量子场论

Posted:   June 07, 2018

Status:   Archived

 Tags :   QFT study-notes

 Categories :   Physics-study-notes

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约定

\newcommand{\d}{\mathrm{d}} \newcommand{\slash}[1]{\not{#1}} \newcommand{\bra }[1]{ \left\langle {#1} \right\vert } \newcommand{\ket }[1]{ \left\vert {#1} \right\rangle } \newcommand{\braket}[2]{\left\langle {#1} \vert {#2} \right\rangle} \newcommand{\Res}[1]{\operatorname{Res} \left( {#1} \right)} \newcommand{\imag}[1]{\operatorname{Im} \left( {#1} \right)} \newcommand{\D}[1]{ \frac{\d}{\d {#1}}} \newcommand{\Partial}[1]{ \frac{\partial}{\partial {#1}}} \newcommand{\Eqn}[1]{\text{Eqn. }\ref{#1} } \newcommand{\idmat}{\mathbb{I}} \newcommand{\Torder}[1]{\mathrm{T} \left\lbrace{#1}\right\rbrace } \newcommand{\expp}[1]{\exp{ \left({#1}\right)}} \newcommand{\Norder}[1]{:\!\!{#1}\!\!:} \newcommand{\Pbracket}[2]{\left[{#1},{#2}\right]} \newcommand{\contract}[1]{\overparen{ {#1} }} \\ \begin{align} p=p^\mu&=(E/c,\vec{p})\\ x^\mu&=(ct,\vec{x})\\ g _ {\mu\nu}&=(+,-,-,-)\\ p=p^\mu&=(E/c,-\vec{p})\\ E _ {\pm p}&=\pm\sqrt{\vec{p}^2+m^2}, \quad E _ {-p}=-E _ p \end{align}

K-G方程

哈密顿量

$p _ \mu p^\mu=E^2-{\vec{p}}^2=m^2$

(由 $E^2={\vec{p}}^2c^2+m^2c^4$ 自然单位制$c=0$)

运动方程

$p^\mu=-i\hbar \frac{\partial}{\partial {x}^\mu}$

$-\frac{\partial}{\partial {x}^\mu} \frac{\partial}{\partial {x} _ \mu}=m^2$

$H \phi = E \phi$

$\left(\Box ^2 + m^2\right)\phi(x)=0$

方程的解

$\left(\Box ^2 + m^2\right)\phi(x)=0$

$\phi(x)=\int \d^4p \operatorname{A}(\vec{p})e^{ipx}\delta(p _ \mu p^\mu-m^2)+\int \d^4p \operatorname{B}(\vec{p})e^{-ipx}\delta(p _ \mu p^\mu-m^2)$

$\delta(f(x))=\sum _ i{\frac{x-a _ i}{\left\vert {f}^\prime(a _ i)\right\vert}}$

\begin{align*} \delta(p _ \mu p^\mu-m^2)&=\delta\left(p _ 0p^0-{\vec{p}}^2-m^2\right)\\ &=\delta\left(p _ 0p^0-({\vec{p}}^2+m^2)\right)\\ &=\delta((p^0)^2-E _ p^2)\\ &=\frac{1}{2\left\vert p^0 \right\vert} \left( \delta(p^0-E _ p)+\delta(p^0+E _ p) \right) \end{align*}

\begin{align*} \int \d^4p \operatorname{A}(\vec{p})e^{ipx}\delta(p _ \mu p^\mu-m^2) &=\int \d^4p \operatorname{A}(\vec{p})e^{ipx}\frac{1}{2\left\vert p^0 \right\vert} \left( \delta(p^0-E _ p)+\delta(p^0+E _ p) \right)\\ &=\int dp^0\int \d^3 {\vec{p} \operatorname{A}(p^0,\vec{p})e^{i(p^0t-\vec{p}\cdot\vec{x})}\frac{1}{2\left\vert p^0 \right\vert} \left( \delta(p^0-E _ p)+\delta(p^0+E _ p) \right)}\\ \end{align*}

\begin{align*} &=\int dp^0\int \d^3 {\vec{p} \operatorname{A}(p^0,\vec{p})e^{i(p^0t-\vec{p}\cdot\vec{x})}\frac{1}{2\left\vert p^0 \right\vert} \delta(p^0-E _ p)}\\ &=\int \d^3 \vec{p}\left({ \operatorname{A}(p^0,\vec{p})e^{i(p^0t-\vec{p}\cdot\vec{x})}\frac{1}{2\left\vert p^0 \right\vert} \int dp^0\delta(p^0-E _ p)}\right)\\ &=\int \d^3 \vec{p}\left( \int dp^0\delta(p^0-E _ p){ \operatorname{A}(p^0,\vec{p})e^{i(p^0t-\vec{p}\cdot\vec{x})}\frac{1}{2\left\vert p^0 \right\vert }} \right)\\ &=\int \d^3 \vec{p} \operatorname{A}(E _ p,\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})}\frac{1}{2\left\vert E _ p \right\vert }\\ &=\int \d^3 \vec{p} \frac{\operatorname{A}(E _ p,\vec{p})}{2 E _ p }e^{i(E _ pt-\vec{p}\cdot\vec{x})}\\ \end{align*}

$\phi(x)=\int \d^3 \vec{p} \left(\frac{\operatorname{A}(E _ p,\vec{p})}{2 E _ p }e^{i(E _ pt-\vec{p}\cdot\vec{x})} + \frac{\operatorname{B}(E _ p,\vec{p})}{2 E _ p }e^{-i(E _ pt-\vec{p}\cdot\vec{x})}\right)$

$\phi(x)=\int \d^3 \vec{p} \left( \frac{a(\vec{p})}{\sqrt{2E _ p}}e^{i(E _ pt-\vec{p}\cdot\vec{x})} + \frac{b(\vec{p})}{\sqrt{2E _ p}} e^{-i(E _ pt-\vec{p}\cdot\vec{x})}\right)$

实数解

$\phi^\dagger(x)=\int \d^3\vec{p}\frac{1}{\sqrt{2E _ p}}\left( b^\dagger(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + a^\dagger(\vec{p}) e^{-i(E _ pt-\vec{p}\cdot\vec{x})}\right)$

\begin{align} a(\vec{p})&=b^\dagger(\vec{p})\\ b(\vec{p})&=a^\dagger(\vec{p}) \end{align}

$\phi(x)=\int \frac{\d^3\vec{p}}{\sqrt{2E _ p}}\left( a(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + a^\dagger(\vec{p}) e^{-i(E _ pt-\vec{p}\cdot\vec{x})}\right)$

