# 法捷耶夫-波波夫鬼粒子

## 法捷耶夫-波波夫方法

${\displaystyle S=\int YM=\int tr(F\wedge *F)=\int F_{\mu \nu }^{a}F^{a\mu \nu }}$

${\displaystyle Z=\int DA\exp(iS(A))}$

${\displaystyle \alpha =\sum _{a}\alpha ^{a}t^{a}\in TG}$

${\displaystyle A\to A_{\alpha }=g(d+A)g^{-1}\approx A+d_{D}\alpha }$

${\displaystyle d_{D}}$ 外共变导数。若${\displaystyle f(A)}$ 规范固定函数，则

${\displaystyle \int D\alpha \ \delta (f(A_{\alpha }))\det({\frac {\delta f(A_{\alpha })}{\delta \alpha }})=1}$

${\displaystyle Z=\int DA\ D\alpha \ \delta (f(A_{\alpha }))\det({\frac {\delta f(A_{\alpha })}{\delta \alpha }})e^{iS(A)}}$

${\displaystyle Z=(\int D\alpha )\ \int DA\ \delta (f(A))\det({\frac {\delta f(A_{\alpha })}{\delta \alpha }})e^{iS(A)}}$

### 电磁理论

${\displaystyle G=U(1)}$ ，这是电磁理论规范变换成为${\displaystyle A_{\alpha }=A+d\alpha }$ ，可以选择

${\displaystyle f(A)=\partial A-\omega }$

${\displaystyle {\frac {\delta f(A_{\alpha })}{\delta \alpha }}=\partial ^{2}}$

${\displaystyle Z=\det(\partial ^{2})(\int D\alpha )\int DA\ \delta (\partial A-\omega )e^{iS(A)}=C\int DA\ \delta (\partial A-\omega )e^{iS(A)}=Z_{\omega }}$

${\displaystyle Z=N(\xi )\int D\omega \ \exp(-i\int \omega ^{2}/2\xi )\ Z_{\omega }}$

${\displaystyle =C'\int DA\ e^{iS(A)}\int D\omega \ \exp(-i\int \omega ^{2}/2\xi )\ \delta (\partial A-\omega )}$

${\displaystyle =C'\int DA\ \exp(iS(A)-i\int (\partial A)^{2}/2\xi )}$

${\displaystyle S(A)=-\int (dA)^{2}=-\int (\partial A)^{2}/4=\int A\partial ^{2}A/4}$

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

• ${\displaystyle \xi =0}$ 是朗道规范
• ${\displaystyle \xi =1}$ 是费恩曼规范

### 杨-米尔斯场论

${\displaystyle f(A^{a})=\partial ^{\mu }A_{\mu }^{a}-\omega ^{a}}$

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

${\displaystyle \xi =1}$ 费恩曼-特·胡夫特规范（Feynman-t' Hooft gauge）。但是这一次雅可比行列式

${\displaystyle \det({\frac {\delta f(A_{\alpha })}{\delta \alpha }})=\det(\partial ^{\mu }D_{\mu })}$

${\displaystyle d_{D}=D_{\mu }dx^{\mu }}$

${\displaystyle \det(\partial ^{\mu }D_{\mu })=\int DcD{\bar {c}}\exp(i\int {\bar {c}}(-\partial ^{\mu }D_{\mu })c)=\int DcD{\bar {c}}\exp(iS(c,{\bar {c}}))}$

${\displaystyle S(c,{\bar {c}})=\int {\bar {c}}_{a}(-\partial ^{\mu }D_{\mu }^{ab})c_{b}=\int {\bar {c}}_{a}(-\delta ^{ac}\partial ^{\mu }\partial _{\mu }-g\partial ^{\mu }f^{abc}A_{\mu }^{b})c_{c}}$

${\displaystyle \langle c_{a}(x){\bar {c}}_{b}(y)\rangle =\int {\frac {d^{4}k}{(2\pi )^{4}}}{\frac {i}{k^{2}}}\delta _{ab}e^{-ik(x-y)}}$

${\displaystyle {\mathcal {L}}=-{\frac {1}{4}}(F_{\mu \nu }^{a})^{2}+{\frac {1}{2\xi }}(\partial A)^{2}+{\bar {\psi }}(iD\!\!\!\!{\big /}-m)\psi +{\bar {c}}(-\partial ^{\mu }D_{\mu })c}$

## 参考

1. ^ W. F. Chen. Quantum Field Theory and Differential Geometry页面存档备份，存于互联网档案馆
2. ^ Peskin, Michael E.; Schroeder, Daniel V.; Martinec, Emil. An Introduction to Quantum Field Theory. Physics Today. 1996-08, 49 (8): 69–72. ISSN 0031-9228. doi:10.1063/1.2807734.

## 阅读

• L. D. Faddeev and V. N. Popov, "Feynman Diagrams for the Yang-Mills Field", Phys. Lett. B25 (1967) 29.
• Peskin Schroeder. Intro to QFT. Ch 9, 16.
• Anthony Zee. QFT in Nutshell.