# 传播子

## 量子力学

${\displaystyle K(x,x';t)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }dk\,e^{ik(x-x')}e^{-i\hbar k^{2}t/(2m)}=\left({\frac {m}{2\pi i\hbar t}}\right)^{1/2}e^{-m(x-x')^{2}/(2i\hbar t)}~.}$

${\displaystyle K(x,x';t)=\left({\frac {m\omega }{2\pi i\hbar \sin \omega t}}\right)^{1/2}\exp \left(-{\frac {m\omega ((x^{2}+x'^{2})\cos \omega t-2xx')}{2i\hbar \sin \omega t}}\right)~.}$

${\displaystyle K(x,x';t,t')=\int Dx(t)\ \exp(i\int _{t}^{t'}L(x,{\dot {x}};t)\ dt)}$

${\displaystyle x(t)=x,\ x(t')=x'}$

L是拉氏量

## 量子场论

### 克莱因-戈尔登方程

${\displaystyle {\tilde {G}}_{F}(p)={\frac {1}{p^{2}-m^{2}+i\epsilon }}.}$

${\displaystyle G_{\mathrm {F} }(x,y)=\lim _{\epsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{p^{2}-m^{2}+i\epsilon }}={\begin{cases}-{\dfrac {1}{4\pi }}\delta (s)+{\dfrac {m}{8\pi {\sqrt {s}}}}H_{1}^{(1)}(m{\sqrt {s}})&{\text{ if }}s\geq 0\\-{\dfrac {im}{4\pi ^{2}{\sqrt {-s}}}}K_{1}(m{\sqrt {-s}})&{\text{if}}s<0.\end{cases}}}$

H是汉克尔函数，K是贝塞尔函数，δ是狄拉克δ函数${\displaystyle s^{2}=x^{\mu }x_{\mu }}$

Feynman传播子使用下面的曲线积分（contour integral，留数定理

Feynman传播子也等于下面的真空期望值

${\displaystyle G_{F}(x-y)=-i\langle 0|T\phi (x)\phi (y)|0\rangle }$

${\displaystyle =-i\langle 0|\theta (x^{0}-y^{0})\phi (x)\phi (y)+\theta (y^{0}-x^{0})\phi (y)\phi (x)|0\rangle }$

### 狄拉克方程

${\displaystyle {\tilde {S}}_{F}(p)={1 \over \gamma ^{\mu }p_{\mu }-m+i\epsilon }={1 \over p\!\!\!/-m+i\epsilon }.}$

${\displaystyle S_{F}(x-y)=\int {{d^{4}p \over (2\pi )^{4}}\,e^{-ip\cdot (x-y)}}\,{(\gamma ^{\mu }p_{\mu }+m) \over p^{2}-m^{2}+i\epsilon }=\left({\gamma ^{\mu }(x-y)_{\mu } \over |x-y|^{5}}+{m \over |x-y|^{3}}\right)J_{1}(m|x-y|).}$

${\displaystyle S_{F}(x-y)=(i\partial \!\!\!/+m)G_{F}(x-y)}$

### 量子电动力学和其他杨-米尔斯场论

${\displaystyle {-ig^{\mu \nu } \over p^{2}+i\epsilon }.}$

${\displaystyle {\frac {g_{\mu \nu }-k_{\mu }k_{\nu }/m^{2}}{k^{2}-m^{2}+i\epsilon }}+{\frac {k_{\mu }k_{\nu }/m^{2}}{k^{2}-m^{2}/\lambda +i\epsilon }}.}$

${\displaystyle D_{\mu \nu }(k)={\frac {-i}{k^{2}+i\epsilon }}(g_{\mu \nu }-(1-\xi ){\frac {k_{\mu }k_{\nu }}{k^{2}}})}$

${\displaystyle \langle A_{\mu }^{a}(x)A_{\nu }^{b}(y)\rangle =D_{\mu \nu }(x-y)^{ab}=\int {\frac {d^{4}k}{(2\pi )^{4}}}{\frac {-ie^{-ik(x-y)}}{k^{2}+i\epsilon }}\delta ^{ab}(g_{\mu \nu }-(1-\xi ){\frac {k_{\mu }k_{\nu }}{k^{2}}})}$

### 引力子

${\displaystyle G_{abcd}(k)={\frac {g_{ac}g_{bd}+g_{bc}g_{ad}-g_{ab}g_{cd}}{k^{2}}}}$

## 参考文献

1. ^ Saddle point approximation页面存档备份，存于互联网档案馆）, planetmath.org
2. ^ IMMERSION OF THE FOURIER TRANSFORM IN A CONTINUOUS GROUP OF FUNCTIONAL TRANSFORMATIONS (PDF). （原始内容存档 (PDF)于2020-05-10）.
3. ^ E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, (1937) 158–164.. Authors list列表中的|first1=缺少|last1= (帮助)
4. ^ Pauli, Wolfgang, 1900-1958. Pauli lectures on physics. Dover edition. Mineola, New York https://www.worldcat.org/oclc/44493172. ISBN 0-486-41457-4. OCLC 44493172. 缺少或|title=为空 (帮助)
5. ^ Huang, Kerson, 1928-. Quantum field theory : from operators to path integrals. New York: Wiley https://www.worldcat.org/oclc/38495059. 1998. ISBN 0-471-14120-8. OCLC 38495059. 缺少或|title=为空 (帮助)
6. ^ Quantum theory of gravitation (PDF). （原始内容存档 (PDF)于2020-07-02）.
7. ^ Graviton and gauge boson propagators in AdSd+1 (PDF). （原始内容存档 (PDF)于2018-07-25）.
8. ^ Zee, Anthony. Quantum Field Theory in Nutshell. Princeton University Press.