场的量子化

graph LR
A[薛定谔方程]
B[相对论哈密顿量]-->C[K-G方程]
A-->C
C-->|傅里叶变换|D[平面波形式解]


傅里叶变换

\begin{align*} \phi(x)&=\int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}}\left( a(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + a^\dagger(\vec{p}) e^{-i(E _ pt-\vec{p}\cdot\vec{x})}\right)\\ &=\int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}} a(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + \int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}} a^\dagger(\vec{p}) e^{-i(E _ pt-\vec{p}\cdot\vec{x})} \\ &=\int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}} a(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + \int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}} a^\dagger( {\color{Red}-\vec{p}}) e^{-i(E _ {\color{Red}-p}t {\color{Red}+\vec{p}}\cdot\vec{x})} \\ &=\int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}} \left(a(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + a^\dagger( -\vec{p}) e^{-i(-E _ pt +\vec{p}\cdot\vec{x})} \right)\\ &=\int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}} \left(a(\vec{p})+a^\dagger( -\vec{p}) \right)e^{ipx}\\ \end{align*}

$\begin{cases} \phi(x)=\int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}} \tilde{\phi}(\vec{p}) e^{ipx}\\\ \tilde{\phi}(\vec{p}) =\int \frac{\d^3\vec{x}}{(2\pi)^{\frac32}\sqrt{2E _ p}} \phi(x) e^{-ipx}\\ \end{cases}$

==这里是三重积分, 四重积分可以转化为三重, 如何证明 文献忘记是哪一个了==

$\pi(x)=\dot{\phi}(x)$

\begin{align*} \pi(x)&=\frac{\partial}{\partial t} \int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}}\left( a(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + a^\dagger(\vec{p}) e^{-i(E _ pt-\vec{p}\cdot\vec{x})}\right) \\ &=\int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}\sqrt{2E _ p}}\frac{\partial}{\partial t} \left( a(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + a^\dagger(\vec{p}) e^{-i(E _ pt-\vec{p}\cdot\vec{x})}\right) \\ &=\int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}}i\sqrt{\frac{E _ p}{2}} \left( -a(\vec{p})e^{i(E _ pt-\vec{p}\cdot\vec{x})} + a^\dagger(\vec{p}) e^{-i(E _ pt-\vec{p}\cdot\vec{x})}\right)\\ \end{align*}

==待补充==

\begin{align} a(\vec{p})&=\frac12\left(\sqrt{2{E} _ p}\tilde{\phi}(\vec{p})+i\sqrt{\frac{2}{E _ p}}\tilde{\pi}(\vec{p})\right)\\ a^\dagger(\vec{p})&=\frac12\left(\sqrt{2{E} _ p}\tilde{\phi}(-\vec{p})-i\sqrt{\frac{2}{E _ p}}\tilde{\pi}(-\vec{p})\right) \end{align}

\begin{align*} \tilde\phi(-\vec{p})&=\tilde\phi^\dagger(\vec{p})\\ \tilde\pi(-\vec{p})&=\tilde\pi^\dagger(\vec{p}) \end{align*}

二次量子化

\begin{align} [a _ p,a^\dagger _ q]&=a _ pa^\dagger _ q-a^\dagger _ qa _ p\\ &=...\\ &=i\delta(\vec{p}-\vec{q}) \end{align}

$H=\int \d^3 x\mathcal{H}$

\begin{align} \mathcal{L}&=\frac12(\partial^\mu\phi)((\partial _ \mu\phi)-\frac12m^2\phi^2\\ \pi(x)&=\frac{\eth \mathcal{L}}{\eth (\partial^0\phi)}=\frac{\eth \mathcal{L}}{\eth (\dot\phi)}=...=\dot\phi(x)\\ \mathcal{H}&=\pi(x)\dot{\phi}(x)-\mathcal{L}\\ &=\pi^2(x)-\frac12({(\dot{\phi}(x))}^2-{(\nabla\phi(x))}^2-m^2{\phi(x)}^2)\\ &=\frac12(\pi^2+{(\nabla\phi)}^2+m^2\phi^2) \end{align}

\begin{align*} \int \d^3 \vec{x} {(\pi(x))}^2 &=\int \d^3\vec{x} \left( \int \frac{\d^3\vec{p}}{(2\pi)^{\frac32}}i\sqrt{\frac{E _ p}{2}} \left( -a(\vec{p})e^{ipx} + a^\dagger(\vec{p}) e^{-ipx}\right)\right) \\& \qquad \cdot \left(\int \frac{\d^3\vec{q}}{(2\pi)^{\frac32}} i\sqrt{\frac{E _ q}{2}} \left( -a(\vec{q})e^{iqx} + a^\dagger(\vec{q}) e^{-iqx}\right) \right)\\ &=\int \d^3 \vec{x} \int \d^3\vec{p} \int \d^3\vec{q} \frac{-1}{2(2\pi)^{3}} \sqrt{E _ pE _ q}\left( -a(\vec{p})e^{ipx} + a^\dagger(\vec{p}) e^{-ipx}\right) \\& \qquad \cdot \left( -a(\vec{q})e^{iqx} + a^\dagger(\vec{q}) e^{-iqx}\right) \\ &=\int \d^3 \vec{x} \int \d^3\vec{p} \int \d^3\vec{q} \frac{-1}{2(2\pi)^{3}} \sqrt{E _ pE _ q}\\ & \qquad \left(a(\vec p)a(\vec q)e^{i(p+q)x}+a^\dagger(\vec p)a^\dagger(\vec q)e^{-i(p+q)x}-a^\dagger(\vec p)a(\vec q)e^{i(q-p)x}-a(\vec p)a^\dagger(\vec q)e^{i(p-q)x}\right)\\ &= \int \d^3\vec{p} \int \d^3\vec{q} \frac{-1}{2(2\pi)^{3}} \sqrt{E _ pE _ q}\int \d^3 \vec{x} \\ & \qquad \left(a(\vec p)a(\vec q)e^{i(p+q)x}+a^\dagger(\vec p)a^\dagger(\vec q)e^{-i(p+q)x}-a^\dagger(\vec p)a(\vec q)e^{i(q-p)x}-a(\vec p)a^\dagger(\vec q)e^{i(p-q)x}\right)\\ &= \int \d^3\vec{p} \int \d^3\vec{q} \frac{- \sqrt{E _ pE _ q}}{2(2\pi)^{3}} \left(a(\vec p)a(\vec q)e^{i(E _ p+E _ q)t}+a^\dagger(\vec p)a^\dagger(\vec q) e^{-i(E _ p+E _ q)t}\right)\delta(\vec p+\vec q)\\ &\qquad - \int \d^3\vec{p} \int \d^3\vec{q} \frac{- \sqrt{E _ pE _ q}}{2(2\pi)^{3}}\left(a^\dagger(\vec p)a(\vec q)e^{i(E _ p-E _ q)t}+a(\vec p)a^\dagger(\vec q)e^{-i(E _ p-E _ q)t}\right)\delta(\vec p-\vec q)\\ &= \int \d^3\vec{p} \frac{-E _ p}{2(2\pi)^{3}} \left(a(\vec p)a(-\vec p)e^{2iE _ pt}+a^\dagger(\vec p)a^\dagger(-\vec p)e^{-2iE _ pt}-a^\dagger(\vec p)a(\vec p)-a(\vec p)a^\dagger(\vec p) \right)\\ \end{align*}

