2}.$
• $\sum_n \gamma_n = 0.$
• Two-level systems as an example: near degeneracy $$\vb*{R}^*$$, $\vb*{V}_+(\vb*{R}) = \Im \frac{\bra{+(\vb*{R})} (\grad H(\vb*{R}^*)) \ket{-(\vb*{R})} \times \bra{-(\vb*{R})}(\grad H(\vb*{R}^*))\ket{+(\vb*{R})}}{(E_+(\vb*{R}) - E_-(\vb*{R}))^2}.$
##### Two-Level Systems
• Generic Hamiltonian: $H = \vb*{\epsilon}(\vb*{R}) \mathbb{1}_{2\times 2} + \vb*{d}(\vb*{R}) \cdot \vb*{\sigma}.$
• Energy levels $E_{\pm} = \epsilon(\vb*{R}) \pm \sqrt{\vb*{d} \cdot \vb*{d}}.$
###### Example: Two-Level Systems
• $$\vb*{d}(\vb*{R})$$ parametrized as $\vb*{d}(\vb*{R}) = d\cdot \vb*{n}(\theta,\phi).$
• Eigenstates: $\ket{-\vb*{R}} = \begin{pmatrix} \sin(\theta/2) e^{-i\phi} \\ -\cos(\theta/2) \end{pmatrix};\quad \ket{+\vb*{R}} = \begin{pmatrix} \cos(\theta/2) e^{-i\phi} \\ \sin(\theta/2) \end{pmatrix}.$
• Not well-defined at the poles.
• Berry curvature: $F_{\theta\phi} = \frac{\sin\theta}{2}.$
• Example: $$d(\vb*{R}) = \vb*{R}$$, under which $\vb*{V}_- = \frac{1}{2} \frac{\vb*{d}}{d^3}.$
• Berry curvature integrated over a closed surface encircling the origin yields the Chern number.
• Degeneracy points in the parameter space act as sources and drains of the Berry curvature.
• Dirac-Weyl fermion: taking a closed path in the $$(k_x,k_y)$$ plane by setting $$k_z = 0$$. $$\gamma = \pm 2\pi$$ if the curve encircles the origin. Otherwise $$\gamma = 0$$.
##### Example: Spin in a Magnetic Field
• Energy levels: $E_n(B) = Bn,\quad n = -S,-S+1,\cdots,+S.$
• Geometric phase: $\gamma_n = -\iint \dd{\vb*{S}} \cdot n\frac{\vb*{B}}{B^3}.$

#### Hall Conductance

##### Linear Response
• Current operator: $\vb*{j}(\vb*{q}) = \sum_{\vb*{k},\alpha,\beta} c^\dagger_{\vb*{k} + \vb*{q}/2}c_{\vb*{k} - \vb*{q}/2} \pdv{h^{\alpha\beta}_{\vb*{k}}}{\vb*{k}}.$
• External Hamiltonian: $H_{\mathrm{ext}} = \sum_{\vb*{q}} \vb*{j}(\vb*{q}) \cdot \vb*{A}(-\vb*{q}).$
• Total current operator: \begin{align*} \vb*{J}(\vb*{x}, t) &= \frac{1}{2}\sum_i \qty[(\vb*{p}_i - e\vb*{A}) \delta(\vb*{x} - \vb*{x}_i) + \delta(\vb*{x} - \vb*{x}_i)(\vb*{p}_i - e\vb*{A})] \\ &= \vb*{j}(\vb*{x}) - ne\vb*{A}(\vb*{x}, t). \end{align*}
• External Hamiltonian to the first order: $H_{\mathrm{ext}} = -e \int \dd{^3 \vb*{x}} \vb*{A}(\vb*{x},t) \vb*{j}(\vb*{x}).$
• Expectation value of current: $\bra{E_N(t)} J_i(x,t) \ket{E_N(t)} = \bra{E_N} j_i(x) \ket{E_N} + \int_{-\infty}^{\infty} \dd{t'} \int \dd{^3 x'} \sum_j R_{ij}(x-x',t-t') A_j(x',t').$
• Response function (Zero temperature): $R_{ij}(x-x',t-t') = -i\Theta(t-t')\bra{E_N} [j_i(x,t), j_j(x',t')] \ket{E_N} - ne\delta_{ij} \delta(x-x')\delta(t-t').$
• Reponse function (Nonzero temperature): $\bra{E_N} [j_i(x,t), j_j(x',t')] \ket{E_N} \rightarrow \frac{\tr\qty{e^{-\beta H_0}[j_i(x,t), j_j(x',t')]}}{Z}.$
• Formulation: $J_\alpha(\omega) = \sigma_{\alpha\beta}(\omega) E_\beta(\omega) = i\omega \sigma_{\alpha\beta}(\omega) A_\beta(\omega).$
###### Zero-Temperature: the Fluctuation-Dissipation Theorem
• Correlation functions: \begin{align*} J_1(\omega) &= \int_{-\infty}^{\infty} \langle A(t) B(0) \rangle e^{i\omega t}\dd{t} = \frac{2\pi}{Z}\sum_{mn} e^{-\beta E_m} \bra{n}B\ket{m} \bra{m}A\ket{n}\delta(E_m - E_n + \omega), \\ J_2(\omega) &= \int_{-\infty}^{\infty} \langle B(0) A(t) \rangle e^{i\omega t}\dd{t} = \frac{2\pi}{Z}\sum_{mn} e^{-\beta E_n} \bra{n}B\ket{m} \bra{m}A\ket{n}\delta(E_m - E_n + \omega), \\ J_2(\omega) &= e^{-\beta \omega} J_1(\omega). \end{align*}{}
• Retarded Green's function: \begin{align*} G^R(t) &= -i\Theta(t) \langle[A(t), B(0)] \rangle \\ &= -i\Theta(t) \cdot \frac{1}{2\pi} \int_{-\infty}^{\infty} J_1(\omega) (1-e^{-\beta\omega}) e^{-i\omega t} \dd{\omega}. \end{align*}{}
• Frequency domain: $G^R(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{J_1(\omega')}{\omega - \omega' + i\eta}(1-e^{-\beta \omega'}) \dd{\omega'}.$
• $\Im(G^R(\omega)) = -\frac{1}{2} (1-e^{-\beta\omega}) J_1(\omega).$
• Time-ordered Green's function: \begin{align*} G^T(t) = -i\langle T[A(t)B(0)] \rangle. \end{align*}{}
• Frequency domain: $G^T(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} J_1(\omega') \qty{\frac{1}{\omega - \omega' + i\eta} - \frac{e^{-\beta\omega'}}{\omega - \omega' - i\eta}} \dd{\omega'}.$
• \begin{align*} \Re G^T(\omega) = \frac{1}{2\pi} \mathcal{P}\int_{-\infty}^{\infty} (1-e^{-\beta \omega'}) \frac{J_1(\omega')}{\omega - \omega'}. \end{align*}{}
• $\Im G^T(\omega) = -\frac{1}{2}(1+e^{-\beta\omega}) J_1(\omega).$
###### Finite Temperature: Matsubara Function
• Temperature function $G^T(\sigma) = \langle T[A(\sigma)B(0)]\rangle = \Theta(\sigma)\langle A(\sigma) B(0) \rangle + \Theta(-\sigma)\langle B(0) A(\sigma) \rangle.$
• Imaginary time-evolution: $A(\sigma) = e^{\sigma H} A e^{-\sigma H}.$
• Periodicity: $G^T(\sigma-\beta) = G^T(\sigma).$
• Frequency domain formulation: $G^T(\omega_m) = -\frac{1}{\beta} \cdot \frac{1}{2\pi} \int_{-\infty}^{\infty} (1-e^{-\beta \omega'}) \frac{J_1(\omega')}{i\omega_m - \omega'}\dd{\omega'}.$ where $\omega_m = \frac{2\pi m}{\beta}.$
• Analytic continuation: $G^R(\omega) = -\beta G^T(\omega_m) |_{i\omega_m = \omega + i\eta}.$
##### Formula for Hall Conductivity
• Hall conductivity (in unit of $$e^2/h$$): \begin{align*} \sigma_{ij\mathrm{Hall}} &= \int \frac{\dd{k_x} \dd{k_y}}{(2\pi)^2} \sum_{a=1}^m (-i)\qty(\bra{\partial_i(a,k)}\partial_j\ket{a,k} - \bra{\partial_j(a,k)\partial_i(a,k)}). \end{align*}
• Summation over all filled bands.
• In terms of the curvature tensor: $\sigma_{xy} = \frac{e^2}{h} \frac{1}{2\pi} \iint \dd{k_x} \dd{k_y} F_{xy}(k),$ where $F_{xy}(k) = \pdv{A_y(k)}{k_x} - \pdv{A_x(k)}{k_y}$ and $A_i = -i \sum_{\mathrm{filled bands}} \bra{a\vb*{k}} \pdv{}{k_i} \ket{a\vb*{k}}.$
• The integral is always a integer.

#### Time-Reversal Symmetry

##### Time-Reversal of Spinless Systems
• TR of creation operators: $T c_j T^{-1} = c_j.$
• TR on band: if the system is time-reversal, then $T h(k) T^{-1} = h(-k).$
• Zero Hall conductance: $F_{ij}(-k) = -F_{ij}(k)$ and therefore the integral vanishes in the BZ.
##### Time-Reversal of Spinful Systems
• The TR operator: $T = e^{-i\pi S_y} K.$
• For spin-$$1/2$$ particles: $e^{-i\pi \sigma_y/2} = -i\sigma_y.$
• Scattering probability: $\bra{\psi} H \ket{T\psi} = 0.$
###### Time-Reversal in Crystal
• Hamiltonian spectrum: $H = \sum_{\vb*{k}} c^\dagger_{\vb*{k}\alpha\sigma} h^{\sigma\sigma'}_{\alpha\beta} c_{\vb*{k}\beta\sigma'}.$
• TR on annhilation operator: relative phase difference due to distinct numbers of particles, \begin{align*} T c_{ja\sigma} T^{-1} &= i (\sigma^{y}_{\sigma'\sigma}) c_{ja\sigma'}, \\ T c^\dagger_{ja\sigma} T^{-1} &= c^\dagger_{ja\sigma'} i (\sigma^y)^T_{\sigma'\sigma}, \\ T c_{\vb*{k}a\sigma} T^{-1} &= i(\sigma^y)_{\sigma\sigma'} c_{-\vb*{k}a\sigma'}, \\ T c^\dagger_{\vb*{k} a\sigma} T^{-1} &= c^\dagger_{-\vb*{k}a\sigma'}i(\sigma^y)^T_{\sigma'\sigma}. \\ \end{align*}{}

#### Smalltalks

##### Topological Insulators
• Trivial insulator: the insulators that, upon slowly turning off the hopping elements and the hybridization between orbitals on different sites, flows adiabatically into the atomic limit.

Expand
 Quantum Field Theory (II) Quantization of Fields

## Quantum Field Theory (III)

Feynman Diagrams

あの日見た1PIの伝播関数は僕たちはまだ知らない

#### Foundations of Field Theory

##### Heisenberg Picture and Interaction Picture
• Heisenberg picture: $\phi(t,\vb*{x}) = e^{iH(t-t_0)} \phi(t_0,\vb*{x}) e^{-iH(t-t_0)}.$
• Interaction picture: $\phi_I(t,\vb*{x}) = e^{iH_0(t-t_0)} \phi(t_0, \vb*{x}) e^{-iH_0(t-t_0)}.$
##### Scalar Field
• Expansion (Schrödinger picture): $\phi(t_0, \vb*{x}) = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \qty(a_{\vb*{p}} e^{i\vb*{p}\cdot \vb*{x}} + a^\dagger_{\vb*{p}} e^{-i\vb*{p}\cdot \vb*{x}}).$
• Expansion (Interaction picture): $\phi_I(t,\vb*{x}) = \left. \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \qty(a_{\vb*{p}} e^{-ip\cdot x} + a^\dagger_{\vb*{p}} e^{ip\cdot x}) \right\vert_{x^0 = t-t_0}.$
##### Dirac Field
• Expansion: \begin{align*} \psi(x) &= \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_s \qty(a^s_{\vb*{p}} u^s(p) e^{-ip\cdot x} + b^{s\dagger}_{\vb*{p}} v^s(p) e^{ip\cdot x}), \\ \overline{\psi}(x) &= \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_s \qty(b^s_{\vb*{p}} \overline{v}^s(p) e^{-ip\cdot x} + a^{s\dagger}_{\vb*{p}} \overline{u}^s(p) e^{ip\cdot x}). \end{align*}{}
##### Electromagnetic Field
• Free electromagnetic field (under the Lorenz gauge): $\partial^2 A_\mu = 0.$
• Each component of $$A$$ obeys the Klein-Gordon equation with $$m=0$$.
• Expansion: $A_\mu(x) = \int \frac{\dd{^3} p}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_{r=0}^3 \qty(a_{\vb*{p}}^r\epsilon^r_\mu{(p)}e^{-ip\cdot x} + a^{r\dagger}_{\vb*{p}} \epsilon^{r*}_{\mu}(p) e^{ip\cdot x}).$
• In the integration we have $$p^2=0$$, and
• $$\epsilon_\mu(p)$$ is a four-vector.
##### Propagator
• $$\ket{\Omega}$$ is usually different from $$\ket{0}$$.
• Note the extra minus sign for Dirac fields.
• Feynman propagator
• Scalar field (definition): $D_F = \bra{0}T\phi(x)\phi(y) \ket{0}.$
• Scalar field (in momentum space): $D_F(x-y) = \int \frac{\dd{^4 p}}{(2\pi)^4} \frac{i}{p^2 - m^2 + i\epsilon} e^{-ip\cdot (x-y)}.$
• Dirac field (definition): $S_F(x-y) = \bra{0}T\psi(x)\overline{\psi}(y)\ket{0}.$
• Dirac field (in momentum space): $S_F(x-y) = \int \frac{\dd{^4 p}}{(2\pi)^4} \frac{i(\unicode{x2215}\kern-.5em {p} + m)}{p^2 - m^2 + i\epsilon} e^{-ip\cdot (x-y)}.$
• Two-point correlation function, or two-point Green's function: $\bra{\Omega}T\phi(x)\phi(y)\ket{\Omega},$ where $$\ket{\Omega}$$ is the ground state of the interacting theory.
• Källén-Lehmann spectral representation: analytic structure of two-point correlation, $\color{red} \bra{\Omega} T\phi(x)\phi(y) \ket{\Omega} = \int_0^\infty \frac{\dd{M^2}}{2\pi} \rho(M^2) D_F(x-y, M^2).$
• Spectral density: $\color{red} \rho(M^2) = \sum_\lambda (2\pi) \delta(M^2 - m_\lambda^2) \abs{\bra{\Omega} \phi(0) \ket{\lambda_0}}^2.$
• $\rho(M^2) = 2\pi\delta(M^2 - m^2) \cdot Z + (\text{nothing else until } M^2 \gtrapprox (2m)^2).$
• The leading contribution is from the single-particle state.
• There are poles from single-particle bounded states near $$M^2 = (2m)^2$$.
• $$Z$$ is referred to as the field-strength renormalization.
• Physical mass: $$m$$ is the exact mass of a single particle, i.e. the exact energy eigenvalue at rest, different from the value of the mass parameter in the Lagrangian.
• Only the physical mass is directly observable.
• Bare mass: $$m_0$$ in the Lagrangian.
• $$\ket{\lambda_{0}}$$: an eigenstate of $$H$$ annihilated by $$\vb*{P}$$.
• Källén-Lehmann spectral representation for Scalar fields: $\int \dd{^4 x} e^{ip\cdot x} \bra{\Omega} T\phi(x) \phi(0) \ket{\Omega} = \frac{iZ}{p^2 - m^2 + i\epsilon} + \int_{\sim 4m^2}^\infty \frac{\dd{M^2}}{2\pi} \rho(M^2) \frac{i}{p^2 - M^2 + i\epsilon}.$
• $$Z_2$$: the probability for the quantum field to create or annihilate an exact one-particle eigenstate of $$H$$, $\sqrt{Z_2} = \bra{\Omega} \phi(0) \ket{p}.$
• Källén-Lehmann spectral representation for Dirac fields: $\int \dd{^4 x} e^{ip\cdot x} \bra{\Omega} T\psi(x)\overline{\psi}(0) \ket{\Omega} = \frac{i Z_2 (\unicode{x2215}\kern-.5em {p} + m)}{p^2 - m^2 + i\epsilon} + \cdots.$
• $$Z_2$$: the probability for the quantum field to create or annihilate an exact one-particle eigenstate of $$H$$, $\sqrt{Z_2} u^s(p) = \bra{\Omega}\psi(0)\ket{p,s}.$
##### Path Integral
• Scalar field: \begin{align*} \color{orange} Z_0[J] &\color{orange}= \int \mathcal{D}\varphi\, e^{i\int \dd{^4 x} [\mathcal{L_0} + J\varphi]} \\ &\color{orange}= \exp\qty[\frac{1}{2}\iint \dd{^4 x} \dd{^4 x'} J(x) S_F(x-x') J(x')]. \end{align*}
• \begin{align*} \color{darkcyan} Z[J] &\color{darkcyan}= e^{i\int \dd{^4} \mathcal{L}_1(\frac{1}{i}\frac{\delta}{\delta J(x)})} Z_0[J]. \end{align*}
• $$\mathcal{L}_1$$ is a perturbation, $\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_1.$
##### Formulation of Scattering Problem

