Quantum Field Theory (I)

Mathematical Prerequisites

物理学のは嫌なので表現論に極振りしたいと思います

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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 \[ C^T = -C,\quad P^T = -P. \]

Lie Groups

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

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