\begin{align} \notag H&=\int \d^3 \vec{x} \mathcal{H} \\ \notag &=\frac12\int \d^3 \vec{p} E _ p \left(a^\dagger(\vec p)a(\vec p)+a(\vec p)a^\dagger(\vec p)\right)\\ \end{align}

零点能与Normal Ordering

\begin{align} H &=\frac12 \int \d^3 \vec{p} E _ p \left(a^\dagger(\vec p)a(\vec p)+a(\vec p)a^\dagger(\vec p)\right)\\ &=\int \d^3 \vec{p} E _ p \left(a^\dagger(\vec p)a(\vec p)+\frac12\left[a(\vec p),a^\dagger(\vec p)\right]\right)\\ &=\int \d^3 \vec{p} E _ p \left(a^\dagger(\vec p)a(\vec p)+\frac12\delta(\vec0) \right)\\ \end{align}

\begin{align} \langle n |H | n\rangle &=\langle n |\int \d^3 \vec{p} E _ p \left(a^\dagger(\vec p)a(\vec p)+\frac12\left[a(\vec p),a^\dagger(\vec p)\right]\right) | n\rangle\\ &=\int \d^3 \vec{p} E _ p \langle n |\left(a^\dagger(\vec p)a(\vec p)+\frac12\left[a(\vec p),a^\dagger(\vec p)\right]\right)| n\rangle \\ &=\int \d^3 \vec{p} E _ p \langle n |a^\dagger(\vec p)a(\vec p)| n\rangle +\int \d^3 \vec{p} E _ p \langle n |\frac12\left[a(\vec p),a^\dagger(\vec p)\right] | n\rangle \\ &=nE _ p+\langle 0 |H | 0\rangle \end{align}

$:H: \equiv \int \d^3 \vec{p} E _ p a^\dagger(\vec p)a(\vec p)$

传播子

传播子与格林函数

 计算在$y$处粒子传播到$x$处的概率幅$\left\langle y x\right\rangle$有:
\begin{align*} \bra{0}\phi(x)\phi(y)\ket{0} &=\bra{0}\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ p}}\int \frac{\d^3 \vec{q}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ q}}(a(\vec{p})e^{-ipx}+a^\dagger(\vec{p}) e^{ipx})(a(\vec{q})e^{-iqy}+a^\dagger(\vec{q}) e^{iqy}) \ket{0}\notag\\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ p}}\int \frac{\d^3 \vec{q}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ q}} \bra{0}(a(\vec{p})e^{-ipx}+a^\dagger(\vec{p}) e^{ipx})(a(\vec{q})e^{-iqy}+a^\dagger(\vec{q}) e^{iqy})\ket{0} \\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ p}}\int \frac{\d^3 \vec{q}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ q}} \\ & \qquad\bra{0}\left(a(\vec{p})a(\vec{q})e^{-iqy-ipx}+a^\dagger(\vec{p})a(\vec{q})e^{-iqy+ipx} +a(\vec{p})a^\dagger(\vec{q})e^{iqy-ipx}+a^\dagger(\vec{p})a^\dagger(\vec{q}) e^{iqy+ipx}\right)\ket{0}\notag\\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ p}}\int \frac{\d^3 \vec{q}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ q}} \bra{0}a(\vec{p})a^\dagger(\vec{q})e^{iqy-ipx}\ket{0}\notag\\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ p}}\int \frac{\d^3 \vec{q}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ q}} \left\langle{\vec p}\vert{\vec q}\right\rangle e^{iqy-ipx}\notag\\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ p}}\int \frac{\d^3 \vec{q}}{(2\pi)^\frac32} \frac{1}{\sqrt{2E _ q}} \delta(\vec p -\vec q) e^{iqy-ipx}\notag\\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{2E _ p} e^{-ip(x-y)}\notag\\ \end{align*}

\begin{align} \bra{0} \Pbracket{\phi(x)}{\phi(y)} \ket{0}&=\bra{0} \phi(x)\phi(y)\ket{0}-\bra{0} \phi(y)\phi(x) \ket{0} \end{align}

Feynman 传播子

Feynman传播子的定义为

$D _ F(x-y)\equiv \bra{0}T\phi(x)\phi(y)\ket{0}\\ T\phi(x)\phi(y)\equiv \begin{cases}D(x-y),\quad t _ x\gt t _ y\\D(y-x) ,\quad t _ x\lt t _ y \end{cases}\\$

\begin{align*} D _ F(x-y)&=\Theta(t _ x-t _ y)\bra{0} \phi(x)\phi(y)\ket{0}-\Theta(t _ y-t _ x)\bra{0} \phi(y)\phi(x) \ket{0}\\ &=\Theta(t _ x-t _ y)\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{2E _ p} e^{-ip(x-y)}+\Theta(t _ y-t _ x)\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{2E _ p} e^{ip(x-y)}\\ &=\Theta(t _ x-t _ y)\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{2E _ p} e^{-iE _ p(t _ x-t _ y)+i\vec{p}\cdot(\vec{x}-\vec{y})}+\Theta(t _ y-t _ x)\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{2E _ p} e^{iE _ p(t _ x-t _ y)-i\vec{p}\cdot(\vec{x}-\vec{y})}\\ \phantom{}\xrightarrow{q=(E _ p,-\vec{p})} &=\Theta(t _ x-t _ y)\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{2E _ p} e^{-iE _ p(t _ x-t _ y)+i\vec{p}\cdot(\vec{x}-\vec{y})}+\Theta(t _ y-t _ x)\int \frac{\d^3 \vec{q}}{(2\pi)^\frac32} \frac{1}{2E _ q} e^{iE _ q(t _ x-t _ y)+i\vec{q}\cdot(\vec{x}-\vec{y})}\\ \phantom{}\xrightarrow{p=q}&=\Theta(t _ x-t _ y)\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{2E _ p} e^{-iE _ p(t _ x-t _ y)+i\vec{p}\cdot(\vec{x}-\vec{y})}+\Theta(t _ y-t _ x)\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} \frac{1}{2E _ p} e^{iE _ p(t _ x-t _ y)+i\vec{p}\cdot(\vec{x}-\vec{y})}\\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} e^{i\vec{p}\cdot(\vec{x}-\vec{y})} \left(\Theta(t _ x-t _ y)\frac{1}{2E _ p} e^{-iE _ p(t _ x-t _ y)}+\Theta(t _ y-t _ x)\frac{1}{2E _ p} e^{iE _ p(t _ x-t _ y)}\right)\\ \end{align*}

