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# 路径积分表述

## 数学方法

### 哈密顿算符在量子力学中的意义

${\displaystyle U(t_{b},t_{a})=e^{-{\frac {i}{\hbar ))(t_{b}-t_{a})H))$

${\displaystyle iG(x_{b},t_{b};x_{a},t_{a})\equiv \left\langle x_{b}\right|U(t_{b},t_{a})\left|x_{a}\right\rangle }$

${\displaystyle U(t_{b},t_{a})=U(t_{b},t)U(t,t_{a})}$

${\displaystyle iG(x_{b},t_{b};x_{a},t_{a})=\int dxiG(x_{b},t_{b};x,t)iG(x,t;x_{a},t_{a})}$

### 时间切片

{\displaystyle {\begin{aligned}\left\langle x_{j}\left|e^{-i{\frac {\Delta }{\hbar ))H({\hat {p)),{\hat {x)))}\right|x_{j-1}\right\rangle &=\int dp_{j}\langle x_{j}|p_{j}\rangle \left\langle p_{j}\left|e^{-i{\frac {\Delta }{\hbar ))H({\hat {p)),{\hat {x)))}\right|x_{j-1}\right\rangle \end{aligned))}

${\displaystyle e^{-i{\frac {\Delta }{\hbar ))H({\hat {p)),{\hat {x)))}=:e^{-i{\frac {\Delta }{\hbar ))H({\hat {p)),{\hat {x)))}:+O(\Delta ^{2})}$

{\displaystyle {\begin{aligned}\left\langle x_{j}\left|e^{-i{\frac {\Delta }{\hbar ))H({\hat {p)),{\hat {x)))}\right|x_{j-1}\right\rangle &=\int {\frac {dp_{j)){2\pi \hbar ))e^{i{\frac {p_{j)){\hbar ))(x_{j}-x_{j-1})}\,e^{-i{\frac {\Delta }{\hbar ))H(p_{j},x_{j-1})}\\&=\int {\frac {dp_{j)){2\pi \hbar ))e^{i{\frac {\Delta }{\hbar ))\left(p_{j}{\frac {x_{j}-x_{j-1)){\Delta ))-H(p_{j},x_{j-1})\right)}\\\end{aligned))}

{\displaystyle {\begin{aligned}iG(x_{b},t_{b};x_{a},t_{a})&=\int dx_{1}\cdots dx_{n-1}\prod _{i=1}^{n-1}dp_{i}\exp \left[{\frac {i}{\hbar ))\sum _{j=1}^{n-1}\Delta \,L\left(t_{j},{\frac {x_{j}+x_{j-1)){2)),{\frac {x_{j}-x_{j-1)){\Delta ))\right)\right]\\&=\int {\mathcal {D))\left[x(t)\right]e^((\frac {i}{\hbar ))S[x(t)]}\end{aligned))}

${\displaystyle S}$是路径${\displaystyle x(t)}$的作用量，拉格朗日量${\displaystyle L(t,x,{\dot {x)))}$的时间积分：

${\displaystyle S=\int L(t,x,{\dot {x)))dt}$

## 简单例子

### 自由粒子

${\displaystyle S=\int {\frac ((\dot {x))^{2)){2))dt}$

${\displaystyle G(x-y;T)=\int _{x(0)=x}^{x(T)=y}e^{-\int _{0}^{T}{\frac ((\dot {x))^{2)){2))dt}{\mathcal {D))x=\int _{x(0)=x}^{x(T)=y}\prod _{t}e^{-{\frac {1}{2))\left({\frac {(x(t+\epsilon )-x(t)}{\epsilon ))\right)^{2}\epsilon }{\mathcal {D))x}$

${\displaystyle {\mathcal {D))x}$是以上时间切成有限片的积分。连乘里每一项都是平均值为${\displaystyle x(t)}$方差为c的高斯函数。多重积分是相邻时间高斯函数${\displaystyle G_{\epsilon ))$的卷积：

${\displaystyle G(x-y;T)=G_{\epsilon }*G_{\epsilon }*G_{\epsilon }\cdots G_{\epsilon ))$

${\displaystyle {\tilde {G))(p;T)={\tilde {G))_{\epsilon }(p)^{T/\epsilon ))$

${\displaystyle {\tilde {G))_{\epsilon }(p)=e^{-\epsilon {\frac {p^{2)){2))))$

${\displaystyle {\tilde {G))(p;T)=e^{-T{\frac {p^{2)){2))))$

${\displaystyle G(x-y;T)\propto e^{-{\frac {(x-y)^{2)){2T))))$

${\displaystyle \int G(x-y;T)dy=1}$

${\displaystyle {\frac {d}{dt))G(x;t)={\frac {\nabla ^{2)){2))G}$

${\displaystyle G(x-y;T)\propto e^{\frac {i(x-y)^{2)){2T))}$

${\displaystyle {\frac {d}{dt))G(x;t)={\frac {i\nabla ^{2)){2))G}$

${\displaystyle \varphi _{t}(x)=\int \varphi _{0}(y)G(x-y;t)dy}$

${\displaystyle G}$一样服从薛定谔方程：

${\displaystyle i{\frac {d}{dt))\varphi _{t}=-{\frac {\nabla ^{2)){2))\varphi _{t}(x)}$

## 量子场论

${\displaystyle Z=\int D\phi \ \exp(iS(\phi ))}$

## 参考资料

1. ^ Chaichian, Masud; Demichev, Andrei Pavlovich. Introduction. Path Integrals in Physics Volume 1: Stochastic Process & Quantum Mechanics. Taylor & Francis. 2001: 1ff. [2016-10-21]. ISBN 0-7503-0801-X. （原始内容存档于2019-05-02）.
2. ^ Dirac, Paul A. M. The Lagrangian in Quantum Mechanics (PDF). Physikalische Zeitschrift der Sowjetunion. 1933, 3: 64–72 [2016-10-21]. （原始内容存档 (PDF)于2017-01-14）.
3. ^ Van Vleck, John H. The correspondence principle in the statistical interpretation of quantum mechanics. Proceedings of the National Academy of Sciences of the United States of America. 1928, 14 (2): 178–188. Bibcode:1928PNAS...14..178V. PMC 1085402. PMID 16577107. doi:10.1073/pnas.14.2.178.