We solve the problem where

• the incident particles occupy a volume of length $$l_B$$ with number density $$\rho_B$$ and velocity (uniformly) $$v$$;
• the target particles occupy a volume of length $$l_A$$ with number density $$\rho_A$$; and
• the area common to the two branches is $$A$$.

We define a few quantities below.

• Cross section: $\color{orange}\sigma = \frac{\text{Number of scattering events}}{\rho_A l_A \rho_B l_B A},$
• Equivalently, \begin{align*} &\text{Number of events} \\ &= \sigma l_A l_B \int \dd{^2 x} \rho_A(x) \rho_B(x) \\ &= \frac{\sigma N_A N_B}{A}. \end{align*}
• Differential cross section: $\color{orange}\frac{\dd{\sigma}}{\dd{^3 p_1}\cdots \dd{^3 p_n}}$
• $$p_1,\cdots,p_n$$ denote the momenta of final states;
• integration over any small $$\dd{^3 p_1}\cdots \dd{^3 p_n}$$, gives the cross section for scattering into the final-state momentum space.
• Decay rate: $\color{orange}\Gamma = \frac{\text{Number of decays per unit time}}{\text{Number of } A \text{ particles present}}.$
• Breit-Wigner formula: near the reasonance energy, the scattering amplitude is given by \begin{align*} & f(E) \propto \frac{1}{E - E_0 + i\Gamma/2}\\ &\xlongequal{\text{relativistic}} \frac{1}{p^2 - m^2 + im\Gamma} \\ &= \frac{1}{2E_{\vb*{p}}(p^0 - E_{\vb*{p}} + i(m/E_{\vb*{p}})\Gamma/2)}, \end{align*}
• and the cross section by $\sigma \propto \frac{1}{(E-E_0)^2 + \Gamma^2/4}.$
##### Cross Section and S-Matrix

We use Heisenberg picture in this subsection.

We reduce the cross section to the $$\mathcal{M}$$ matrix.

• $$B$$ denotes the incident particles.
• Normalization: $\ket{\vb*{k}} = \sqrt{2E_{\vb*{k}}}a_{\vb*{k}}^\dagger\ket{0}.$
• $$\bra{\phi}\ket{\phi} = 1$$ if $\int \frac{\dd{^3 k}}{(2\pi)^3} \abs{\phi(\vb*{k})}^2 = 1.$
• Expansion of in states: plane wave to in state, $\ket{\phi_A \phi_B}_{\text{in}} = \int \frac{\dd{^3 k_A}}{(2\pi)^3} \int \frac{\dd{^3 k_B}}{(2\pi)^3} \frac{\phi_A(\vb*{k}_A) \phi_B(\vb*{k}_B) e^{-i\vb*{b}\cdot \vb*{k}_B}}{\sqrt{(2E_A) (2E_B)}}\ket{\vb*{k}_A \vb*{k}_B}_{\text{in}}.$
• Constructed in the far past.
• $$\phi_{B}(\vb*{k}_B)$$ is the state of the incident particle, with $$\vb*{b} = 0$$.
• Expansion of out states: plane wave to out state, $_{\text{out}}\bra{\phi_1\phi_2\cdots} = \qty(\prod_f \int \frac{\dd{^3 p_f}}{(2\pi)^3} \frac{\phi_f (\vb*{p}_f)}{\sqrt{2E_f}}) {_{\text{out}}\bra{\vb*{p}_1 \vb*{p}_2 \cdots}}.$
• Constructed in the far future.
• $$S$$-matrix: \begin{align*} &\color{orange}\phantom{{}={}} _{\text{out}}\bra{\vb*{p}_1\vb*{p}_2\cdots}\ket{\vb*{k}_A \vb*{k}_B}_{\text{in}} \\ &\color{orange}= \bra{\vb*{p}_1\vb*{p}_2\cdots} e^{-iH(2T)} \ket{\vb*{k}_A \vb*{k}_B} \\ &\color{orange}= \bra{\vb*{p}_1\vb*{p}_2\cdots} S \ket{\vb*{k}_A \vb*{k}_B}. \end{align*}
• $$T$$-matrix: $\color{orange} S = \mathbb{1} + iT$
• Invariant matrix element $$\mathcal{M}$$: brakets between in and out plane waves, $\color{orange} \bra{\vb*{p}_1\vb*{p}_2\cdots} iT \ket{\vb*{k}_A \vb*{k}_B} = (2\pi)^4 \delta^{(4)}\qty(k_A + k_B - \sum p_f)\cdot i\mathcal{M}\qty(k_A,k_B \rightarrow p_f),$
• The $$\delta^{(4)}$$ is imposed by the conservation of momentum.
• The probability of getting a certain out state (depending on the impact parameter): brakets of in and out states to probability, $\mathcal{P}(AB\rightarrow 1,2,\cdots,n) = \qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3}\frac{1}{2E_f})\abs{{_{\text{out}}\bra{\vb*{p}_1\cdots \vb*{p}_n}\ket{\phi_A \phi_B}_{\text{in}} }}^2.$
• Cross section: probability to cross section, summing over all impact parameters, $\sigma = \int \dd{^2 b} \mathcal{P}(\vb*{b}).$
• Putting together: \begin{align*} \dd{\sigma} &= \qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3}\frac{1}{2E_f}) \frac{\abs{\mathcal{M}(p_A,p_B\rightarrow \qty{p_f})}^2}{2E_A 2E_B \abs{v_A - v_B}} \int \frac{\dd{^3 k_A}}{(2\pi)^3} \int \frac{\dd{^3 p_B}}{(2\pi)^3} \\ &{\phantom{{}={}}} \times \abs{\phi_A \qty(\vb*{k}_A)}^2 \abs{\phi_B\qty(\vb*{k}_B)}^2 (2\pi)^4 \delta^{(4)}\qty(k_A + k_B - \sum p_f). \end{align*}{}
• Conclusion: $$\mathcal{M}$$ matrix to cross section, \begin{align*} \color{red}\dd{\sigma} &= \color{red}\frac{1}{2E_A 2E_B \abs{v_A - v_B}} \qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3}\frac{1}{2E_f}) \\ &{\phantom{{}={}}} \color{red}\times \abs{\mathcal{M}(p_A,p_B\rightarrow \qty{p_f})}^2 (2\pi)^4 \delta^{(4)}\qty(p_A+p_B - \sum p_f). \end{align*}{}
• We demanded the wave packet be smooth (not so sharp).

When doing integration to get the total cross section or decay rate for a final state of $$n$$ particles, one should either restrict the integration to inequivalent configurations or divide the result by $$n!$$.

• Two-body scattering:
• In the center-of-mass frame, $$\vb*{p_1} = -\vb*{p}_2$$.
• $$E_{\mathrm{cm}}$$ denote the total energy.
• Cross section of two-particles: $\color{red} \qty(\dv{\sigma}{\Omega})_{\text{CM}} = \frac{1}{2E_A 2E_B \abs{v_A - v_B}} \frac{\abs{\vb*{p}_1}}{(2\pi)^2 4E_{\text{cm}}}\abs{\mathcal{M}(p_A,p_B \rightarrow p_1,p_2)}^2.$
• Identical mass: $\color{red} \qty(\dv{\sigma}{\Omega})_{\text{CM}} = \frac{\abs{\mathcal{M}}^2}{64\pi^2 E^2_{\text{cm}}}.$
• Decay rate: $\color{red} \dd{\Gamma} = \frac{1}{2m_A}\qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3} \frac{1}{2E_f}) \abs{\mathcal{M}(m_A \rightarrow \qty{p_f})}^2 (2\pi)^4 \delta^{(4)}\qty(p_A - \sum p_f).$
###### Remark on Decay Rate
• $$M^2(p^2)$$ denotes the sum of all 1PI insertions into the $$\phi^4$$ propagator. Let $$m$$ be defined by $m^2 - m_0^2 - \Re M^2(m^2) = 0.$
• The two-point correlation is found to be $\frac{i Z}{p^2 - m^2 - iZ \Im M^2(p^2)}.$
• Decay rate identified to be $\Gamma = -\frac{Z}{m} \Im M^2(m^2).$
• cf. Breit-Wigner formula $\sigma \propto \abs{\frac{1}{p^2 - m^2 + im\Gamma}}^2.$
##### Born Approximation

Born approximation: $\bra{p'}iT\ket{p} = -i\tilde{V}\qty(\vb*{q})(2\pi)\delta \qty(E_{\vb*{p}'} - E_{\vb*{p}})$ where $$\vb*{q} = \vb*{p}' - \vb*{p}$$.

• Assuming nonrelativistic normalization: $\bra{p'}\ket{p} = \delta(p'-p).$
##### LSZ Reduction Formula

LSZ Reduction Formula \color{red} \begin{align*} & \prod_1^n \int \dd{^4 x_i} e^{ip_i\cdot x_i} \prod_1^m \int \dd{^4 y_j} e^{-ik_i \cdot y_j} \bra{\Omega} T\qty{\phi(x_1) \cdots \phi(x_n) \phi(y_1) \cdots \phi(y_m)} \ket{\Omega} \\ & \underset{\substack{p_i^0 \rightarrow E_{\vb*{p}_i} \\ k_j^0 \rightarrow E_{\vb*{p}_j}}}{\sim} \qty( \prod_1^n \frac{\sqrt{Z} i}{p_i^2 - m^2 + i\epsilon} )\qty( \prod_1^m \frac{\sqrt{Z} i}{k_j^2 - m^2 + i\epsilon} ) \bra{\vb*{p}_1 \cdots \vb*{p}_n} S \ket{\vb*{k_1} \cdots \vb*{k}_m}. \end{align*}

##### The Optical Theorem
• From the unitarity of $$S$$: \begin{align*} &\phantom{{}={}}-i[\mathcal{M}(k_1k_2 \rightarrow p_1p_2) - \mathcal{M}^*(p_1p_2 \rightarrow k_1k_2)] \\ &= \sum_n \qty(\prod_{i=1}^n \int \frac{\dd{^3 q_i}}{(2\pi)^3} \frac{1}{2E_i}) \mathcal{M}^*(p_1p_2 \rightarrow \qty{q_i}) \mathcal{M}(k_1k_2 \rightarrow \qty{q_i}) \times (2\pi)^4 \delta^{(4)}(k_1 + k_2 - \sum_i q_i). \end{align*}{}
• Abbreviated as $-i[\mathcal{M}(a\rightarrow b) - \mathcal{M}^*(b\rightarrow a)] = \sum_f \int \dd{\Pi_f} \mathcal{M}^*(b\rightarrow f) \mathcal{M}(a\rightarrow f).$
• With the factors in the formula for cross section, we obtain the optical theorem, $\color{red} \Im \mathcal{M}(k_1, k_2 \rightarrow k_1, k_2) = 2 E_{\mathrm{cm}} p_{\mathrm{cm}} \sigma_{\mathrm{tot}}(k_1,k_2 \rightarrow \mathrm{anything}).$
• Feynman diagrams yields imaginary part only when the virtual particles in the diagram go on-shell.
• Discontinuity: above the energy threshold, $\operatorname{Disc} \mathcal{M}(s) = 2i \Im \mathcal{M}(s+i\epsilon).$

#### Perturbation Expansion of Correlation Functions

The perturbed Hamiltonian has the form $H = H_0 + H_{\text{int}} = H_0 + H_{\text{Klein-Gordon}}.$

In the $$\phi^4$$ theory $H = H_0 + H_{\text{int}} = H_{\text{Klein-Gordon}} + \int \dd{^3 x} \frac{\lambda}{4!} \phi^4(x).$

