Ze Chen
Mathematical
Quantum Field Theory (I)
Mathematical Prerequisites

Quantum Field Theory (II)

Quantization of Fields

波動、真空に存在する特殊無摂動量子場。 対処法1、\(a_{\vb*{p}}\)を以てこれを消滅する。 対処法2、経路積分して、伝播函数を求める。


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 \[ m^2 \rightarrow m^2 - i\epsilon. \]

Faddeev-Popv Trick
  • 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)

See On the Quantization of the 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, \[ \ket{B} = \sqrt{2M} \int \frac{\dd{^3 k}}{(2\pi)^3} \tilde{\psi}(\vb*{k}) \frac{1}{\sqrt{2m}} \frac{1}{\sqrt{2m}} \ket{\vb*{k}\uparrow, -\vb*{k}\uparrow}. \]
    • \[ \tilde{\psi}(\vb*{k}) = \int \dd{^3 x} e^{i\vb*{k}\cdot \vb*{r}} \psi(\vb*{r}). \]
Feynman
Quantum Field Theory (III)
Feynman Diagrams

2021/3/28 10:57:50

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