查看“︁传播子”︁的源代码

在[[量子力學]]以及[[量子场论]]中的'''传播子'''（propagator；'''核子，kernel'''），是描述[[粒子]]在特定時間由一處移動到另一處的[[機率幅]]，或是粒子以特定能量及動量移動的[[機率幅]]。传播子也是场的[[运动方程]]的[[格林函数]]。物理学家使用核子计算[[费曼图]]以及[[散射]]过程的概率。

== 量子力学 ==
自由粒子（[[波包]]）的核子是<ref>[http://planetmath.org/encyclopedia/SaddlePointApproximation.html Saddle point approximation] {{Wayback|url=http://planetmath.org/encyclopedia/SaddlePointApproximation.html |date=20150918183231 }}, planetmath.org</ref>

{{Equation box 1|border|indent=:|equation=<math>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)} ~.</math>|cellpadding=6|border colour=#0073CF|bgcolor=#F9FFF7}}

[[量子諧振子]]的{{Internal link helper/en|Mehler核子|Mehler kernel}}<ref>{{Cite web|title=IMMERSION OF THE FOURIER TRANSFORM IN A CONTINUOUS GROUP OF FUNCTIONAL TRANSFORMATIONS|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf|accessdate=|author=|date=|format=|publisher=|language=|archive-date=2020-05-10|archive-url=https://web.archive.org/web/20200510085417/https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf|dead-url=no}}</ref><ref>{{Cite book|title=E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, (1937) 158–164.|last=|first=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf|publisher=|year=|isbn=|location=|pages=}}</ref><ref>{{Cite book|edition=Dover edition|chapter=Pauli lectures on physics|url=https://www.worldcat.org/oclc/44493172|location=Mineola, New York|isbn=0-486-41457-4|oclc=44493172|last=Pauli, Wolfgang, 1900-1958.}}</ref>

{{Equation box 1|border|indent=:|equation=<math>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) ~.</math>|cellpadding=6|border colour=#0073CF|bgcolor=#F9FFF7}}

通过[[泛函积分]]，核子等于

<math>K(x, x'; t, t') = \int Dx(t) \ \exp(i\int_{t}^{t'}L(x, \dot{x}; t) \ dt)</math>

<math>x(t) = x, \ x(t') = x'</math>

L是[[拉氏量]]。

== 量子场论 ==

=== [[Klein-Gordon方程|克莱因-戈尔登方程]] ===
克戈场论（Klein-Gordon）的Feynman传播子

<math>\tilde{G}_F(p) = \frac{1}{p^2 -  m^2 + i\epsilon}. </math>

据黄教授说，这是<ref>{{Cite book|chapter=Quantum field theory : from operators to path integrals|url=https://www.worldcat.org/oclc/38495059|publisher=Wiley|date=1998|location=New York|isbn=0-471-14120-8|oclc=38495059|last=Huang, Kerson, 1928-}}</ref>

<math> G_\mathrm{F}(x,y)
= \lim_{\epsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p \, \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{i m}{ 4 \pi^2 \sqrt{-s}} K_1(m \sqrt{-s}) & \text{if} s < 0.
\end{cases} </math>

H是[[汉克尔函数]]，K是[[贝塞尔函数]]，δ是[[狄拉克δ函数]]，<math>s^2 = x^{\mu} x_{\mu}</math>。

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

[[Image:FeynmanPropagatorPath.svg]]



Feynman传播子也等于下面的[[真空期望值]]：

<math>G_F(x-y) = -i\langle 0|T\phi(x)\phi(y)| 0 \rangle</math>

<math>= -i\langle 0|\theta(x^0-y^0)\phi(x)\phi(y) + \theta(y^0-x^0)\phi(y)\phi(x)| 0 \rangle</math>

上面T是[[路径排序]]算子，<math>\theta</math>是[[单位阶跃函数]]。

=== [[狄拉克方程]] ===
<math>\tilde{S}_F(p) = {1 \over \gamma^\mu p_\mu - m + i\epsilon} = {1 \over p\!\!\!/ - m + i\epsilon}. </math>

<math>S_F(x-y) = \int{{d^4 p\over (2\pi)^4} \, e^{-i p \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|).
</math>

传播子也是格林函数

<math>S_F(x-y) = (i \partial\!\!\!/ + m) G_F(x-y)</math>

这描述[[费米子]]、[[电子]]。

=== [[量子电动力学]]和其他[[杨-米尔斯场论]] ===
{{Main|规范场论}}
[[光子]]传播子是

<math>{-i g^{\mu\nu} \over p^2 + i\epsilon }.</math>

<math> \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}.</math>

<math>D_{\mu \nu}(k) = \frac{-i}{k^2 + i\epsilon} (g_{\mu \nu} - (1-\xi)\frac{k_{\mu}k_{\nu}}{k^2})  </math>

也阅读[[FP鬼子]]，给予[[膠子]]传播子或杨米尔斯传播子：

<math>\langle A_{\mu}^a(x)A_{\nu}^b(y)\rangle =D_{\mu \nu}(x-y)^{ab} = \int \frac{d^4k}{(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})  </math>

选择<math>\xi</math> 需要[[规范固定]]。

=== [[引力子]] ===
重力子的传播子是<ref>{{Cite web|title=Quantum theory of gravitation|url=https://dspace.library.uu.nl/bitstream/handle/1874/4837/Quantum_theory_of_gravitation.pdf?sequence=2&isAllowed=y|accessdate=|author=|date=|format=|publisher=|language=|archive-date=2020-07-02|archive-url=https://web.archive.org/web/20200702193522/https://dspace.library.uu.nl/bitstream/handle/1874/4837/Quantum_theory_of_gravitation.pdf?sequence=2&isAllowed=y|dead-url=no}}</ref><ref>{{Cite web|title=Graviton and gauge boson propagators in AdSd+1|url=http://cds.cern.ch/record/378516/files/9902042.pdf|accessdate=|author=|date=|format=|publisher=|language=|archive-date=2018-07-25|archive-url=https://web.archive.org/web/20180725014800/http://cds.cern.ch/record/378516/files/9902042.pdf|dead-url=no}}</ref><ref>{{Cite book|title=Quantum Field Theory in Nutshell|last=Zee, Anthony|first=|publisher=|year=|isbn=|location=Princeton University Press|pages=}}</ref>

<math>G_{abcd}(k) = \frac{g_{ac}g_{bd}+g_{bc}g_{ad} - g_{ab}g_{cd}}{k^2}</math>

==相关条目==
* [[量子场论]]
* [[格林函数]]
* [[路径积分表述]]
*[[费恩曼图]]

== 参考文献 ==
{{Reflist}}

== 阅读 ==

* Feynman Hibbs. Path integrals.
*Peskin Schroeder. Intro QFT.
*Huang. QFT.
*https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf {{Wayback|url=https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf |date=20201111195944 }} {{Wayback|url=https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf |date=20201111195944 }}<nowiki/>（Srendiecki QFT）
* [[徐一鸿]] Anthony Zee. QFT in Nutshell.

{{量子场论}}
[[Category:量子力学]]
[[Category:量子场论]]
[[Category:理论物理]]
[[Category:数学物理]]