##### Obtaining the Field Operator%%% Obtaining $$\phi$$
• $$\phi$$ under Schrödinger picture: $\color{darkcyan}\phi(t,\vb*{x}) = U^\dagger(t,t_0)\phi_I(t,\vb*{x})U(t,t_0)$
• Time evolution operator under the interaction picture: $\color{orange}U(t,t_0) = e^{iH_0(t-t_0)}e^{-iH(t-t_0)}$
• Equation of motion: $i\pdv{}{t}U(t,t_0) = H_I(t) U(t,t_0).$
• Interaction Hamiltonian: $\color{orange}H_I(t) = e^{iH_0(t-t_0)} (H_{\mathrm{int}}) e^{-iH_0 (t-t_0)}.$
• Dyson series: \begin{align*} U(t,t_0) &= \mathbb{1} + (-i) \int_{t_0}^t \dd{t_1} H_I(t_1) + (-i)^2 \int_{t_0}^{t} \dd{t_1} \int_{t_0}^{t_1} \dd{t_2} H_I(t_1) H_I(t_2) + (-i)^3 \int_{t_0}^{t} \dd{t_1} \int_{t_0}^{t_1} \dd{t_2} \int_{t_0}^{t_2} \dd{t_3} H_I(t_1) H_I(t_2) H_I(t_3) + \dots \\ &= 1 + (-i) \int_{t_0}^t \dd{t_1} H_I(t_1) + \frac{(-i)^2}{2!} \int_{t_0}^t \dd{t_1} \dd{t_2} T\qty{H_I(t_1)H_I(t_2)} + \cdots \\ \color{red}U(t,t') &\color{red}= T\qty{\exp\qty[-i\int_{t'}^t\dd{t''}H_I(t'')]}. \end{align*}
• $$U(t,t')$$ is only defined for $$t>t'$$.
• For $$t_1 \ge t_2 \ge t_3$$,
• $U(t_1,t_2) U(t_2,t_3) = U(t_1,t_3),$
• $U(t_1,t_3)[U(t_2,t_3)]^\dagger = U(t_1,t_2).$
##### Obtaining the Vacuum State%%% Obtaining $$\ket{\Omega}{}$$
• $\ket{\Omega} = e^{-iHT} \ket{0}.$
• $\color{red} T \rightarrow \infty(1-i\epsilon).$
• $E_0 = \bra{\Omega}H\ket{\Omega}.$
• $H_0\ket{0} = 0.$
• $U(t,t') = e^{iH_0(t-t_0)} e^{-iH(t-t')}e^{-iH_0(t'-t_0)}.$
• $$\Omega$$ from vacuum:
• $\ket{\Omega} = \qty(e^{-iE_0(t_0 - (-T))}\bra{\Omega}\ket{0})^{-1} U(t_0,-T)\ket{0}.$
• $\bra{\Omega} = \bra{0}U(T,t_0) \qty(e^{-iE_0(T-t_0)}\bra{0}\ket{\Omega})^{-1}.$
• Normalization: $\bra{\Omega}\ket{\Omega} = 1.$
Derivation

With $e^{-iHT}\ket{0} = e^{-iE_0 T} \ket{\Omega}\bra{\Omega}\ket{0} + \sum_{n\neq 0} e^{-iE_n T}\ket{n}\bra{n} \ket{0},$ we find that $\ket{\Omega} = \qty(e^{-iE_0 T}\bra{\Omega}\ket{0})^{-1} e^{-iHT}\ket{0},$ which simplifies to $\ket{\Omega} = \qty(e^{-iE_0(t_0 - (-T))}\bra{\Omega}\ket{0}) U(t_0, -T)\ket{0}.$

What are the Consequences of the Imaginary Part of $$T$$?

After applying the momentum space Feynman rules, we are facing with $\int_{-T}^T \dd{z^0} \int \dd{^3 z} e^{-i(p_1 + p_2 + p_3 - p_4)\cdot z}.$ To prevent the integral from blowing up, we demand $p^0 \propto (1+i\epsilon),$ which is equivalent to the Feynman prescription of propagator.

##### Obtaining the Correlation Function (Canonical Quantization)
• Two-point correlation function: $\color{red} \bra{\Omega}T\qty{\phi(x)\phi(y)}\ket{\Omega}= \dfrac{\bra{0}T\qty{\phi_I(x)\phi_I(y)\exp\qty[-i\int_{-T}^{T}\dd{t}H_I(t)]}\ket{0}}{\bra{0}T\qty{\exp\qty[-i\int_{-T}^T \dd{t} H_I(t)]}\ket{0}}.$ For each factor inserted to the LHS, we inserted an identical one on the RHS.
##### Obtaining the Correlation Function (Path Integral)
• Path integral formulation: $\color{red} \bra{\Omega} T\phi_H(x_1) \phi_H(x_2) \ket{\Omega} = \dfrac{\int \mathcal{D}\phi\, \phi(x_1) \phi(x_2) \exp\qty[i\int_{-T}^T \dd{^4 x}\mathcal{L}]}{\int \mathcal{D}\phi\, \exp\qty[i\int_{-T}^T \dd{^4 x}\mathcal{L}]}.$
• From the generating functional: $\color{red} \bra{\Omega} T\phi(x_1)\phi(x_2) \ket{\Omega} = \frac{1}{Z_0}\left.\qty(-i\frac{\delta}{\delta J(x_1)})\qty(-i\frac{\delta}{\delta J(x_2)}) Z[J]\right|_{J=0}.$

Due to an unfortunate collision of notation, $$Z_0$$ may stands for both the generating functional $$Z_0[J_A,J_B,\cdots]$$ or the constant $Z_0 = Z_0[0,0,\cdots].$

1. Write down the perturbation $$\mathcal{L}_1$$, which may contain various kinds of fields.
• It's important that the fields occur in product form, e.g. $\mathcal{L}_1 = g\phi_A \phi_B,$ so that we may write the perturbation as products of variations of the generating functional.
2. Obtain the unperturbed generating functional $$Z_0[J_A, J_B, \cdots]$$. This should admit the form \begin{align*} Z_0[J_A,J_B,\cdots] &= \qty(\int \mathcal{D}\varphi_A \int \mathcal{D} \varphi_B \cdots) \exp\qty[i\int \dd{^4 x} \qty[\mathcal{L}_A + J_A \phi_A]] \exp\qty[i\int \dd{^4 x} \qty[\mathcal{L}_B + J_B \phi_B]] \\ &= Z_0 \cdot \exp\qty[-\frac{1}{2}\iint \dd{^4 x} \dd{^4 x'} J_A(x) D^A_F(x-x') J_A(x')] \exp\qty[-\frac{1}{2}\iint \dd{^4 x \dd{^4 x'}} J_B(x) D^B_F(x-x') J_B(x')]\cdots. \end{align*}
3. To obtain the perturbed generating functional, we
1. replace each item in the perturbation as a functional derivative with respect to the corresponding source: $g\phi_A \phi_B \rightarrow g\qty(-i\frac{\delta}{\delta J_A})\qty(-i\frac{\delta}{\delta J_B}).$
• Extra care should be taken for the possible minus sign for Dirac fields.
2. then exponentiate it and let it act on the unperturbed generating functional: \begin{align*} Z[J_A, J_B,\cdots] &= Z_0 \cdot \exp\qty[\int \dd{^4 x} ig\qty(-i\frac{\delta}{\delta J_A(x)})\qty(-i\frac{\delta}{\delta J_B(x)})]Z_0[J_A,J_B,\cdots] \\ &= Z_0 e^{iW[J_A, J_B, \cdots]}. \end{align*}{}
4. To obtain the correlation amplitude, we apply functional derivatives to the perturbed generating functional: \begin{align*} \color{red} \bra{\Omega} T\phi_A(x_1) \phi_A(x_2) \ket{\Omega} &\color{red} = \frac{Z_0}{Z[0,0,\cdots]} \left.\qty(-i \frac{\delta}{\delta J_A(x_1)})\qty(-i \frac{\delta}{\delta J_A(x_2)}) \cdot \exp\qty[\int \dd{^4 x} ig\qty(-i\frac{\delta}{\delta J_A(x)})\qty(-i\frac{\delta}{\delta J_B(x)})]Z_0[J_A,J_B,\cdots]\right\vert_{J_A = J_B = \cdots = 0} \\ &\color{red} = \left.\qty(-i \frac{\delta}{\delta J_A(x_1)})\qty(-i \frac{\delta}{\delta J_A(x_2)}) \cdot i W[J_A, J_B, \cdots]\right\vert_{J_A = J_B = \cdots = 0}. \end{align*}{}
##### Obtaining the T-Matrix
• T-matrix: \begin{align*} &\bra{\vb*{p}_1 \cdots \vb*{p}_n} iT \ket{\vb*{p}_A \vb*{p}_B} \\ &= \qty({_0\bra{\vb*{p}_1 \cdots \vb*{p}_1}T\qty(\exp\qty[-i\int_{-T}^T \dd{t} H_I(t)])\ket{\vb*{p}_A \vb*{p}_B}_0})_{\substack{\text{connected,}\\\text{amputated}}}. \end{align*}

#### Wick's Theorem: Correlation Amplitude to Feynman Propagators

In contractions, $$\phi$$ is always under the interaction picture even without the subscript $$I$$.

• $$\phi^+$$ and $$\phi^-$$ denote the annihilation and creation parts of $$\phi$$ respectively, i.e.
• $\color{warning} \phi^+_I(x) = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} a_{\vb*{p}} e^{-ip\cdot x};$
• $\color{warning} \phi^-_I(x) = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} a^\dagger_{\vb*{p}} e^{+ip\cdot x}.$

The time-ordered product of a Dirac field contains a factor of the sign of the permutation, e.g. $T\qty(\psi_1 \psi_2 \psi_3 \psi_4) = (-1)^3 \psi_3 \psi_1 \psi_4 \psi_2$ if $x_3^0 > x_1^0 > x_4^0 > x_2^0.$

• $$N(XYZ)$$ that turns $$XYZ$$ into normal order,
• Scalar field:$\color{warning} N(a_{\vb*{p}}a_{\vb*{k}}^\dagger a_{\vb*{q}}) = a^\dagger_{\vb*{k}}a_{\vb*{p}}a_{\vb*{q}}.$
• Dirac field:$\color{warning}N(a_{\vb*{p}} a_{\vb*{q}} a_{\vb*{r}}^\dagger) = (-1)^2 a_{\vb*{r}}^\dagger a_{\vb*{p}} a_{\vb*{q}} = (-1)^3 a_{\vb*{r}}^\dagger a_{\vb*{q}} a_{\vb*{p}}.$
• Wick contraction: different from the convention in most literatures, here we use color to denote contraction, ${\color{blue}\phi}(x){\color{blue}\phi}(y) \color{orange} = T\qty{\phi(x)\phi(y)} - N\qty{\phi(x)\phi(y)}.$ More generally, ${\color{blue} A}{\color{blue} B} = T\qty{AB} - N\qty{AB}.$
• For a scalar field:${\color{blue}\phi}(x){\color{blue}\phi}(y) \color{orange} = \begin{cases} \qty[\phi^+(x), \phi^-(y)] & \text{for } x^0 > y^0; \\ \qty[\phi^+(y), \phi^-(x)] & \text{for } y^0 > x^0. \\ \end{cases}$
• The contraction yields the Feynman propagator, ${\color{blue}\phi}(x){\color{blue}\phi}(y) = D_F(x-y).$
• Surprisingly, the contraction yields a number, without bra-keting with $$\ket{0}$$.
• For a Dirac field:${\color{blue}\psi}(x) {\color{blue} \overline{\psi}}(y) = \begin{cases} \phantom{-}\qty{\psi^+(x), \overline{\psi}^-(y)}, & \text{for } x^0 > y^0; \\ -\qty{\overline{\psi}^+(y),\psi^-(x)}, & \text{for } x^0 < y^0. \end{cases}$
• The contraction yields the Feynman propagator:${\color{blue}\psi}(x) {\color{blue} \overline{\psi}}(y) = S_F(x-y),$
• or vanishes: ${\color{blue}\psi}{\color{blue}\psi} = {\color{blue}\overline{\psi}}{\color{blue}\overline{\psi}} = 0.$
• The following notation is used, although it seems that contraction should not be put in $$N\qty{\cdots}$$:
• Scalar field: $N\qty{{\color{blue}\phi_1}\phi_2{\color{blue}\phi_3}\phi_4} = D_F(x_1 - x_3)\cdot N\qty{\phi_2\phi_4}.$
• Dirac field: $N({\color{blue}\psi_1} \psi_2 {\color{blue}\overline{\psi}_3} \overline{\psi}_4) = -{\color{blue}\psi_1}{\color{blue}\overline{\psi}_3} N(\psi_2 \overline{\psi}_4) = -S_F(x_1 - x_3)N(\psi_2 \overline{\psi}_4).$

With the normal ordering we consider only either the $$a_{\vb*{p}}$$ or $$a_{\vb*{p}}^\dagger$$ part in the expansion of $$\color{blue}\phi_I$$.

• Contracting field with creation and annihilation operators:
• ${\color{blue}\phi_I}(x)\ket{\color{blue}\vb*{p}} = {\color{blue}\phi_I}(x){\color{blue}a^\dagger_{\vb*{p}}}\ket{0} = e^{-ip\cdot x}\ket{0}{ }.$
• $\bra{\color{blue}\vb*{p}}{\color{blue}\phi_I}(x) = \bra{0}{\color{blue} a_{\vb*{p}}}{\color{blue}\phi_I}(x) = \bra{0} e^{+ip\cdot x}{ }.$
• Mixing different kinds of fields: Do I contract a Klein-Gordon field with a Dirac field?
• I don't know. I am fabricating shit here. No. If the two fields decouple in the unperturbed Hamiltonian, then their creation and annihilation operators commutes, and therefore we don't give a fuck to their relative ordering. See also

Wick's theorem: \begin{align*} & \color{red} T\qty{\phi(x_1)\phi(x_2)\cdots \phi(x_m)} \\ & \color{red} = N\qty{\phi(x_1)\phi(x_2)\cdots \phi(x_m) + \text{all possible contractions}}. \end{align*}

For $$m=4$$ we have \begin{align*} & T\qty{\phi_1\phi_2\phi_3\phi_4} \\ &= N\big\{ \phi_1\phi_2\phi_3\phi_4 \\ & + {\color{blue}\phi_1}{\color{blue}\phi_2}\phi_3\phi_4 + {\color{blue}\phi_1}\phi_2{\color{blue}\phi_3}\phi_4 + {\color{blue}\phi_1}\phi_2\phi_3{\color{blue}\phi_4} \\ & + \phi_1{\color{blue}\phi_2}{\color{blue}\phi_3}\phi_4 + \phi_1{\color{blue}\phi_2}\phi_3{\color{blue}\phi_4} + \phi_1\phi_2{\color{blue}\phi_3}{\color{blue}\phi_4} \\ & + {\color{blue}\phi_1}{\color{blue}\phi_2}{\color{magenta}\phi_3}{\color{magenta}\phi_4} + {\color{blue}\phi_1}{\color{magenta}\phi_2}{\color{blue}\phi_3}{\color{magenta}\phi_4} + {\color{blue}\phi_1}{\color{magenta}\phi_2}{\color{magenta}\phi_3}{\color{blue}\phi_4}\big\}. \end{align*}

Since the normal order put annihilation on the right, we have \begin{align*} &\bra{0}T\qty{\phi_1\phi_2\phi_3\phi_4}\ket{0} \\ &= D_F(x_1 - x_2) D_F(x_3 - x_4) \\ &+ D_F(x_1 - x_3) D_F(x_2 - x_4) \\ &+ D_F(x_1 - x_4) D_F(x_2 - x_3). \end{align*}

Only fully paired contractions are present in$\color{red}\bra{0}T\qty{\cdots}\ket{0}.$

#### Feynman Diagrams: Wick Contractions / Functional Derivatives Visualized

The $$m=4$$ example may be rewritten using Feynman diagrams as

$$\bra{0}T\qty{\phi_1\phi_2\phi_3\phi_4}\ket{0} =$$ $$+$$ $$+$$ .