\begin{align*} f(p _ 0)&= \frac{1}{p^2-m^2}e^{-ip(x-y)} \\ &=\frac{1}{(p _ 0)^2-({\vec{p}}^2+m^2)}e^{-ip _ 0(t _ x-t _ y)}e^{i\vec{p}\cdot(\vec{x}-\vec{y})} \\ &=\frac{e^{-ip _ 0(t _ x-t _ y)}}{(p _ 0)^2-{E _ p}^2}e^{i\vec{p}\cdot(\vec{x}-\vec{y})} \\ &=e^{i\vec{p}\cdot(\vec{x}-\vec{y})}\left(\frac{-\frac{1}{2E _ p}e^{-ip _ 0(t _ x-t _ y)}}{p _ 0-E _ p}+\frac{\frac{1}{2E _ p}e^{-ip _ 0(t _ x-t _ y)}}{p _ 0+E _ p}\right) \end{align*}

\begin{align} \Res{f(E _ p)}&=\frac{1}{2E _ p}e^{-iE _ p(t _ x-t _ y)}\\ \Res{f(-E _ p)}&=-\frac{1}{2E _ p}e^{iE _ p(t _ x-t _ y)} \end{align}

\begin{align*} &\quad \oint _ {C _ \pm} f(p _ 0) \d p _ 0 \\ &=e^{i\vec{p}\cdot(\vec{x}-\vec{y})}\oint _ {C _ \pm} \d p _ 0 \left(\frac{-\frac{1}{2E _ p}e^{-ip _ 0(t _ x-t _ y)}}{p _ 0-E _ p}+\frac{\frac{1}{2E _ p}e^{-ip _ 0(t _ x-t _ y)}}{p _ 0+E _ p}\right) \\ &= e^{-i\vec{p}\cdot(\vec{x}-\vec{y})} 2\pi i \begin{cases} -\Res{f(E _ p)}&\quad t _ x-t _ y \gt 0, & C _ {+,{\color{Red}\circlearrowright }}: \imag{p _ 0} \lt 0\\ \Res{f(-E _ p)} &\quad t _ x-t _ y \lt 0, & C _ {-,{\color{Red} \circlearrowleft} } : \imag{p _ 0} \gt 0\end{cases} \\ &=2i\pi e^{-i\vec{p}\cdot(\vec{x}-\vec{y})}\begin{cases}- \frac{1}{2E _ p}e^{iE _ p(t _ x-t _ y)}&\quad t _ x-t _ y \gt 0 \\ - \frac{1}{2E _ p}e^{-iE _ p(t _ x-t _ y)}&\quad t _ x-t _ y \lt 0\end{cases} \\ &= 2i\pi e^{-i\vec{p}\cdot(\vec{x}-\vec{y})}(-1)\begin{cases}\frac{1}{2E _ p} e^{-iE _ p(t _ x-t _ y)}, & C _ +\\ \frac{1}{2E _ p} e^{iE _ p(t _ x-t _ y)}, & C _ -\end{cases} \end{align*}

\begin{align} &\phantom{=} \Theta(t _ x-t _ y)\frac{1}{2E _ p} e^{-iE _ p(t _ x-t _ y)}+\Theta(t _ y-t _ x)\frac{1}{2E _ p} e^{iE _ p(t _ x-t _ y)}\notag\\ &=\Theta(t _ x-t _ y)\frac{1}{-2\pi i} \oint _ {C _ +} f(p _ 0) \d p _ 0 +\Theta(t _ y-t _ x) \frac{1}{-2\pi i}\oint _ {C _ -} f(p _ 0) \d p _ 0 \notag\\ &=\frac{1}{-2\pi i}\left(\Theta(t _ x-t _ y)\oint _ {C _ 0+C _ {l}} f(p _ 0) \d p _ 0+\Theta(t _ y-t _ x)\oint _ {C _ 0+C _ {u}} f(p _ 0) \d p _ 0\right) \notag\\ &=\frac{1}{-2\pi i}\left(\Theta(t _ x-t _ y)+\Theta(t _ y-t _ x)\right)\oint _ {C _ 0} f(p _ 0) \d p _ 0 +\frac{1}{-2\pi i}\Theta(t _ x-t _ y)\oint _ {C _ {l}} f(p _ 0) \d p _ 0\notag \\ &\qquad +\frac{1}{-2\pi i}\Theta(t _ y-t _ x)\oint _ {C _ {u}} f(p _ 0) \d p _ 0 \notag\\ \xrightarrow[\oint _ {C _ {l,u}} f(p _ 0) \d p _ 0\rightarrow 0]{R\rightarrow\infty, }&=\frac{1}{-2\pi i}\oint _ {C _ 0} f(p _ 0) \d p _ 0 \label{residualFeynman} \end{align}

\begin{align} D _ F(x-y) &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32} e^{i\vec{p}\cdot(\vec{x}-\vec{y})} \left(\Theta(t _ x-t _ y)\frac{1}{2E _ p} e^{-iE _ p(t _ x-t _ y)}+\Theta(t _ y-t _ x)\frac{1}{2E _ p} e^{iE _ p(t _ x-t _ y)}\right)\notag\\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32}\frac{1}{-2\pi i} \oint _ {C _ 0} \d p _ 0\frac{1}{p^2-m^2}e^{-ip(x-y)}\notag\\ &=\int \frac{\d^3 \vec{p}}{(2\pi)^\frac32}\frac{1}{2\pi } \oint _ {C _ 0} \d p _ 0\frac{i}{p^2-m^2}e^{-ip(x-y)}\notag\\ &=\oint _ {C _ 0}\frac{\d^4 p}{(2\pi)^\frac32}\frac{1}{2\pi } \frac{i}{p^2-m^2}e^{-ip(x-y)}\\ \end{align}

\begin{align} D _ F(x-y)&\equiv\lim _ {\varepsilon\rightarrow0}{\int\frac{\d^4 p}{2\pi)^4} \frac{i}{p^2-m^2+i\varepsilon}e^{-ip(x-y)}}\\ \end{align}

