Ze Chen
 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

2021/3/28 10:57:50

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