• Line $$=$$ propagator $$=$$ contraction $$=$$ whatever you multiply.
• Vertex:
• In position space: vertex $$=$$ integration, $$\displaystyle \int \dd{^4 x}$$.
• In momentum space: vertex $$=$$ intergration $$\displaystyle \int \dd{^4 p}$$ and $$\delta(\sum p)$$.
##### Correlation Amplitude to Diagram (Canonical Quantization)

How do we convert the expression for $$\bra{\Omega}T\qty{\phi(x)\phi(y)}\ket{\Omega}{}$$ into diagram?

• Converting the numerator to diagram:
• The numerator is given by $\bra{0}T\qty{\phi_I(x)\phi_I(y)\exp\qty[-i\int_{-T}^{T}\dd{t}H_I(t)]}\ket{0}.$
• Expand the $$\exp$$ via Taylor series. Note that $$H_I$$ consists of fields ($$\varphi, \psi$$, etc.): $\bra{0} T\qty{ \phi(x)\phi(y) + \phi(x)\phi(y)\qty[-i\int \dd{t} H_I(t)] + \cdots } \ket{0}.$
• Apply Wick's theorem to each item after doing the Taylor expansion.
• Retain only full contractions.
• Convert contractions into Feynman propagators.
• Conclusion: $\color{red} \bra{0}T\qty[\phi(x_1)\phi(x_2)\cdots \phi(x_n)]\ket{0} = \qty(\sum \text{ all possible diagrams with } n \text{ external points}).$
• Taking the denominator into account:
• It can be shown that the disconnected diagrams (disconnected from the two external points) contribute a factor that cancels the denominator in the equation for $$\bra{\Omega}T\qty{\phi(x)\phi(y)}\ket{\Omega}{}$$.

Conclusion. \begin{align*} &\phantom{{}={}} \color{red}\bra{\Omega}T\qty[\phi(x_1)\phi(x_2)\cdots \phi(x_n)]\ket{\Omega} \\ &\color{red}= \qty(\sum \text{ connected diagrams with } n \text{ external points}). \end{align*}

For the $$\phi^4$$ perturbation to the first order we find \begin{align*} & \bra{0}T\qty{\phi(x)\phi(y)\qty(\frac{-i\lambda}{4!})\int \dd{^4 z}\phi(z)\phi(z)\phi(z)\phi(z)}\ket{0} \\ & = 3\cdot \qty(\frac{-i\lambda}{4!})D_F(x-y) \int \dd{^4 z} D_F(z-z)D_F(z-z) \\ & + 12 \cdot \qty(\frac{-i\lambda}{4!}) \int \dd{^4 z} D_F(x-z)D_F(y-z)D_F(z-z). \end{align*} This may be rewritten using Feynman diagrams as

$$+$$ .

One of the contractions in the $$\lambda^3$$ term is given by \begin{align*} & \bra{0}{\color{blue}\phi}(x){\color{magenta}\phi}(y) \frac{1}{3!} \qty(\frac{-i\lambda}{4!})^3 \int \dd{^4 z} {\color{blue}\phi}{\color{cyan}\phi}{\color{cyan}\phi}{\color{green}\phi} \int \dd{^4 w} {\color{green}\phi}{\color{orange}\phi}{\color{red}\phi}{\color{magenta}\phi} \int \dd{^4 u} {\color{red}\phi}{\color{orange}\phi}{\color{purple}\phi}{\color{purple}\phi} \ket{0} \\ &= \frac{1}{3!}\qty(\frac{-i\lambda}{4!})^3 \int \dd{^4 z} \dd{^4 w} \dd{^4 u} D_F(x-z) D_F(z-z) D_F (z-w) D_F(w-y) D_F^2(w-u) D_F(u-u), \end{align*} which accounts for $10368 = 3! \times 4\cdot 3 \times 4\cdot 3\cdot 2 \times 4\cdot 3 \times 1/2$ contractions. The Feynman diagram is

.

Symmetry Factor

To obtain the overall constant of a diagram in the $$\phi^4$$ theory, we apply the following recipe:

• We drop the factor $$\displaystyle \frac{1}{k!}\qty(\frac{1}{4!})^k$$ in front of each item, i.e. multiply by $$k! (4!)^k$$, which
• copies this diagram for $$k!$$ times: taking into accound the exchange of vertices;
• and then copies for $$(4!)^k$$ times: on each vertex we arbitrarily permute the four incoming lines.
• Then we divide the item by the symmetry factor:
• The permutation above may not result in a new diagram, and therefore we should divide by a symmtry factor to prevent these diagrams being counted multiple times.
• In the above example, the following permutations does not yield a new diagram, which contribute a symmetry factor $$S = 2\cdot 2\cdot 2 = 8$$.
• exchanging the two ends of the loop on $$z$$,
• exchanging the two ends of the loop on $$u$$,
• exchanging the two propagators connecting $$w$$ and $$u$$.
• Lines in the diagrams are refered to as propagators.
• Internal points ($$z$$, $$u$$, $$w$$, etc.) are called vertices.
##### Correlation Amplitude to Diagram (Path Integral)
• We apply the Taylor expansion to \begin{align*} Z[J_A, J_B,\cdots] &= Z_0 \cdot \exp\qty[\int \dd{^4 x} ig\qty(-i\frac{\delta}{\delta J_A(x)})\qty(-i\frac{\delta}{\delta J_B(x)})]Z_0[J_A,J_B,\cdots] \\ &= Z_0 \cdot \sum_{V=0}^\infty \qty(\int \dd{^4 x} ig\qty(-i\frac{\delta}{\delta J_A(x)})\qty(-i\frac{\delta}{\delta J_B(x)}))^V \sum_{P=0}^\infty \qty( {-\frac{1}{2}\iint \dd{^4 x} \dd{^4 x'} J_A(x) D^A_F(x-x') J_A(x') -\frac{1}{2}\iint \dd{^4 x \dd{^4 x'}} J_B(x) D^B_F(x-x') J_B(x')} )^P. \end{align*}
• For each integral in $$\sum_P$$, we associate with it a propagator (line).
• For each integral in $$\sum_V$$, we associate with it an internal vertex.
• For each source not killed, we assocaite with it an external vertex.
• We are not interested in the full form of $$Z$$. With $$Z=e^{iW}$$ we identify $$iW$$ with the contribution from all connected diagrams.
• We count only connected diagrams.
• To obtain the correlation amplitude, we apply $\qty(-i \frac{\delta}{\delta J})$ on $$iW$$. Every instance of such functional derivative kills a source, giving rise to an external vertex.
##### T-Matrix to Diagram

In the equation for the $$T$$-matrix \begin{align*} &\bra{\vb*{p}_1 \cdots \vb*{p}_n} iT \ket{\vb*{p}_A \vb*{p}_B} \\ &= \qty({_0\bra{\vb*{p}_1 \cdots \vb*{p}_1}T\qty(\exp\qty[-i\int_{-T}^T \dd{t} H_I(t)])\ket{\vb*{p}_A \vb*{p}_B}_0})_{\substack{\text{connected,}\\\text{amputated}}}, \end{align*}{ } the terminologies are explained below.

• Connected: only the fully connected (i.e. all vertices are connected to each other) diagrams are counted.
• Amputated: performed an operation that starts from the tip of each external leg and cut at the last point at which removing a single propagator would separates the leg from the rest of the diagram.

Conclusion. \begin{align*} &\phantom{{}={}}\color{red} i\mathcal{M} \cdot (2\pi)^4 \delta^{(4)}\qty(p_A + p_B - \sum p_f) \\ &\color{red}= \begin{pmatrix} \text{sum of all connected, amputated Feynman} \\ \text{diagrams with } p_A, p_B \text{ incoming, } p_f \text{ outgoing} \end{pmatrix}. \end{align*}

Conclusion. $\color{red} i\mathcal{M} = \begin{pmatrix}\text{sum of all connected, amputated diagrams}\\ \text{in the momentum space}\end{pmatrix}.$

The two rules above should be revised for loop diagrams.

• $$S$$-matrix and Feynman diagram. $\bra{\vb*{p}_1 \cdots \vb*{p}_n} S \ket{\vb*{k}_1 \vb*{k}_2} = \qty(\sqrt{Z})^{n+2} \qty[\text{Amputated}].$
##### Determination of Symmetry Factor

The symmetry factor counts the number of ways of interchanging components without changing the diagram.

The symmetry factor of is $$2\cdot 2\cdot 2 = 8$$.

##### Disconnected Diagrams
• A diagram is disconnected if it is disconnected from all external points.
• A diagram connected to some of the external points is deemed as connected.
• Disconnected diagrams are called vacuum bubbles.
• Let $$V_i$$ denote the disconnected diagrams as well as their values.
• The $$V_i$$'s are not isomorphic to each other.
• The diagrams representing the two-point correlation function yield the sum \begin{align*} &\phantom{{}={}}\sum_{\text{connected}} \sum_{\qty{n_i}} \qty(\begin{array}{c} \text{value of} \\ \text{connected piece} \end{array}) \times \qty(\prod_i \frac{1}{n_i!} {(V_i)}^{n_i}) \\ &= \qty{\sum \text{connected}} \times \sum_{\qty{n_i}} \qty(\prod_i \frac{1}{n_i!}{(V_i)^{n_i}}) \\ &= \qty{\sum \text{connected}} \times \exp\qty{\sum_i V_i}. \end{align*}
• The contribution to the denominator in the two-point correlation function is exactly $\exp\qty{\sum_i V_i}.$
• Therefore, we may take only connected diagrams in our calculation.
• Each disconnected diagram contains a factor $(2\pi)^4 \delta^{(4)}(0) = 2T\cdot V.$
##### Constructing Feynmann Rules (Position Space)
1. Write down the perturbation $$H_I$$, which may contain various kinds of fields $$\phi$$, $$\psi$$, etc.
2. Obtain the Feynman propagators: $$D_F$$, $$S_F$$, etc.
• Ideally, this is equivalent to asking (from the recipe via canonical quantization): what are $${\color{blue}\phi}(x){\color{blue}\phi}(y)$$, $${\color{blue}\psi}(x){\color{blue}\overline{\psi}}(y)$$, etc.?
• The contractions should be functions of $$(x-y)$$ only thanks to the homogeneity of spacetime.
3. Follow the procedure specified by Correlation Amplitude to Diagram to write down the diagrams.
4. For each propagator, associate with it a factor equal to the contraction it represents, $${\color{blue}\phi}(x){\color{blue}\phi}(y)$$, $${\color{blue}\psi}(x){\color{blue}\overline{\psi}}(y)$$, etc.
5. For each vertex, associate with it an integration $(-i\lambda)\int \dd{^4 z}.$
• We should have associated $-i\frac{\lambda}{4!} \int \dd{^4 z}$ for the $$\displaystyle \frac{\lambda}{4!} \phi^4$$ perturbation. However, to better take symmetries of diagrams into account, we kill the $$1/4!$$ factor to make copies of the diagram for all permutations of lines.
6. For each external point, associate a factor $$1$$.
7. Divide by the symmetry factor.
##### Constructing Feynmann Rules (Momentum Space)
1. Contruct the position space Feynman rule first.
2. Rewrite the contractions in the momentum space.
• ${\color{blue}\phi}(x){\color{blue}\phi}(y) = {\color{magenta}\int \frac{\dd{^4 p}}{(2\pi)^4}} e^{-ip\cdot({\color{darkcyan}x}-{\color{darkgreen}y})} {\color{orange}\tilde{\Delta}(p)}.$
3. The diagram is left intact. However, each propagator should be assigned a momentum (with direction). $x - y \rightarrow p.$
4. We associate each propagator with the factor $$\color{orange}\tilde{\Delta}(p)$$:
• ${\color{blue}\phi}(x){\color{blue}\phi}(y) \rightarrow {\color{orange}\tilde{\Delta}(p)}.$
5. For each vertex, kill the $$\displaystyle \int \dd{^4 z}$$ in the position space Feynmann rules: $(-i\lambda) \int \dd{^4 z} \rightarrow -i\lambda \cdots.$
6. The integral in $$z$$ should be replaced by a $$\delta$$-function representing the conservation of momentum: $(-i\lambda) \int \dd{^4 z} \rightarrow (-i\lambda) {\color{darkgreen}(2\pi)^4 \delta\qty(\sum p)}.$
7. For each external points, associate a factor $$\color{darkcyan} e^{-ip\cdot x}$$ if the momentum points towards $$x$$, and $$e^{+ip\cdot x}$$ if pointing away from $$x$$.
8. Integrate over each momentum: $\color{magenta} \int \frac{\dd{^4 p}}{(2\pi)^4}.$
9. Divide by the symmetry factor.
##### Feynman Rules for Scattering (Position Space)
1. The procedure for position space Feynman rules applies when the contraction does not involve in and out states.
2. In addition to contraction between fields, we have to obtain the factor for contraction between field and in and out states. ${\color{blue} \phi_I}(x)\ket{\color{blue} \vb*{p}} = {\color{orange} \tilde{\phi}_I(p) e^{-ip\cdot x}}\ket{0},$
• Contractions between in and out states themselves, i.e. ${\color{blue}a_{\vb*{q}}a^\dagger_{\vb*{p}}} = {\color{orange} \delta^{(4)}}(p-q),$ does not occur in (fully) connected diagrams.
• Always a line there when we are doing contraction and obtaining something as a factor!
##### Feynman Rules for Scattering (Momentum Space)
1. Construct the position space Feynman rule for scattering first.
2. The procedure for momentum space Feynman rules applies when the contraction does not involve in and out states.
3. Replace $${\color{orange} \tilde{\phi}_I(p) e^{-ip\cdot x}}$$ by its Fourier transform: ${\color{orange} \tilde{\phi}_I(p) e^{-ip\cdot x}} \rightarrow {\color{orange} \tilde{\phi}_I(p)}.$
4. External $$p$$'s should not be integrated.