格林函数

\begin{align*} &\phantom{=}\left(\partial _ \mu\partial^\mu+m^2\right)D _ F(x-y)\\ &=\left(\partial _ \mu\partial^\mu+m^2\right)\oint _ {C _ 0}\frac{\d^4 p}{(2\pi)^\frac32}\frac{1}{2\pi } \frac{i}{p^2-m^2}e^{-ip(x-y)}\\ &=\oint _ {C _ 0}\frac{\d^4 p}{(2\pi)^\frac32}\frac{1}{2\pi }\frac{i}{p^2-m^2} \left(\partial _ \mu\partial^\mu+m^2\right)e^{-ip(x-y)}\\ &=\oint _ {C _ 0}\frac{\d^4 p}{(2\pi)^\frac32}\frac{1}{2\pi }\frac{i}{p^2-m^2} \left(-p^2+m^2\right)e^{-ip(x-y)}\\ &=\oint _ {C _ 0}\frac{\d^4 p}{(2\pi)^\frac32}\frac{-i}{2\pi }e^{-ip(x-y)}\\ &=\frac{-i}{2\pi }\delta(x-y) \end{align*}

Dirac方程

==希望我有时间可以直接从:要描述spin=1/2粒子开始写方程 角度来写==

哈密顿量

$H^2=p^2+m^2\\ \rightarrow H=\gamma^0(\vec{\gamma}\cdot\vec{p}+m)$

$\gamma$矩阵的性质

$\gamma$矩阵的上下标的来源, 其实是因为他们是矢量. 把它们当作标量, 才导致旋量这一变换十分复杂的量. 详见[^gamma as vector]

运动方程

$i\hbar \frac{\partial}{\partial t}\psi(x)=\gamma^0(-i\vec{\gamma}\cdot \vec{\nabla}+m)\psi(x)$

$(i\gamma^\mu \partial _ \mu-m)\psi(x)=0$

方程的解

\begin{align} (p^\mu\gamma _ \mu-m)u(\vec p ) &= 0\\ (p^\mu\gamma _ \mu+m)v(\vec p ) &= 0 \end{align}

场的量子化

Dirac场的拉格朗日量

$\mathcal{L}=\bar{\psi}(i\not\partial -m)\psi$

Dirac场的量子化

$\psi(x)=\int \d^3\vec{p}$

相互作用

graph LR
A[转移矩阵]-->|相互作用绘景|B[渐进形式]
B-->|逐级展开|C[wick定理]
C-->D[Feynman图]


相互作用的形式1

$\mathcal{L}=\frac 12 (\partial \phi)^2-V _ 0(\phi)$

$V _ 0(\phi)=\frac12 {m _ 0}^2\phi^2$

$V(\phi) =\underbrace{\frac 12 {m _ 0}^2\phi^2} _ {V _ 0}+\underbrace{\frac{1}{3!}g\phi^3+\frac{1}{4!}\lambda\phi^4+\cdots } _ {V _ {interaction}}$

$\mathcal{L}=\mathcal{L} _ 0+\mathcal{L} _ {int}\\ \mathcal{L} _ {int}=-V _ {int}$

$(\partial^2+m^2)\phi =V _ {int}\neq 0$

相互作用哈密顿量

$H _ I=\underbrace{\underbrace{H _ 0} _ \text{Hisenburg Picture}+\underbrace{H _ {int}} _ \text{Schrodinger Picture}} _ \text{Interaction Picture}$

\begin{align} \ket{\psi} _ I&=e^{iH _ 0t}\ket{\psi} _ S && \text{S for Schrodinger}\\ \hat{O} _ I&=e^{iH _ 0t}\hat{O} _ H e^{-iH _ 0t} &&\text{H for Hisenburg}\\ \end{align}

$i \D{t}\ket{\psi(t)} _ S=H\ket{\psi(t)} _ S$

\begin{align} i \D{t}\left( e^{-iH _ 0t}\ket{\psi(t)} _ I\right)&=\left(H _ 0+H _ {int}\right)e^{-iH _ 0t}\ket{\psi(t)} _ I\notag\\ H _ 0 e^{-iH _ 0t}\ket{\psi(t)} _ I+e^{-iH _ 0t}\D{t} \ket{\psi(t)} _ I&=H _ 0 e^{-iH _ 0t}\ket{\psi(t)} _ I+H _ {int} e^{-iH _ 0t}\ket{\psi(t)} _ I\notag\\ e^{-iH _ 0t}\D{t} \ket{\psi(t)} _ I&=H _ {int} e^{-iH _ 0t}\ket{\psi(t)} _ I\notag\\ \D{t} \ket{\psi(t)} _ I&=e^{iH _ 0t}H _ {int} e^{-iH _ 0t}\ket{\psi(t)} _ I\notag\\ \D{t} \ket{\psi(t)} _ I&={(H _ {int})} _ I(t)\ket{\psi(t)} _ I \label{simplifedSchrodingerEqn} \\ \end{align}

$\ket{\psi(t)} _ I=U(t,t _ 0)\ket{\psi(t _ 0)} _ I \label{defU}$

$i\D{t}U(t,t _ 0)=H _ I(t)U(t,t _ 0) \label{diffEqnforU}$

$U(t,t _ 0)=U(t _ 0,t _ 0)-i\int _ {t _ 0}^{t}H _ I(\tau _ 1)U(\tau _ 1,t _ 0)\d \tau _ 1 \label{intEqnforU}$

\begin{align*} U^{(1)}(t,t _ 0)&=\idmat-i\int _ {t _ 0}^{t}H _ I(\tau _ 1)\underbrace{U^{(0)}(\tau _ 1,t _ 0)} _ {=\idmat}\d \tau _ 1\\ U^{(2)}(t,t _ 0)&=\idmat-i\int _ {t _ 0}^{t}H _ I(\tau _ 2)U^{(1)}(\tau _ 2,t _ 0)\d \tau _ 2\\ &= \idmat-i\int _ {t _ 0}^{t}H _ I(\tau _ 2) \left(\idmat -i\int _ {t _ 0}^{\tau _ 2}H _ I(\tau _ 1)\underbrace{U^{(0)}(\tau _ 1,t _ 0)} _ {=\idmat}\d \tau _ 1\right) \d\tau _ 2\\ &=\idmat + (-i)\int _ {t _ 0}^{t}H _ I(\tau _ 2)\d \tau _ 2 + (-i)^2 \int _ {t _ 0}^{t} \d \tau _ 2 \int _ {t _ 0}^{\tau _ 2} \d \tau _ 1 H _ I(\tau _ 2) H _ I(\tau _ 1)\\ &=\idmat + (-i)\int _ {t _ 0}^{t}H _ I(\tau _ 1)\d \tau _ 1 + (-i)^2 \int _ {t _ 0}^{t} \d \tau _ 1 \int _ {t _ 0}^{\tau _ 1} \d \tau _ 2 H _ I(\tau _ 1) H _ I(\tau _ 2) \qquad \rm{interchange\ \tau _ 1\ and\ \tau _ 2}\\ & \phantom{=}\vdots \\ U^{(\infty)}(t,t _ 0)&=\idmat+\sum _ {n=1}^{\infty}\left((-i)^n\int _ {t _ 0}^{t} \d \tau _ 1\int _ {t _ 0}^{\tau _ 1} \d \tau _ 2 \cdots \int _ {t _ 0}^{\tau _ n} \d \tau _ n H _ I(\tau _ 1) H _ I(\tau _ 2) \cdots H _ I(\tau _ n)\right)\\ \end{align*}