Rules for scattering in the position space is for $i\mathcal{M} \cdot (2\pi)^4 \delta^{(4)}\qty(p_A + p_B - \sum p_f),$ while in the momentum space for $i\mathcal{M}.$

##### Fermion Trivia
• We denote
• scalar particles by dashed lines,
• fermions by solid lines, and
• photons by dashed lines.
• Direction of momentum:
• On internal fermion lines, the momentum must be assigned in the direction of particle-number flow.
• On external fermion lines, the momentum is assigned in the direction of particle number for fermions while oppsite for antifermions.
• Symmetry of diagrams:
• We always drop the $$\displaystyle \frac{1}{k!}$$ factor from $$\exp$$, since it cancels out the permutation of vertices.
• The diagrams of Yukawa theory has no symmetry factors, because the three fields cannot replace one for another.
• Determining the overall sign of a diagram:
• We should not allow any intruder operators between a Dirac field operator and a creation or annihilation operator.
• However, pure complex numbers ($$D_F$$, $$S_F$$, etc.) are ok.
• For a specific contraction, we permute the field operators to eliminate the intruders.
• Fully contracted operators between are ok.
• Tricks:
• $${\color{blue} \overline{\psi}\psi}$$ commutes with any operator.
• Closed loop: ${\color{darkcyan} \overline{\psi}}{\color{blue}\psi}{\color{blue} \overline{\psi}}{\color{darkcyan}\psi} = - \operatorname{tr}{[S_F S_F]}.$

#### Example: Quartic Interaction

$H_{\mathrm{int}} = \int \dd{^3 x} \frac{\lambda}{4!} \phi^4(\vb*{x}).$

##### Position Space Feynman Rules (Quartic Interaction)
1. For each propagator, $$=D_F(x-y)$$;
2. For each vertex, $$\displaystyle =(-i\lambda) \int \dd{^4 z}$$;
3. For each external point, $$=1$$;
4. Divide by the symmetry factor.
##### Momentum Space Feynman Rules (Quartic Interaction)
1. For each propagator, $$\displaystyle = \frac{i}{p^2 - m^2 + i\epsilon}$$;
2. For each vertex, $$\displaystyle = -i\lambda$$;
3. For each external point, $$\displaystyle = e^{-ip\cdot x}$$;
4. Impose momentum conservation at each vertex;
5. Integrate over each undetermined momentum: $$\displaystyle \int \frac{\dd{^4 p}}{(2\pi)^4}$$;
6. Divide by the symmetry factor.

In the integration on $$p$$, we should take $$p^0$$ as $$p^0\propto (1+i\epsilon)$$, i.e. using the Ferynman prescription.

##### Position Space Feynman Rules for Scattering (Quartic Interaction)
• ${\color{blue}\phi_I}(x)\ket{\color{blue}\vb*{p}} = e^{-ip\cdot x}\ket{0}{ };$
• $\bra{\color{blue}\vb*{p}}{\color{blue}\phi_I}(x) = \bra{0} e^{+ip\cdot x}{ }.$
1. For each propagator, $$=D_F(x-y)$$;
2. For each vertex, $$\displaystyle =(-i\lambda) \int \dd{^4 z}$$;
3. For each external line, $$=e^{-ip\cdot x}$$;
4. Divide by the symmetry factor.

We evalute \begin{align*} &\bra{\vb*{p}_1 \cdots \vb*{p}_n} iT \ket{\vb*{p}_A \vb*{p}_B} \\ &= \qty({_0\bra{\vb*{p}_1 \cdots \vb*{p}_1}T\qty(\exp\qty[-i\int_{-T}^T \dd{t} H_I(t)])\ket{\vb*{p}_A \vb*{p}_B}_0})_{\substack{\text{connected,}\\\text{amputated}}}, \end{align*}{ } to the zeroth and first order below, ignoring the prescription connected and amputated first.

Zeroth order: trivial because of the conservation of momentum. \begin{align*} _0\bra{\vb*{p}_1 \vb*{p}_2}\ket{\vb*{p}_A \vb*{p}_B}_0 &= \sqrt{2E_1 2E_2 2E_A 2E_B} \bra{0} a_1 a_2 a_A^\dagger a_B^\dagger \ket{0} \\ &= 2E_A 2E_B (2\pi)^6 \qty(\delta^{(3)}\qty(\vb*{p}_A - \vb*{p}_1) \delta^{(3)}\qty(\vb*{p}_B - \vb*{p}_2) + \delta^{(3)}\qty(\vb*{p}_A - \vb*{p}_2)\delta^{(3)}\qty(\vb*{p}_B - \vb*{p}_1)). \end{align*}

$$+$$

First order: using the Wick's theorem we find \begin{align*} & _0\bra{\vb*{p}_1 \vb*{p}_2} T\qty({-i\frac{\lambda}{4!}\int \dd{^4 x}\phi_I^4 (x)}) \ket{\vb*{p}_A \vb*{p}_B}_0 \\ &= {_0 \bra{\vb*{p}_1 \vb*{p}_2} N\qty({-i\frac{\lambda}{4!}\int \dd{^4 x}\phi_I^4(x) + \text{contractions}}) \ket{\vb*{p_A}\vb*{p}_B}_0}. \end{align*}

• Contraction of type $${\color{blue}\phi}{\color{blue}\phi}{\color{magenta}\phi}{\color{magenta}\phi}$$ (trivial again):

$$\times \Bigg($$ $$+$$ $$\Bigg)$$

• Contraction of type $${\color{blue}\phi}{\color{blue}\phi}{\phi}{\phi}$$: the two uncontracted $$\phi$$'s should have one contracted to the bra and one to the ket. Trivial again.

$$+$$ $$+$$ $$+$$

• Contraction of type $$\phi\phi\phi\phi$$, i.e. all $$\phi$$'s are contracted to the bra and ket: \begin{align*} &(4!)\cdot \qty(-i\frac{\lambda}{4!})\int \dd{^4 x} e^{-i(p_A + p_B - p_1 - p_2)\cdot x} \\ &= -i\lambda (2\pi)^4 \delta^{(4)}\qty(p_A + p_B - p_1 - p_2). \end{align*}

With the definition of $$\mathcal{M}$$ we find that $$\mathcal{M} = -\lambda$$.

##### Momentum Space Feynman Rules for Scattering (Quartic Interaction)
1. For each propagator, $$\displaystyle = \frac{i}{p^2 - m^2 + i\epsilon}$$;
2. For each vertex, $$\displaystyle = -i\lambda$$;
3. For each external point, $$\displaystyle = 1$$;
4. Impose momentum conservation at each vertex;
5. Integrate over each undetermined momentum: $$\displaystyle \int \frac{\dd{^4 p}}{(2\pi)^4}$$;
6. Divide by the symmetry factor.

#### Example: Yukawa Theory

\begin{align*} H &= H_{\text{Dirac}} + H_{\text{Klein-Gordon}} + H_{\text{int}} \\ &= H_{\text{Dirac}} + H_{\text{Klein-Gordon}} + \int \dd{^3 x} g\overline{\psi}\psi\phi. \end{align*}

##### Momentum Space Feynman Rules for Scattering (Yukawa Theory)
1. Propagators:

$${\color{blue}\phi}(x){\color{blue}\phi}(y) =$$ $$\displaystyle = \frac{i}{q^2 - m_\phi^2 + i\epsilon}{};$$

$${\color{blue}\psi}(x){\color{blue}\overline{\psi}}(y) =$$ $$\displaystyle = \frac{i(\unicode{x2215}\kern-.5em {p} + m)}{p^2 - m^2 + i\epsilon}{}.$$

1. Vertices: $$=-ig$$.
2. External leg contractions:
• $${\color{blue}\phi}\ket{\color{blue}\vb*{q}} =$$ $$=1$$;
• $$\bra{\color{blue}\vb*{q}}{\color{blue}\phi} =$$ $$=1$$;
• $${\color{blue}\psi}\underbrace{\ket{\color{blue}\vb*{p},s}}_{\text{fermion}} =$$ $$=u^s(p)$$;
• $$\underbrace{\bra{\color{blue}\vb*{p},s}}_{\text{fermion}}{\color{blue}\overline{\psi}} =$$ $$=\overline{u}^s(p)$$;
• $${\color{blue}\overline{\psi}}\underbrace{\ket{\color{blue} \vb*{k},s}}_{\text{antifermion}} =$$ $$= \overline{v}^s(k)$$;
• $$\underbrace{\bra{\color{blue}\vb*{k},s}}_{\text{antifermion}} {\color{blue}\psi} =$$ $$=v^s(k)$$.
3. Impose momentum conservation at each vertex.
4. Integrate over undetermined loop momentum.
5. Figure out the overall sign of the diagram.

We consider $\mathrm{fermion}(p) + \mathrm{fermion}(k) \longrightarrow \mathrm{fermion}(p') + \mathrm{fermion}(k').$

• Lowest order: $_0\bra{\vb*{p}',\vb*{k}'} T\qty({\frac{1}{2!}(-ig) \int \dd{^4 x} \overline{\psi}_I \psi_I \phi_I (-ig) \int \dd{^4 y} \overline{\psi}_I \psi_I \phi_I}) \ket{\vb*{p},\vb*{k}}_0.$
• Implicitly we assocaite spins $$s, r, s', r'$$ to the momenta.
• We drop the factor $$1/2!$$ for any diagrams since we could interchange $$x$$ and $$y$$.
• Contractions:
• $\bra{0}{\color{blue} a_{\vb*{k}'}}{\color{darkcyan} a_{\vb*{p}'}} {\color{blue} \overline{\psi}_x} {\color{magenta} \psi_x} {\color{darkcyan} \overline{\psi}_y} {\color{orange} \psi_y} {\color{orange}a_{\vb*{p}}^\dagger} {\color{magenta} a_{\vb*{k}}^\dagger}\ket{0}.$
• Move $${\color{darkcyan} \psi_y}$$ two spaces to the left.
• Factor: $$(-1)^2 = +1$$.
• $\bra{0}{\color{darkcyan} a_{\vb*{k}'}}{\color{blue} a_{\vb*{p}'}} {\color{blue} \overline{\psi}_x} {\color{magenta} \psi_x} {\color{darkcyan} \overline{\psi}_y} {\color{orange} \psi_y} {\color{orange}a_{\vb*{p}}^\dagger} {\color{magenta} a_{\vb*{k}}^\dagger}\ket{0}.$
• Move $${\color{darkcyan} \psi_y}$$ one spaces to the left.
• Factor: $$-1$$.

Conclusion:

$$i\mathcal{M} =$$ $$+$$ \begin{align*} &= (-ig^2) \qty( \overline{u}\qty(p') u\qty(p) \frac{1}{(p'-p)^2 - m_\phi^2} \overline{u}\qty(k') u\qty(k) - \overline{u}\qty(p') u\qty(k) \frac{1}{(p'-k)^2 - m_\phi^2} \overline{u}\qty(k') u\qty(p) ). \end{align*}

Typo: wavy lines should be replaced with dashed lines here.

##### The Yukawa Potential
• For distinguishable particles, only the first diagram contributes.
• $i\mathcal{M} \approx \frac{ig^2}{\abs{\vb*{p}' - \vb*{p}}^2 + m_\phi^2} 2m\delta^{ss'} 2m\delta^{rr'}.$
• With the Born approximation we find $V(r) = -\frac{g^2}{4\pi}\frac{1}{r}e^{-m_\phi r}.$

#### Example: Quantum Electrodynamics

$H_{\text{int}} = \int \dd{^3 x} e\overline{\psi}\gamma^\mu \psi A_\mu.$

##### Momentum Space Feynman Rules for Scattering (Quantum Electrodynamics)
1. Fermion rules from the Yukawa theory apply here.
2. New vertex: $$= -ie\gamma^\mu$$.
3. Photon propagator: $$\displaystyle = \frac{-ig_{\mu\nu}}{q^2 + i\epsilon}$$.
4. External photon lines:
• $${\color{blue}A_\mu}\ket{\color{blue}\vb*{p}}=$$ $$=\epsilon_\mu(p)$$.
• $$\bra{\color{blue}\vb*{p}}{\color{blue}A_\mu}=$$ $$=\epsilon^*_\mu(p)$$.
• Initial and final state photons should be transversely polarized.
• $\epsilon^\mu = (0, \vb*{\epsilon}).$
• $\vb*{p}\cdot \vb*{\epsilon} = 0.$

We consider $\mathrm{fermion}(p) + \mathrm{fermion}(k) \longrightarrow \mathrm{fermion}(p') + \mathrm{fermion}(k').$

$$i\mathcal{M} =$$ \begin{align*} &= (-ie)^2 \overline{u}\qty(p') \gamma^\mu u\qty(p) \frac{-ig_{\mu\nu}}{\qty(p'-p)^2}\overline{u}\qty(k')\gamma^\nu u\qty(k). \end{align*}

##### The Coulomb Potential
• $i\mathcal{M} \approx \frac{-ie^2}{\abs{\vb*{p}' - \vb*{p}}^2}(2m\xi'^\dagger \xi)_p (2m\xi'^\dagger \xi)_k.$
• Comparing this to the Yukawa case we obtain a repulsive Coulomb potential $V(r) = \frac{e^2}{4\pi r} = \frac{\alpha}{r}.$
• For scattering between an electron and an anti-electron, we get an attractive potential.
 Quantum Field Theory (IV) Renormalization

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A few websites for pronunciations:

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## Curves in Projective Space

This post is not yet done.

#### Projective Space

Let $$[(x_0,y_0,z_0)]$$ be a point in $$P_2(k)$$. We define an affine local coordinate about it by $\varphi(\Phi^{-1}(x,y,1)) = (x,y)$ where $$\Phi\in\mathrm{GL}(3,k)$$ and $$(x_0,y_0,z_0) = \Phi^{-1}(0,0,1)$$.

A homogeneous polynomial $$F$$ may be made into a function on $$P_2(k)$$ by $f(x,y) = F(\Phi^{-1}(x,y,1)).$

With $$\Phi = \mathbb{1}$$ we get the canonical coordinate $(x,y,z) \rightarrow (\frac{x}{z}, \frac{y}{z}).$

With $\Phi = \begin{pmatrix} 1 & 0 & -x_0 \\ 0 & 1 & -y_0 \\ 0 & 0 & 1 \end{pmatrix}$ we get a shift $(x,y,1) \rightarrow (x-x_0, y-y_0).$ Under this prescription, we have $f(x,y) = F(x+x_0, y+y_0, 1).$

With $\Phi = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$ We get a permutation $(x,1,w) \rightarrow (w,x).$ Under this prescription, we have $f(x,y) = F(y,1,x).$

#### Curves and Tangents

A plane curve or a projective plane curve is a nonzero homogeneous polynomial $$F\in k[x,y,z]_d$$ for some $$d>0$$. The locus $F(K) = \set{(x,y,w) | F(x,y,w) = 0}.$ where $$K$$ is an extension of $$k$$ is well defined.

In a specific affine coordinate about $$[(x_0,y_0,z_0)]$$ we have $f(x,y) = F(\Phi^{-1}(x,y,1)) = f_0 + f_1 + \cdots + f_d.$ If $$[(x_0,y_0,w_0)]\in F(K)$$ then $$f_0 = 0$$. $$[(x_0,y_0,w_0)]$$ is said to be a singular point if $$f_1$$ is the zero polynomial.

$$F$$ over $$k$$ is said to be nonsingular or smooth if $$F$$ is nonsingular at every point of $$F(\overline{k})$$ where $$\overline{k}$$ is the algebraic closure of $$\vb*{k}$$.

Let $F(x,y,w) = x^2y + xyw + w^3$ and $$w_0=1$$. With $$\Phi$$ defined as in this example we obtain \begin{align*} f(x,y) &= (x_0^2 y_0 + x_0 y_0 + 1) \\ &{\phantom{{}={}}} + (x_0^2 y + 2x_0y_0x + x_0y + y_0x) \\ &{\phantom{{}={}}} + (y_0x^2 + 2x_0xy + xy) \\ &{\phantom{{}={}}} + (x^2y). \end{align*} If $$(x_0,y_0,w_0)\in F(K)$$ then $$f_0=0$$. $$(x_0,y_0,w_0)$$ is singular only if $$(x_0,y_0) = (0,0)$$ or $$(x_0,y_0) = (-1,0)$$, none of which is in $$F(K)$$.

Proposition. Suppose $$F$$ and $$G$$ are two plane curves. If $$(x_0,y_0,z_0)$$ is in $$F(k)\cap G(k)$$ then $$(x_0,y_0,z_0)$$ is a singular point of $$FG$$.

Bézout's Theorem. Let $$F$$ and $$G$$ be plane curves of degree $$m$$ and $$n$$ respectively, Then $$F(\overline{k})$$ and $$G(\overline{k})$$ is nonempty. If it has more than $$mn$$ points then $$F$$ and $$G$$ have as a common factor some homogeneous polynomial of degree $$>0$$.

Corollary. If $$F$$ is a plane curve and reducible over $$\overline{k}$$. Then the factors are homogeneous polynomials, and $$F$$ is singular.

$$f_1$$ may be lift back to a homogeneous function of degree $$1$$ on $$(x,y,w)$$ and is called the tangent line to $$F$$ at $$(x_0,y_0,z_0)$$.

Proposition. Let $$F$$ be a plane curve over $$k$$. If $$(x_0,y_0,z_0)$$ is on the curve, then $$(x_0,y_0,z_0)$$ is nonsingular if and only if at least one of $$\displaystyle \pdv{F}{x}, \pdv{F}{y}, \pdv{F}{w}$$ is nonzero at $$(x_0,y_0,z_0)$$. The tangent line is given by $L = \qty(\pdv{F}{x})_{(x_0,y_0,z_0)} x + \qty(\pdv{F}{y})_{(x_0,y_0,z_0)} y + \qty(\pdv{F}{z})_{(x_0,y_0,z_0)} z.$

Let $$Q$$ denote the homogeneous quadratic polynomial derived from $$f_2$$.

Proposition. Suppose the characteristic of $$k$$ is not $$2$$. Let $$F$$ be a plane curve over $$k$$ of degree $$d$$ and $$(x_0,y_0,z_0)$$ be a nonsingular point on the curve. Then $2Q(x,y,w) = \begin{pmatrix} x & y & w \end{pmatrix} H \begin{pmatrix} x\\ y\\ w \end{pmatrix} - 2(d-1) L'_\Phi(x,y,w) L(x,y,w)$ where $$H$$ is the Hessian matrix of $$F$$ at $$(x_0,y_0,z_0)$$ and $$L'_\Phi(x,y,w)$$ is a line with $$L'_\Phi(x_0,y_0,w_0) = 1$$.

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## Dirac Equation

#### Free Particles

##### The Equation and the Solution

We demand that $$\vb*{\alpha}$$ and $$\beta$$ satisfy the commutation relations \begin{align*} \beta^2 &= \mathbb{1}; \\ \qty{\alpha_i,\beta} &= 0; \\ \qty{\alpha_i,\alpha_j} &= \mathbb{1}\delta_{ij}. \end{align*} A conventional representation is given by $\vb*{\alpha} = \begin{pmatrix} 0 & \vb*{\sigma} \\ \vb*{\sigma} & 0 \end{pmatrix};\quad \beta = \begin{pmatrix} \mathbb{1} & 0 \\ 0 & -\mathbb{1} \end{pmatrix}.$ Now we have the Dirac equation $i\hbar\pdv{{\psi}}{t} = (c\vb*{\alpha}\cdot \vb*{p} + \beta mc^2){\psi}.$

###### Probability Current

Under the interpretation $\rho = \psi^\dagger \psi;\quad \vb*{j} = \psi^\dagger \vb*{\alpha} \psi;$ the continuity equation $\pdv{\rho}{t} + \div \vb*{j} = 0$ holds.

#### Electromagnetic Interaction

The Dirac equation of a particle in a electromagnetic field is written as $i\hbar\pdv{\psi}{t} = \qty[c\vb*{\alpha}\cdot\qty(\vb*{p} - q\vb*{A}/c)+\beta mc^2 + q\phi]\psi.$

##### In Pure Magnetic Field

In the following we set $$\phi = 0$$. Writing $$\psi(t) = \psi e^{-iEt/\hbar}$$ we get $(c\vb*{\alpha}\cdot \vb*{\pi} + \beta mc^2)\psi = E\psi,$ where $$\vb*{\pi} = \vb*{p} - q\vb*{A}/c$$ is the kinetic momentum.

Let $$\displaystyle \psi = \begin{pmatrix}\chi \\ \Phi\end{pmatrix}{}$$ where $$\chi$$ is called the large component and $$\Phi$$ the small component. The Dirac equation is rewritten as

\begin{align*} (\vb*{\sigma}\cdot \vb*{A})(\vb*{\sigma}\cdot \vb*{B}) &= \vb*{A}\cdot \vb*{B} + i\vb*{\sigma}\cdot \vb*{A}\times \vb*{B};\\ \vb*{\pi} \times \vb*{\pi} &= \frac{iq\hbar}{c}\vb*{B}. \end{align*}

\begin{align*} \chi &= \qty(\frac{c\vb*{\sigma}\cdot \vb*{\pi}}{E-mc^2})\Phi; \\ \Phi &= \qty(\frac{c\vb*{\sigma}\cdot \vb*{\pi}}{E+mc^2})\chi. \end{align*}

Let $$E_S = E - mc^2$$ denote the energy that appeared in the Schrödinger equation. In the nonrelativistic limit we get $\qty[\frac{(\vb*{p} - q\vb*{A}/c)^2}{2m} - \frac{q\hbar}{2mc}\vb*{\sigma}\cdot \vb*{B}] \chi = E_S \chi.$

##### Hydrogen Fine Structure

Let $$V$$ denote the Coulomb potential in the hydrogen atom. The Dirac equation is rewritten as \begin{align*} \Phi &= \qty(\frac{c\vb*{\sigma}\cdot \vb*{p}}{E-V+mc^2})\chi; \\ (E-V-mc^2) &= c\vb*{\sigma}\cdot \vb*{p}\qty[\frac{1}{E-V+mc^2}]c\vb*{\sigma}\cdot \vb*{p}\chi. \end{align*} In the nonrelativistic limit we keep the $$(E_S - V)$$ on the denominator on the RHS to the first order and find $E_S \chi = H\chi = \qty[\frac{\vb*{p}^2}{2m} + V - \frac{\vb*{\sigma}\cdot \vb*{P}(E_S -V)\vb*{\sigma}\cdot \vb*{P}}{4m^2c^2}]\chi.$ To retain the conservation of probability and to modify $$H$$ into a Hermitian operator, we solve for $\chi_S = \qty(1+\frac{\vb*{p}^2}{8m^2c^2})\chi$ instead of $$\chi$$. The equation then becomes (working to order $$\alpha^4$$) $E_S \chi_S = H_S \chi_S = \qty(H + \qty[\frac{\vb*{p}^2}{8m^2c^2},V]).$ Approximate to order $$\alpha^4$$ we find \begin{align*} H_S &= \frac{\vb*{p}^2}{2m} + V - \frac{\vb*{p}^4}{8m^3c^2} - \frac{i\vb*{\sigma}\cdot \vb*{p}\times \qty[\vb*{p}, V]}{4m^2c^2} + \qty{-\frac{\vb*{p}\cdot \qty[\vb*{p},V]}{4m^2c^2}+\frac{\qty[\vb*{p}\cdot \vb*{p},V]}{8m^2c^2}} \\ &= \frac{\vb*{p}^2}{2m} + V - \frac{\vb*{p}^4}{8m^3c^2} + \frac{e^2}{2m^2c^2r^2}\vb*{S}\cdot \vb*{L} + \frac{e^2\hbar^2 \pi}{2m^2 c^2}\delta^3(\vb*{r}) \end{align*} These items accounts for $$T$$, $$V$$, relativistic $$T$$, spin-orbit interaction, and the Darwin term sequentially.

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## Red-Black Tree

#### Insertion

/**
* Invoked if both children of a node are black.
*/
void handleReorient( const Comparable & item )
{
current->color = RED;
current->left->color = BLACK;
current->right->color = BLACK;

if ( parent->color = RED ) // rotate
{
grand->color = RED;

if ( item < grand->element != item < parent->element ) // rotate twice
parent = rotate( item, grand );
current = rotate( item, great );
current->color = BLACK;
}
}


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## Scattering Theory

#### Asymptotic Solutions

The solution of scattering by a potential $$V(r)$$ has the form $\psi_k \xrightarrow[r\rightarrow\infty]{} e^{ikz} + f(\theta,\varphi)\frac{e^{ikr}}{r},$ where $$f$$ is called the scattering amplitude. The cross section is given by $\dv{\sigma}{\Omega} = \abs{f(\theta,\varphi)}^2.$

#### Approximations

##### Time-Dependent Approximation

Fermi's Golden Rule states that the transition rate is given by $\Gamma_{i\rightarrow f} = \frac{2\pi}{\hbar}\abs{\bra{f}V\ket{i}}^2\rho(E_f).$

The S Matrix is the operator defined by $S = \lim_{\substack{t_f \rightarrow \infty\\ t_i\rightarrow -\infty}} U(t_f,t_i).$

With the Fermi's golden rule the transition rate is found to be \begin{align*} R_{i\rightarrow \dd{\Omega}} &= \frac{2\pi}{\hbar}\abs{\bra{\vb*{p}_f}V\ket{\vb*{p}_i}}^2 \mu p_i \dd{\Omega}. \end{align*}

The incoming probability current is given by $j_i = \frac{\hbar k}{\mu}\qty(\frac{1}{2\pi\hbar})^3,$ from which we find that the differential cross section is given by $\dv{\sigma}{\Omega}\dd{\Omega} = \frac{R_{i\rightarrow \dd{\Omega}}}{j_i},$ i.e. $$\label{eq:diff_cross_sec} \dv{\sigma}{\Omega} = \abs{\frac{\mu}{2\pi\hbar^2}\int e^{-i\vb*{q}\cdot \vb*{r}'}V(\vb*{r}')\dd{^3 \vb*{r}'}}^2,$$ where $$\hbar \vb*{q} = \vb*{p}_f - \vb*{p}_i$$.

I am I little bit confused by Shankar's normalization. One may arrive at exactly the same result as in \eqref{eq:diff_cross_sec} with the conventional box normalization.

From \eqref{eq:diff_cross_sec} we infer that (up to a phase factor) $f(\theta,\varphi) = -\frac{\mu}{2\pi\hbar^2} \int e^{-i\vb*{q}\cdot\vb*{r}'}V(\vb*{r}')\dd{^3\vb*{r}'}.$

In particular, for a central potential $$V(r)$$ we have $f(\theta) = -\frac{2\mu}{\hbar^2}\int \frac{\sin qr'}{q}V(r')r'\dd{r'}.$

For the Yukawa potential $V(r) = \frac{g e^{-\mu_0 r}}{r}{}$ we find $\dv{\sigma}{\Omega} = \frac{4\mu^2 g^2}{\hbar^4\qty[\mu_0^2 + 4k^2\sin^2\qty(\theta/2)]^2}.$ Let $$g = Ze^2$$ and $$\mu_0 = 0$$, we find the differential cross section for Coulomb scattering $\dv{\sigma}{\Omega} = \frac{(Ze^2)^2}{16E^2\sin^4\qty(\theta/2)}.$

##### Born Approximation

We rewrite the scattering problem as $(\grad^2 + k^2)\psi_k = \frac{2\mu}{\hbar^2}V(r)\psi_k.$ The solution may be obtained using the Green's function by $\psi_{k}(r) = e^{i\vb*{k}\cdot\vb*{r}} + \frac{2\mu}{\hbar^2} \int G^0(\vb*{r},\vb*{r}')V(\vb*{r}')\psi_k(\vb*{r}')\dd{^3 \vb*{r}'},$ where $$G^0$$ is the solution to $(\grad^2 + k^2) G^0(\vb*{r},\vb*{r}') = \delta^3(\vb*{r} - \vb*{r}').$ By demanding $$G(\vb*{r},\vb*{r}')$$ be out-going we obtain $G(\vb*{r},\vb*{r}') = -\frac{e^{ik\abs{\vb*{r} - \vb*{r}'}}}{4\pi \abs{\vb*{r} - \vb*{r}'}}.$

Minutia in Obtaining the Green's Function

The operator $$\grad^2 + k^2$$ is not invertible since $$e^{ikz}$$ is in the null space thereof. Adding a $$i\epsilon$$ we obtain a operator $$\grad^2 + k^2 + i\epsilon$$ invertible in the physical Hibert space since elements in its null space diverges at infinity. A positive $$\epsilon$$ happened to yield the retarded Green's function, while a negative $$\epsilon$$ yields the advanced Green's functions.

To the first order we recover $f(\theta,\varphi) = -\frac{2\mu}{4\pi\hbar^2} \int e^{-i(\vb*{k}_f - \vb*{k}_i)\cdot \vb*{r}'}V(\vb*{r}')\dd{^3\vb*{r}'}.$

##### The Partial Wave Expansion

For the spherical Bessel function we have the asymptotic formula \begin{align*} j_l(kr) &\xrightarrow[r\rightarrow\infty]{}\frac{\sin(kr - l\pi/2)}{kr}; \\ n_l(kr) &\xrightarrow[r\rightarrow\infty]{}-\frac{\cos(kr - l\pi/2)}{kr}. \end{align*}

We expand $$f$$ in terms of Legendre Polynomials $f(\theta,k) = \sum_{l=0}^\infty (2l+1)a_l(k) P_l(\cos\theta)$ as well as the incident wave $e^{ikz} = \sum_{l=0}^\infty i^l (2l+1) j_l(kr) P_l(\cos\theta),$ which consists of an incoming and an outgoing spherical wave. By allowing a phase shift in $R_l(l) = \frac{U_l(r)}{r} \xrightarrow[r\rightarrow\infty]{} \frac{A_l \sin \qty[kr - l\pi/2 + \delta_l(k)]}{r}$ and demanding the amplitude of incoming spherical wave be the same as in $$e^{ikz}$$ we find $\psi_{k}(\vb*{r}) = e^{ikz} + \qty[\sum_{l=0}^\infty (2l+1)\qty(\frac{e^{2i\delta_l} - 1}{2ik})P_l(\cos\theta)]\frac{e^{ikr}}{r}.$ Note that from the ansatz of $$R_l$$ the S matrix in the basis of $$(k,l,m)$$ where $$m=0$$ is found to be $S_l(k) = e^{2i\delta_l(k)}.$ From the expression of $$\psi_k$$ we deduce $a_l(k) = \frac{e^{2i\delta_l} - 1}{2ik} = \frac{e^{i\delta_l}\sin\delta_l}{k},$ whence we obtain the total cross section $\sigma = \frac{4\pi}{k^2} \sum_{k=0}^\infty (2l+1)\sin^2\delta_l = \frac{4\pi}{k^2}\Im f(0),$ where the last equality gives the optical theorem.