$H _ I(x)H _ I(y)=e^{iH _ 0x}H _ {int}e^{-iH _ 0x}e^{iH _ 0y}H _ {int}e^{-iH _ 0y}=e^{iH _ 0y}H _ {int}e^{-iH _ 0y}e^{iH _ 0x}H _ {int}e^{-iH _ 0x}=H _ I(y)H _ I(x)$

$U(t,t _ 0)\equiv U^{(\infty)}(t,t _ 0)=\idmat+\sum _ {n=1}^{\infty}\left((-i)^n\int _ {t _ 0}^{t} \d \tau _ 1\int _ {t _ 0}^{\tau _ 1} \d \tau _ 2 \cdots \int _ {t _ 0}^{\tau _ n} \d \tau _ n H _ I(\tau _ 1) H _ I(\tau _ 2) \cdots H _ I(\tau _ n)\right)\\ \label{UnonTorder}$

\begin{align*} \int _ {t _ 0}^{t} \d \tau _ 1\int^{\tau _ 1} _ {t _ 0} \d \tau _ 2 H _ I(\tau _ 1) H _ I(\tau _ 2) &=\int _ {t _ 0}^{t} \d \tau _ 2\int^{\tau _ 2} _ {t _ 0} -\d \tau _ 1 H _ I(\tau _ 2) H _ I(-\tau _ 1) & \tau _ 1\rightarrow \tau _ 2 , \ \tau _ 2\rightarrow -\tau _ 1 \\ &=\int _ {t _ 0}^{t} \d \tau _ 2\int _ {t _ 0}^{\tau _ 2} \d \tau _ 1 H _ I(\tau _ 2)H _ I(\tau _ 1) \\ \end{align*}

$\Torder{H _ I(\tau _ 2)H _ I(\tau _ 1)}=\begin{cases}H _ I(\tau _ 2)H _ I(\tau _ 1),\quad t _ x\gt t _ y\\H _ I(\tau _ 1)H _ I(\tau _ 2) ,\quad t _ x\lt t _ y \end{cases}$

\begin{align*} \underbrace{\int _ {t _ 0}^{t} \d \tau _ 1\int^{\tau _ 1} _ {t _ 0}} _ {t _ 0\lt\tau _ 2\lt \tau _ 1\lt t} \d \tau _ 2 H _ I(\tau _ 1) H _ I(\tau _ 2) &=\underbrace{\int _ {t _ 0}^{t} \d \tau _ 2\int _ {t _ 0}^{\tau _ 2}} _ {t _ 0\lt\tau _ 1\lt \tau _ 2\lt t} \d \tau _ 1 H _ I(\tau _ 2)H _ I(\tau _ 1) \\ \underbrace{\int _ {t _ 0}^{t} \d \tau _ 1\int^{\tau _ 1} _ {t _ 0} \d \tau _ 2 \underbrace{\Torder{H _ I(\tau _ 1) H _ I(\tau _ 2)}} _ {t _ 0\lt\tau _ 2\lt \tau _ 1\lt t} } _ \text {upper triangle} &=\underbrace{\int _ {t _ 0}^{t} \d \tau _ 2\int _ {t _ 0}^{\tau _ 2} \d \tau _ 1 \underbrace{\Torder{H _ I(\tau _ 1)H _ I(\tau _ 2)}} _ {t _ 0\lt\tau _ 1\lt \tau _ 2\lt t} } _ \text{lower triangle} \end{align*}

\begin{align*} \underbrace{\int _ {t _ 0}^{t} \d \tau _ 2\int _ {t _ 0}^{t} \d \tau _ 1 \Torder{H _ I(\tau _ 1)H _ I(\tau _ 2)}} _ \text{entire square} &=2\underbrace{\int _ {t _ 0}^{t} \d \tau _ 2\int _ {t _ 0}^{\tau _ 2} \d \tau _ 1 \underbrace{\Torder{H _ I(\tau _ 1)H _ I(\tau _ 2)}} _ {t _ 0\lt\tau _ 1\lt \tau _ 2\lt t} } _ \text{lower triangle, top-bottom} \\ &=\underbrace{2\int _ {t _ 0}^{t} \d \tau _ 1\int^{\tau _ 1} _ {t _ 0} \d \tau _ 2 H _ I(\tau _ 1) H _ I(\tau _ 2)} _ \text{lower triangle, left-right} \end{align*}

\begin{align} &\quad(-i)^n\int _ {t _ 0}^{t} \d \tau _ 1\int _ {t _ 0}^{\tau _ 1} \d \tau _ 2 \cdots \int _ {t _ 0}^{\tau _ n} \d \tau _ n H _ I(\tau _ 1) H _ I(\tau _ 2)\cdots H _ I(\tau _ n)\notag\\ & =\frac{(-i)^n}{n!}\int _ {t _ 0}^{t} \d \tau _ 1\int _ {t _ 0}^{t} \d \tau _ 2 \cdots \int _ {t _ 0}^{t} \d \tau _ n \Torder{H _ I(\tau _ 1) H _ I(\tau _ 2)\cdots H _ I(\tau _ n)}\notag\\ &= \frac{(-i)^n}{n!}\int _ {t _ 0}^{t} \d \tau _ 1\int _ {t _ 0}^{t} \d \tau _ 2 \cdots \int _ {t _ 0}^{t} \d \tau _ n \Torder{H _ I(\tau _ 1) H _ I(\tau _ 2)\cdots H _ I(\tau _ n)}\notag\\ &= \frac{(-i)^n}{n!}\Torder{\int _ {t _ 0}^{t} \d \tau _ 1\int _ {t _ 0}^{t} \d \tau _ 2 \cdots \int _ {t _ 0}^{t} \d \tau _ nH _ I(\tau _ 1) H _ I(\tau _ 2)\cdots H _ I(\tau _ n)}\notag\\ &= \frac{(-i)^n}{n!}\Torder{\left(\int _ {t _ 0}^{t} \d \tau _ 1H _ I(\tau _ 1) \right)\left(\int _ {t _ 0}^{t} \d \tau _ 2 H _ I(\tau _ 2)\right)\cdots \left( \int _ {t _ 0}^{t} \d \tau _ n H _ I(\tau _ n)\right)}\notag\\ &=\Torder{\frac{(-i)^n}{n!}\left(\int _ {t _ 0}^{t} \d \tau H _ I(\tau) \right)^n} \label{TorderProdH} \end{align}

\begin{align*} U(t,t _ 0)&=\idmat+\sum _ {n=1}^{\infty}\left((-i)^n\int _ {t _ 0}^{t} \d \tau _ 1\int _ {t _ 0}^{\tau _ 1} \d \tau _ 2 \cdots \int _ {t _ 0}^{\tau _ n} \d \tau _ n H _ I(\tau _ 1) H _ I(\tau _ 2) \cdots H _ I(\tau _ n)\right)\\ &=\idmat + \sum _ {n=1}^{\infty}\left(\Torder{\frac{(-i)^n}{n!}\left(\int _ {t _ 0}^{t} \d \tau H _ I(\tau) \right)^n}\right)\\ &=\Torder{\idmat + \sum _ {n=1}^{\infty}\left(\frac{(-i)^n}{n!}\left(\int _ {t _ 0}^{t} \d \tau H _ I(\tau) \right)^n\right)}\\ &=\Torder{\expp{-i\int _ {t _ 0}^{t} \d \tau H _ I(\tau) }}\\ \end{align*}

S矩阵

$\ket{\psi(t)} _ I=U(t,t _ 0)\ket{\psi(t _ 0)} _ I \notag$

$\ket{\psi(\infty)}=U(\infty,-\infty)\ket{\psi(-\infty)}$

$S=U(\infty,-\infty)=\Torder{\expp{-i\int _ {-\infty}^{\infty} \d \tau H _ I(\tau) }}\\$

$S=\Torder{\expp{-i\int _ {-\infty}^{\infty} \d \tau \int \d^3 \vec{x} \mathcal{H} _ I(\vec{x},\tau) }} =\Torder{\expp{-i\int _ {-\infty}^{\infty} \d^4 x \mathcal{H} _ I(x) }}$

Wick 定理

$S$矩阵含有编时乘积, 但哈密顿量密度是Normal Order的. 将编时乘积转换为Normal Ordering的过程由Wick 定理给出.

举例计算

$\bra{0}\Torder{\phi(x) \phi(y)}\ket{0}=\Theta(t _ x-t _ y) \bra{0}\phi(x)\phi(y)\ket{0} + \Theta(t _ y-t _ x) \bra{0}\phi(y)\phi(x)\ket{0}$

\begin{align} &\phi(x)=\phi _ +(x)+\phi _ -(x)\\ &\phi _ +(x)\ket{0}=0\\ &\bra{0}\phi _ -(x)=0 \end{align}

$\bra{0}\phi(x)\phi(y)\ket{0} = \bra{0}\phi _ +(x)\phi _ +(y) + \phi _ +(x) \phi _ -(y)+ \phi _ -(x)\phi _ +(y) + \phi _ -(x)\phi _ -(y)\ket{0}$

Normal Order之后得到

\begin{align*} \Norder{\phi(x)\phi(y)} &= \phi _ +(x)\phi _ +(y) + \phi _ -(x)\phi _ +(y)+ \phi _ -(x)\phi _ +(y) + \phi _ -(x)\phi _ -(y)\\ &= \phi _ +(x)\phi _ +(y) + {\color{Red}\phi _ +(x)\phi _ -(y)}+ \phi _ -(x)\phi _ +(y) + \phi _ -(x)\phi _ -(y) + {\color{Green}\phi _ -(x)\phi _ +(y)} - {\color{Red}\phi _ +(x)\phi _ -(y)}\\ &=\phi(x)\phi(y) + \phi _ -(x)\phi _ +(y)-\phi _ +(x)\phi _ -(y)\\ &=\phi(x)\phi(y) +\Pbracket{\phi _ -(x)}{\phi _ +(y)} \end{align*}

\begin{align*} \Torder{\phi(x) \phi(y)}&=\Theta(t _ x-t _ y) \phi(x)\phi(y)+\Theta(t _ y-t _ x)\phi(y)\phi(x)\\ &=\Theta(t _ x-t _ y)\left(\Norder{\phi(x)\phi(y)}+\Pbracket{\phi _ +(x)}{\phi _ -(y)}\right)+ \Theta(t _ y-t _ x)\left(\Norder{\phi(x)\phi(y)}+\Pbracket{\phi _ +(x)}{\phi _ -(y)}\right)\\ &=\Norder{\phi(x)\phi(y)} +\Theta(t _ x-t _ y) \Pbracket{\phi _ +(x)}{\phi _ -(y)} + \Theta(t _ y-t _ x) \Pbracket{\phi _ +(x)}{\phi _ -(y)} \end{align*}

\begin{align*} \bra{0}\Torder{\phi(x) \phi(y)}\ket{0}&=\bra{0}\Norder{\phi(x)\phi(y)}\ket{0}+\bra{0}\Big(\Theta(t _ x-t _ y) \Pbracket{\phi _ +(x)}{\phi _ -(y)} + \Theta(t _ y-t _ x) \Pbracket{\phi _ +(x)}{\phi _ -(y)}\Big)\ket{0}\\ &=\bra{0}\Norder{\phi(x)\phi(y)}\ket{0}+D _ F(x-y) \end{align*}

$\bra{0}\Norder{\phi(x)\phi(y)}\ket{0}\equiv0$

\begin{align*} \bra{0}\Torder{\phi(x) \phi(y)}\ket{0}&=D _ F(x-y) \end{align*}

$\bra{0}\Torder{\phi(x) \phi(y)}\ket{0} =\contract{\phi(x) \phi} (y)$

\begin{align*} \bra{0}\Torder{\phi(x _ 1) \phi(x _ 2) \phi(x _ 3) }\ket{0}&=\bra{0}\contract{\phi(x _ 1) \phi}(x _ 2)\phi(x _ 3)\ket{0}+\bra{0}\phi(x _ 1)\contract{\phi(x _ 2) \phi}(x _ 3)\ket{0}+\bra{0}\contract{\phi(x _ 1) \phi(x _ 2) \phi}(x _ 3\ket{0})\\ &=\bra{0}\Norder{\phi(x _ 1)}\ket{0}D _ F(x _ 2-x _ 3)+\bra{0}\Norder{\phi(x _ 3)}\ket{0}D _ F(x _ 1-x _ 2)+\bra{0}\Norder{\phi(x _ 2)}\ket{0}D _ F(x _ 1-x _ 3)\\ &=0 \end{align*}