###### The Hard Sphere

For the potential $V(r) = \begin{cases} \infty, & \text{if }r<r_0,\\ 0, & \text{if }r>r_0 \end{cases}$ we have the free particle solution $R_l(r) = A_l j_l(kr) + B_l n_l(kr)$ for $$r>r_0$$ subjected to the boundary condition $$R_l(r_0) = 0$$. The phase shift is found by using the asymptotic formulae of $$j_l$$ and $$n_l$$ to be $\delta_l = \arctan \frac{j_l(kr_0)}{n_l(kr_0)}.$ In particular, for $$k\ll 1$$ we have $$\delta_l \approx (kr_0)^{2l+1}$$.

Expand

## Partial Differential Equations in Physics

#### Integeral Transforms

By knowledgebly applying integral transforms, some integrals could be evaluated much more easily.

##### The Variational Method for the Helium Ground State

We should see below that with the help of Fourier transformation, the evaluation of the Coulomb integral between the two electrons becomes much easier.

We use the following trial wavefunction $f(\vb*{r}_1,\vb*{r}_2) = \frac{Z^3}{a_0^3\pi}e^{-Zr_1/a_0}e^{-Zr_2/a_0}.$ With the Fourier transforms \begin{align*} \int \dd{^3\vb*{r}} e^{-\alpha r} e^{-i\vb*{k}\cdot \vb*{r}} &= \frac{8\pi \alpha}{(k^2+\alpha^2)^2}; \\ \int \dd{^3\vb*{r}} \frac{1}{r} e^{-i\vb*{k}\cdot \vb*{r}} &= \frac{4\pi}{k^2} \end{align*} we obtain \begin{align*} \int \dd{^3\vb*{r}_1}\dd{^3\vb*{r}_2} &= \qty[e^{-\alpha r}*\frac{1}{r}*e^{-\alpha r}](0) \\ &= \mathcal{F}^{-1}\qty[\frac{8\pi \alpha\cdot 8\pi\alpha\cdot 4\pi}{(k^2+\alpha^2)^2\cdot k^2}](0) \\ &= \frac{1}{(2\pi)^3}\int_0^\infty \dd{k}\cdot 4\pi \cdot \frac{256\pi^3\alpha^2}{(k^2+\alpha^2)^4} \\ &= \frac{20\pi^2}{\alpha^5}. \end{align*} The energy is found to be $E = (2c^2-8c+\frac{5}{4}c)E_0$ where $$-E_0$$ is the ground state energy of the hydrogen atom.

Minimizing $$E$$ we find that $E_{\mathrm{min}} = -\frac{729}{128}E_0 = -77.5\;\text{eV}$ with $$\displaystyle Z = \frac{27}{16} = 1.69$$.

Expand

A useful Japanese dictionary that contains etymology is found here.

Expand

The environment variable may be assigned by export root_path=root@zechen.site/root/MyPersonalWebsite/public

Expand

## Analytic Number Theory (I)

#### Some Elementary Results

##### $$\sum p^{-1}$$ Diverges

The infinitude of prime numbers had been well known since Euclid. A stronger result is that the following series diverges: $\sum_p \frac{1}{p} \rightarrow \infty.$

An Elementary Proof

Suppose the series above converges, i.e. there is a $$q$$ such that $\sum_{p>q} \frac{1}{p} < \frac{1}{2}.$ Then we let $$p_1,\cdots,p_m$$ denotes all prime numbers less than $$q$$, and let $Q = p_1 \cdots p_m.$ Now since $$\{1+nQ\}$$ are not divisible by $$p_1,\cdots,p_m$$, we find that $\sum_{n=1}^r \frac{1}{1+nQ} \le \sum_{i=1}^\infty (\sum_{p>q} \frac{1}{p})^i$ since every item on the LHS is included in the summation on the RHS. This yields a contradiction.

#### Arithmetic Functions

We define $$\operatorname{id}_k(n) = n^k$$, and $$\operatorname{id}(n) = n$$.

##### The Dirichlet Convolution and Derivative

Definition. $$\displaystyle (f * g)(n) = \sum_{d | n} f(d) g\qty(\frac{n}{d}).$$

Under this definition, the commutation law and the association law hold. $f*g = g*f,\quad (f*g)*h = f*(g*h).$ The Dirichlet inverse of $$f$$ is denoted by $$f^{-1}$$ and satisfies $$f^{-1}*f = I$$. If $$f(1) \neq 0$$, then the inverse exists and is unique.

Definition. The derivative of a arithmetic function is defined by $$f'(n) = f(n)\log n$$, for $$n\ge 1$$.

Many properties of the derivative in calculus hold for the arithmetic derivative as well.

• $$(f+g)' = f' + g'$$.
• $$(f*g)' = f'*g + f*g'$$.
• $$(f^{-1})' = -f' * (f * f)^{-1}$$.
##### Multiplicative Functions

Multiplicative functions make up a subgroup of arithmetic functions under the binary operation defined by the Dirichlet convolution, i.e. if $$f*g$$ is multiplicative if both $$f$$ and $$g$$ are, and $$f^{-1}$$ is multiplicative if $$f$$ is.

Examples of multiplcative functions includes

• $$1(n)$$ (completely multiplicative),
• $$\operatorname{id}_k(n)$$ (completely multiplicative),
• $$\lambda(n)$$ (complete multiplicative),
• $$\varphi(n)$$,
• $$\mu(n)$$,
• $$\sigma_k(n)$$.

Theorem. Let $$f$$ be multiplicative. Then $$f$$ is completely multiplicative if and only if $$f^{-1}(n) = \mu(n) f(n)$$.

Theorem. If $$f$$ is multiplicative, then $\sum_{d|n} \mu(d) f(d) = \prod_{p|n} (1-f(p)).$

##### The Number Theoretic $$\delta$$-Function

Definition. $$\displaystyle I(n) = \qty(n) = \begin{cases} 1, & \text{if } n = 1; \\ 0, & \text{if } n > 1.\end{cases}{}$$

This is the analogy of the $$\delta$$-function in number theory. $I * f = f * I = f.$

##### The Möbius Function $$\mu(n)$$

Definition. $$\mu(1) = 1$$, and for $$n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}>1$$ we define $\mu(n) = \begin{cases} (-1)^k, & \text{if } \alpha_1 = \cdots = \alpha_k = 1; \\ 0, & \text{otherwise}. \end{cases}{}$

Theorem. If $$n\ge 1$$ we have $(1 * \mu)(n) = \sum_{d | n} \mu(d) = I(n) = \qty[\frac{1}{n}] = \begin{cases} 1, & \text{if } n = 1; \\ 0, & \text{if } n > 1. \end{cases}{}$

##### The Euler's Totient Function $$\varphi(n)$$

Definition. $$\varphi(n)$$ is defined to be the number of of integers in $$1,\cdots,n$$ that's coprime with $$n$$.

Theorem. If $$n\ge 1$$ we have $(1 * \varphi)(n) = \sum_{d | n} \varphi(d) = n.$

Theorem. If $$n\ge 1$$ we have $(\mu*\operatorname{id})(n) = \sum_{d | n}\mu(d)\frac{n}{d} = \varphi(n).$

Theorem. $$\varphi$$ could be written as the product form $\varphi(n) = n \prod_{p | n} \qty(1-\frac{1}{p}).$

Some properties of the totient function are given below.

• For a prime number $$p$$, $$\varphi(p^\alpha) = p^\alpha - p^{\alpha-1}{}$$.
• $$\displaystyle\varphi(mn) = \varphi(m)\varphi(n)\qty(\frac{d}{\varphi(d)})$$ where $$d = \gcd(m,n)$$.
• In particular, if $$\gcd(m,n) = 1$$ then $$\varphi(mn) = \varphi(m)\varphi(n)$$.
• If $$a|b$$ then $$\varphi(a)|\varphi(b)$$.
• For $$n\ge 3$$, $$\varphi(n)$$ is even. If $$n$$ has $$r$$ different prime factors, then $$2^r | \varphi(n)$$.
##### The Möbius Inversion Formula

Theorem. If $$f = 1 * g$$, then $$g = f * \mu$$.

##### The von Mangoldt Function $$\Lambda(n)$$

Definition. $\displaystyle \Lambda(n) = \begin{cases}\log p, & \text{if } n = p^m, \text{ where } p \text{ is a prime}, \text{ and } m\ge 1; \\ 0, &\text{otherwise}.\end{cases}{}$

Theorem. If $$n\ge 1$$, $$\displaystyle (1')(n) = \log n = 1 * \Lambda = \sum_{d|n} \Lambda(d)$$.

Theorem. If $$n\ge 1$$, $$\displaystyle \Lambda(n) = \mu * (1') = -\sum_{d|n} \mu(d) \log d$$.

##### The Liouville Function $$\lambda(n)$$

Definition. $$\lambda(1) = 1$$, and for $$n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}>1$$ we define $\lambda(n) = (-1)^{\alpha_1 + \cdots + \alpha_k}.$

Theorem. For $$n>1$$, we have $(1 * \lambda)(n) = \sum_{d|n} \lambda(d) = \begin{cases} 1, &\text{if } n \text{ is a perfect square}; \\ 0, & \text{otherwise}. \end{cases}$ We have also $$\lambda^{-1}(n) = \mu^2(n)$$.

##### The Divisor Functions $$\sigma_k(n)$$

Definition. $$\sigma_k(n) = \operatorname{id}_k * 1 = \sum_{d|n} d^k$$.

In particular, we denote $$\sigma_0(n)$$ by $$d(n)$$, which counts the number of divisors, and $$\sigma_1(n)$$ by $$\sigma(n)$$, which is the sum of all divisors.

Theorem. The inversion of the divisor function is given by $\sigma_k^{-1} = (\mu \operatorname{id}_k)*\mu = \sum_{d|n} d^k \mu(d) \mu\qty(\frac{n}{d}).$

##### Generalized Convolution

In the following context, $$F$$ is a function on $$(0,+\infty)$$ that satisfies $$F(x) = 0$$ for $$x\in (0,1)$$.

Definition. $$\displaystyle (\alpha\circ F)(x) = \sum_{n\le x} \alpha(n) F\qty(\frac{x}{n})$$.

Theorem. Let $$\alpha$$ and $$\beta$$ be arithmetic functions. Then $\alpha\circ(\beta\circ F) = (\alpha*\beta)\circ F.$

Theorem. The generalized inversion formula is given by $G = \alpha \circ F \Leftrightarrow F = \alpha^{-1}\circ G.$

Theorem. For a completely multiplcative $$\alpha$$ we have $G = \alpha\circ F \Leftrightarrow F = (\mu\alpha)\circ G.$

##### Bell Series

Definition. Let $$f$$ be a arithmetic function. We define the Bell series of $$f$$ modulo $$p$$ by the formal power series $f_p(x) = \sum_{n=0}^\infty f(p^n) x^n.$

Examples of Bell series are listed below.

• $$\mu_p(x) = 1-x$$.
• $$\displaystyle\varphi_p(x) = \frac{1-x}{1-px}{}$$.
• If $$f$$ is completely multiplicative, we have $\displaystyle f_p(x) = \frac{1}{1-f(p)x}.$
• $$I_p(x) = 1$$.
• $$\displaystyle 1_p(x) = \frac{1}{1-x}$$.
• $$\displaystyle \operatorname{id}_p^\alpha(x) = \frac{1}{1-p^\alpha x}$$.
• $$\displaystyle \lambda_p(x) = \frac{1}{1+x}$$.

Theorem. Let $$h = f*g$$. Then $h_p(x) = f_p(x) g_p(x).$

##### Summary

We have the following table of convolution relations of various arithmetic functions.

 $$f*g$$ $$I$$ $$1$$ $$\operatorname{id}{}$$ $$\varphi$$ $$\mu$$ $$I$$ $$I$$ $$1$$ $$\operatorname{id}{}$$ $$\varphi$$ $$\mu$$ $$1$$ $$1$$ $$d$$ $$\sigma$$ $$\operatorname{id}{}$$ $$I$$ $$\operatorname{id}{}$$ $$\operatorname{id}{}$$ $$\sigma$$ $$\sigma\cdot \operatorname{id}{}$$ $$\varphi$$ $$\varphi$$ $$\varphi$$ $$\operatorname{id}{}$$ $$\mu$$ $$\mu$$ $$I$$ $$\varphi$$

Expand

## Index

Welcome to my home page. I am Ze Chen from the University of Science and Technology of China and have written some notes on the topics that I found interesting. Feel free to leave a comment on my posts, or to create your own posts. You may speak anonymously, or after signing up (takes only 20 seconds, no email verification required) if you wish.

If you spot any mistakes or have advice on improvements, comment under this post or under the post where the mistake lies. Your feedback is cherished and will be really helpful in improving the user experience of all visitors.

The following is an index of the notes I've written.

#### Physics

##### Quantum Field Theory
 Quantum Field Theory (I) Mathematical Prerequisites
 Quantum Field Theory (II) Quantization of Fields
 Quantum Field Theory (III) Feynman Diagrams
 Quantum Field Theory (IV) Renormalization
 Quantum Field Theory (V) Applications of Feynman Diagrams
##### Mathematical Physics
 Representation of Finite Groups ねぇ、あなたは何色になりたい？

#### Quizzes

Just for fun. Don't take them seriously.

#### FAQ

Also a reminder to myself.

The following code

%%% NAV
%%% NAV-TAG Tag1
content 1.
%%% NAV-TAG-END
%%% NAV-TAG Tag2
content 2.
%%% NAV-TAG-END
%%% NAV-END


renders

content 1.

content 2.

##### How do I add a <summary> tag?

The following code

+++ Title
content.
+++


renders

Title

content.

##### How do I add a <span> with style?

Due to a conflict between plugins, the syntax for inserting a <span> is a little complicated.

[content]{}{ style="color: red;" }


renders content.

##### How do I preview my post?

By pressing <Ctrl> + <Enter> or by clicking the button.

##### How to open a specific post in a new tab?

If there are a few posts on your page, you may click the button to open a specific post in a new tab.

##### What are the pros and cons of writing notes in Markdown?

Pros Markdown interfaces better with HTML. It's crude on its own but is capable of utilizing the powerful functions provided by HTML and JavaScript. The ability to render <summary> tag is one of the examples where Markdown (with a lot of plugins) is able to render some interactive controls that are impossible to be implemented with LaTeX.

Cons LaTeX offers much more abundant packages to typeset diagrams (circuits diagrams, Feynman diagrams, commutative diagrams, etc.). These figures are compiled with LaTeX before uploaded and inserted into my notes.

##### How did I set up a Jupyter Notebook server?

I followed the procedure described here.

##### How do I remove a post?

Go to the Control Panel.

Scribble.findOneAndRemove({uuid:"e9dc6366-7f90-43ac-9ac4-4857f2746fc0"});


#### A TODO List

The following a list of improvements yet to be done on this site.

• Make the behavior of auto-collapsing more reasonable.
• Span.
• Preview behavior at the quiz editing page.

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## Bash Tricks (I)

#### Text Processing Tools: grep, cut, sed, and awk

##### grep

grep stands for "global regular expression print", which basically searches for the lines that contains the pattern given. Let the file hosts be given below, copied from here.

# Copyright (c) 1993-2009 Microsoft Corp.
#
# This is a sample HOSTS file.
#
# For example:
#
#      102.54.94.97     rhino.acme.com          # source server
#       38.25.63.10     x.acme.com              # x client host
# localhost name resolution is handled within DNS itself.
# 127.0.0.1       localhost
# ::1             localhost
192.168.168.200 doug.LoweWriter.com             # Doug’s computer
192.168.168.201 server1.LoweWriter.com s1       # Main server
192.168.168.202 debbie.LoweWriter.com           # Debbie’s computer
192.168.168.203 printer1.LoweWriter.com p1      # HP Laser Printer


Now we are going to find out all the lines that contains .com. The following command

cat hosts | grep ".com"


prints

#      102.54.94.97     rhino.acme.com          # source server
#       38.25.63.10     x.acme.com              # x client host
192.168.168.200 doug.LoweWriter.com             # Doug’s computer
192.168.168.201 server1.LoweWriter.com s1       # Main server
192.168.168.202 debbie.LoweWriter.com           # Debbie’s computer
192.168.168.203 printer1.LoweWriter.com p1      # HP Laser Printer


It accidentally includes the lines commented out as well. To exclude these, we use the regular expression with grep by the -E option.

cat hosts | grep -E "^[^#].*\.com"


The ouput now becomes

192.168.168.200 doug.LoweWriter.com             # Doug’s computer
192.168.168.201 server1.LoweWriter.com s1       # Main server
192.168.168.202 debbie.LoweWriter.com           # Debbie’s computer
192.168.168.203 printer1.LoweWriter.com p1      # HP Laser Printer


##### cut

Now we want the IP addresses and the domain names only, leaving out the comments at the end of lines. The following cut command does this job,

cat hosts | grep -E "^[^#].*\.com" | cut -d " " -f 1-2


which prints

192.168.168.200 doug.LoweWriter.com
192.168.168.201 server1.LoweWriter.com
192.168.168.202 debbie.LoweWriter.com
192.168.168.203 printer1.LoweWriter.com


cut -d " " doesn't work like String.split() in most programming languages.

It should be noted that, with cut -d " ", every single space is counted as a delimiter. For example,

cat hosts | grep -E "^[^#].*\.com" | cut -d " " -f 1-3


prints

192.168.168.200 doug.LoweWriter.com
192.168.168.201 server1.LoweWriter.com s1
192.168.168.202 debbie.LoweWriter.com
192.168.168.203 printer1.LoweWriter.com p1


##### sed

sed stands for "stream editor", with which we are able to make certain replacements. For example

cat hosts | grep -E "^[^#].*\.com" | cut -d " " -f 1-2 | sed 's/192\.168/255.255/'


prints

255.255.168.200 doug.LoweWriter.com
255.255.168.201 server1.LoweWriter.com
255.255.168.202 debbie.LoweWriter.com
255.255.168.203 printer1.LoweWriter.com


##### awk

awk stands for "Aho, Weinberger, Kernighan", i.e. the authors. It's among the most powerful text processing tools. We may swap the two columns with it, using

cat hosts | grep -E "^[^#].*\.com" | cut -d " " -f 1-2 | awk '{ t = $1;$1 = $2;$2 = t; print }'


which prints

doug.LoweWriter.com 192.168.168.200
server1.LoweWriter.com 192.168.168.201
debbie.LoweWriter.com 192.168.168.202
printer1.LoweWriter.com 192.168.168.203


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## Comparison of Languages (II)

#### by language I mean programming language...

This post is devoted to numerical computations and focuses on numpy and MATLAB.

##### Basic Array Operations

Array Creation and Array Transpose In MATLAB, arrays are first-class objects. With numpy, however, arrays should be created with the correct constructor.

import numpy as np

arr = np.array([[2, 3, 5, 7], [11, 13, 17, 19]])


arr stores now $$\begin{pmatrix}2 & 3 & 5 & 7 \\ 11 & 13 & 17 & 19\end{pmatrix}{}$$.

arr = arr.transpose()


arr stores now $$\begin{pmatrix}2 & 11 \\ 3 & 13 \\ 5 & 17 \\ 7 & 19\end{pmatrix}{}$$.

arr = [2, 3, 5, 7; 11, 13, 17, 19];


arr stores now $$\begin{pmatrix}2 & 3 & 5 & 7 \\ 11 & 13 & 17 & 19\end{pmatrix}{ }$$.

arr = arr';


arr stores now $$\begin{pmatrix}2 & 11 \\ 3 & 13 \\ 5 & 17 \\ 7 & 19\end{pmatrix}{}$$.

Expand

## Comparison of Languages (I)

#### by language I mean programming language...

As a cyber-multilinguist, I frequently get confused by how different languages implement a certain operation. It is String.startsWith() in Java while String.startswith() in python. I have to look up the document to find out the correct spelling everytime. Sometimes the problem is even worse, especially when it comes to dealing with regex. Languages adopt their own idiomatic usages and it's the programmers' duty to take note.

#### Syntax

##### Destructuring Assignment

What do I do if the function returns a tuple or an array?

int a[2] = {1, 2};

auto [x,y] = a; // creates e[2], copies a into e, then x refers to e[0], y refers to e[1]
auto& [xr, yr] = a; // xr refers to a[0], yr refers to a[1]

// ---

float x{};
char  y{};
int   z{};

std::tuple<float&,char&&,int> tpl(x, std::move(y), z);
const auto& [a, b, c] = tpl;
// a names a structured binding that refers to x; decltype(a) is float&
// b names a structured binding that refers to y; decltype(b) is char&&
// c names a structured binding that refers to the 3rd element of tpl; decltype(c) is const int

// ---

struct S {
mutable int x1 : 2;
volatile double y1;
};
S f();

const auto [x, y] = f(); // x is an int lvalue identifying the 2-bit bit field
// y is a const volatile double lvalue

a, b = [10, 20]
print(a) # 10
print(b) # 20

a_t, b_t = (10, 20)
print(a_t) # 10
print(b_t) # 20

c, _, *d = [10, 15, 20, 30]
print(c) # 10
print(d) # [20, 30]

c_t, _, *d_t = (10, 15, 20, 30)
print(c_t) # 10
print(d_t) # [20, 30]

a, b = b, a
print(a) # 20
print(b) # 10

let a, b;
[a, b] = [10, 20];
console.log(a); // 10
console.log(b); // 20

let c, d;
[c, , ...d] = [10, 15, 20, 30];
console.log(c); // 10
console.log(d); // [20, 30]

[a, b] = [b, a];
console.log(a); // 20
console.log(b); // 10


JavaScript supports object destructuring, which could be quite useful. However, since this feature exists only in one language, we don't put it here.

Another useful syntax of JavaScript is the object initializer, which allows shorthands like

let a = 1, b = 2, c = 3;
let d = {a, b, c};

##### Template String

How do I print a variable in a string?

from string import Template
t = Template('Hey, $name!') let ts = t.substitute(name='Bob') print(ts) # 'Hey, Bob!' a, b = 1, 2 print(f"sum {a+b}") # sum 3  let a = 1; let b = 2; console.log("sum${a+b}") # sum 3


#### Basic Libraries

##### Array

In C++, there is vector as well as the build-in array. vector is the first choice in most circumstances. Also note that in many functional programming languages, using loop on arrays may be deprecated. Seek map or reduce whenever possible!

Push Array push is among the most frequent operations.

std::vector<int> arr;
arr.push_back(10032);

arr = []
arr.append(10032)

let arr = []
arr.push(10032)

arr = [1, 2, 3];
arr = [arr, 10032];
brr = [1, 2, 3]';
brr = [brr; 10032];


##### Regex

Pattern Search What do you do if you are given a string like "length = 16" ?

import re

config = "length = 16"
regex  = re.compile(r'(?P<key>[a-z]+)([\s])*=([\s])*(?P<val>\d+)')
groups = re.match(regex, config).groupdict()

print(groups["key"]) # "length"
print(groups["val"]) # "16"

let config = "length = 16";
let regex  = /(?<key>[a-z]+)([\s])*=([\s])*(?<val>[\d]+)/;
let groups = config.match(regex).groups;

console.log(groups.key); // "length"
console.log(groups.val); // "16"

##### JSON

How do you dump an object into JSON, and vice versa?

import json

# data = {'foo', 'bar'}
print(json.dumps(data))
# {"foo": "bar"}

let data = JSON.parse('{"foo": "bar"}');
// data = {foo: "bar"}
console.log(JSON.stringify(data));
// {"foo":"bar"}

##### Request

How do you send a request and read the response?

from urllib import request
from bs4 import BeautifulSoup
import sys

req = request.Request(
"http://zechen.site/home",
data=None,
'User-Agent': 'Mozilla/5.0 (Macintosh; Intel Mac OS X 10_10_1) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/39.0.2171.95 Safari/537.36'
}
)

try:
page = request.urlopen(req)
soup =  BeautifulSoup(page, "html.parser")
print(soup.select("#signInNotice"))
except Exception as err:
print(err, file=sys.stderr)

const urllib = require('urllib');
const jQuery = require( "jquery" );
const { JSDOM } = require( "jsdom" );

urllib.request('http://zechen.site/home', {
method: 'GET',
'User-Agent': 'Mozilla/5.0 (Macintosh; Intel Mac OS X 10_10_1) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/39.0.2171.95 Safari/537.36'
}
}).then(function (result) {
const { window } = new JSDOM(result.data);
const $= jQuery(window); console.log($("#signInNotice").html());
}).catch(function (err) {
console.error(err);
});


##### Number From String

For conversion in JavaScript, see parseInt vs unary plus, when to use which?

std::stoi('3.14')

+'3.14'


Expand

## Density Functional Theory (I)

Background on Solid State Physics

#### Thermodynamics Background

##### Volume or Pressure as the Most Fundamental Variable

Let $$\Omega$$ denote the volume of a unit cell. We have \begin{align*} E &= E(\Omega), \\ P &= -\dv{E}{\Omega}, \\ B &= -\Omega \dv{P}{\Omega} = \Omega \dv[2]{E}{\Omega}, \end{align*} where $$B$$ denotes the bulk modulus.

To predict the equilibrium volume $$\Omega^0$$, i.e. where $$E$$ attains its minimum, we may either

• calculate $$E$$ for several values of $$\Omega$$, and fit to an analytic form, e.g. the Murnaghan equation; or
• calculate $$P$$ from a generalization of the virial theorem, and $$B$$ from response functions.

Expand

## A Certain Tokugawa's Schrödinger

### 某徳川家的薛定谔

A quick review on quantum mechanics, including theories of Hilbert space and angular momentum, relativistic quantum mechanics, and some widgets of algebra.

#### Vol.I   Basic Theory

##### Density Matrix
 Definition $$\displaystyle \rho = \sum_j p_j \ket{\psi_j} \bra{\psi_j}{}$$ Properties $$\trace \rho = 1,\quad \rho \ge 0,\quad \rho^\dagger = \rho$$ Expectation Value $$\langle A \rangle = \trace (\rho A)$$ Evolution $$\displaystyle i\hbar\pdv{\rho}{t} = [{\color{red}{H, \rho}}],\quad \rho(t) = e^{-iHt/\hbar}\rho(0)e^{iHt/\hbar}{}$$
Proof of the time-evolution

$\dv{t} \ket{\psi}\bra{\psi}{} = \qty(\dv{t} \ket{\psi})\bra{\psi}{} + \ket{\psi}\qty(\dv{t} \bra{\psi}){} = \qty[H,\ket{\psi}\bra{\psi}{}].$

##### Second Quantization
 Normalization $$\begin{cases}a^\dagger_i\ket{\cdots,n_i - 1,\cdots} = \sqrt{n_i}\ket{\cdots,n_i,\cdots}{} \\ a_i\ket{\cdots,n_i,\cdots} = \sqrt{n_i}\ket{\cdots,n_i - 1,\cdots}{} \\ N_i = a_i^\dagger a_i \end{cases}{}$$ Commutators Boson Fermion $$[a_i,a_i^\dagger] = \delta_{ij}{}$$ $$\{a_i,a_j^\dagger\} = \delta_{ij}{}$$

#### Vol.II   $$\mathrm{SO}(3)$$ and Angular Momentum

##### Matrix Representations

Spin-1/2

$S_x = \frac{\hbar}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}{}, \quad S_y = \frac{\hbar}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}{},\quad S_z = \frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}{}$

$\ket{\vb*{S}\cdot \hat{\vb*{n}}; +} = \cos(\frac{\theta}{2})e^{-i\varphi/2}\ket{+} + \sin(\frac{\theta}{2})e^{-i\varphi/2}\ket{-}{}$

##### Spherical Tensors

Wigner-Eckart Theorem

$\bra{\alpha',j'm'}T^{(k)}_q\ket{\alpha,jm} = \bra{jk;mq}\ket{jk;j'm'}\frac{\langle \alpha'j'\vert\vert T^{(k)} \vert\vert \alpha j\rangle}{\sqrt{2j+1}}{}$

Expand

## A Certain Analog/Digital Index

### 某模/数的禁书目录

A quick review on analog and digital circuit.

#### Vol.I(a)   Combinational Logic

##### TTL
 Not Gate Nor Gate Nand Gate AOI Gate

Expand

## Japanese Verb Pairs

This post is not yet done.

かす ける

にが げる

らす れる

やす える

かす

かわかす かわ

よろこばす よろこ

らす

らす

らす