一般的Wick 定理

\begin{align*} \bra{0}\Torder{\phi(x _ 1) \phi(x _ 2) \cdots \phi(x _ n) }\ket{0} &=\bra{0}\Norder{\phi(x _ 1) \phi(x _ 2) \cdots \phi(x _ n) }\ket{0}\\ &+\bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \phi(x _ 3) \cdots \phi(x _ n)+\text{all permutaion} \bigg]\\ &+\bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \phi(x _ n)+\text{all permutaion} \bigg]\\ &+\cdots\\ &+\begin{cases} \bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \contract{\phi(x _ {n-1})\phi}(x _ n)+\text{all permutaion} \bigg] &n \text{ even} \\ \bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \contract{\phi(x _ {n-2})\phi}(x _ {n-1})\phi(x _ n)+\text{all permutaion} \bigg] & n \text{ odd} \end{cases} \end{align*}

\begin{align*} &\bra{0}\Torder{\phi(x _ 1) \phi(x _ 2) \cdots \phi(x _ n) }\ket{0}\\ &=\bra{0}\Norder{\phi(x _ 1) \phi(x _ 2) \cdots \phi(x _ n) }\ket{0}\\ &+\bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \phi(x _ 3) \cdots \phi(x _ n)+\text{all permutaion} \bigg]\\ &+\bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \phi(x _ n)+\text{all permutaion} \bigg]\\ &+\cdots\\ &+\begin{cases} \bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \contract{\phi(x _ {n-1})\phi}(x _ n)+\text{all permutaion} \bigg] &n \text{ even} \\ \bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \contract{\phi(x _ {n-2})\phi}(x _ {n-1})\phi(x _ n)+\text{all permutaion} \bigg] & n \text{ odd} \end{cases} \end{align*} \begin{align*} &\bra{0}\Torder{\phi(x _ 1) \phi(x _ 2) \cdots \phi(x _ n) \phi(x _ {n+1}) }\ket{0}\\ &=\bra{0}\Norder{\phi(x _ 1) \phi(x _ 2) \cdots \phi(x _ n) \phi(x _ {n+1}) }\ket{0}\\ &+\bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \phi(x _ 3) \cdots \phi(x _ n)+\text{all permutaion} \bigg]\phi(x _ {n+1}) \\ &+\bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \phi(x _ n)+\text{all permutaion} \bigg]\phi(x _ {n+1}) \\ &+\cdots\\ &+\begin{cases} \bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \contract{\phi(x _ {n-1})\phi}(x _ n)+\text{all permutaion} \bigg]\phi(x _ {n+1}) &n \text{ even} \\ \bigg[\contract{\phi(x _ 1) \phi}(x _ 2) \contract{\phi(x _ 3) \phi}(x _ 4) \cdots \contract{\phi(x _ {n-2})\phi}(x _ {n-1})\phi(x _ n)+\text{all permutaion} \bigg]\phi(x _ {n+1}) & n \text{ odd} \end{cases} \end{align*}

自由真空与相互作用真空

\begin{align} H _ 0\ket{0}&=0\\ H _ 0\ket{\Omega}&=E _ \Omega \ket{\Omega} \end{align}

\begin{align} e^{-iHT}\ket{0}&=e^{-iHT}\sum _ n{\ket{n}\bra{n}}\ket{0}\notag\\ &=\sum _ ne^{-iE _ nT}\ket{n}\braket{n}{0}\notag \\ &=e^{-iE _ \Omega T}\ket{\Omega}\braket{\Omega}{0}+\sum _ {n\neq \ket{\Omega}}e^{-iE _ nT}\ket{n}\braket{n}{0} \label{oandOmega} \\ \end{align}

==为什么要这样取极限, 直接$\infty$不行吗, 这样是精确的还是近似?==

\begin{align*} \lim _ {T\rightarrow \infty(1-i\varepsilon)}e^{-iHT}\ket{0}&=\lim _ {T\rightarrow \infty(1-i\varepsilon)} e^{-iE _ \Omega T}\ket{\Omega}\braket{\Omega}{0}+\lim _ {T\rightarrow \infty(1-i\varepsilon)}\sum _ {n\neq \ket{\Omega}}e^{-iE _ nT}\ket{n}\braket{n}{0} \\ \lim _ {T\rightarrow \infty(1-i\varepsilon)}e^{-iHT}\ket{0}&=\lim _ {T\rightarrow \infty(1-i\varepsilon)} e^{-iE _ \Omega T}\ket{\Omega}\braket{\Omega}{0}+0\\ \left(\lim _ {T\rightarrow \infty(1-i\varepsilon)}e^{-iHT}\right)\ket{0}&=\left(\lim _ {T\rightarrow \infty(1-i\varepsilon)} e^{-iE _ \Omega T} \right)\ket{\Omega}\braket{\Omega}{0}\\ \end{align*}

$\ket{\Omega}=\lim _ { T\rightarrow \infty(1-i\varepsilon)} \frac{e^{-iHT}\ket{0}}{ e^{-iE _ \Omega T} \braket{\Omega}{0}}$

==…==

$\bra{\Omega}\Torder{\prod _ i{\phi(x _ i)}}\ket{\Omega}=\lim _ {T\rightarrow \infty(1-i\varepsilon)} \frac{\bra{0}\Torder{\prod _ i{\phi(x _ i)}e^{-i\int _ {-T}^{T}dtH _ I(t)}}\ket{0}}{\bra{0}\Torder{\ e^{-i\int _ {-T}^{T}dtH _ I(t)}}\ket{0}}$

相互作用的传播子

$\bra{\Omega}T\phi(x)\phi(y)\ket{\Omega}$

• 相互作用发生在$t=0$时刻
• 相互作用的粒子来自于$t=-\infty$ , 此时的粒子处于自由状态
• 相互作用产生的粒子到$t=\infty$时的粒子也处于自由状态

参考文献

1. About the notion of the self-interaction of a field, ACuriousMind (https://physics.stackexchange.com/users/50583/acuriousmind),Physics Stack Exchange, URL:https://physics.stackexchange.com/q/277861 version: 2016-09-02,https://physics.stackexchange.com/q/277861  2

2. skriptQFT1  2