tic c-number kernel $G_s^{AB}=G^{AB}+\gamma^{AB}$. We assume the identity

\_s\^A\_s\^B=\^A\^B+G\^[AB]{}+\^A\^B+\^A\^B+\^[AB]{} The required expectation values are

\_H\^C\_H\^E=\^C\^E+\^C\^E

\_H\^C\_H\^D\_H\^E&=&\_H\^C(\_H\^D\_H\^E)&=&\^C\^D\^E+\^CG\^[DE]{}+\^DG\^[CE]{}+\^EG\^[CD]{}+\^C\^[DE]{}Observe that this implies that $\left\langle \varphi^C\gamma^{DE}\right\rangle$ is totally symmetric.

\_H\^C\_H\^D\_H\^E\_H\^F&=&(\_H\^C\_H\^D)(\_H\^E\_H\^F)&=&(\^C\^D+G\^[CD]{})(\^E\^F+G\^[EF]{})&+&\^C&+&\^D&+&\^E\^F\^[CD]{}+\^F\^E\^[CD]{}+\^[CD]{}\^[EF]{}We can now relate the derivatives of the mean fields with respect to the sources to stochastic averages

\^[C,E]{}=i\^C\^E

G\^[CD,E]{}=i\^C\^[DE]{}

\^[C,(EF)]{}=i2{\^C\^[EF]{}+\^EG\^[CF]{}+\^FG\^[CE]{}}

G\^[CD,(EF)]{}=i2{\^E\^F\^[CD]{}+\^F\^E\^[CD]{}+\^[CD]{}\^[EF]{}} Whereby we get the identities

\^C\^E+\_[,A(CD)]{}\^[CD]{}\^E=i\_A\^E

&&{\^C\^[EF]{}+\^EG\^[CF]{}+\^FG\^[CE]{}}&+&\_[,A(CD)]{}{\^E\^F\^[CD]{}+\^F\^E\^[CD]{}+\^[CD]{}\^[EF]{}}=2i\_[(AB)]{}\^[(EF)]{}\^B which reduces to

\^C\^[EF]{}+\_[,A(CD)]{}\^[CD]{}\^[EF]{}=0

\_[,(AB)C]{}\^C\^E+\_[,(AB)(CD)]{}\^[CD]{}\^E=0

\_[,(AB)C]{}\^C\^[EF]{}+\_[,(AB)(CD)]{}\^[CD]{}\^[EF]{}=i2{\_A\^E\_B\^F+\_A\^F\_B\^E} Assuming the mean field equation $\Gamma_{,\left(AC\right)}=0$ these equation suggest a stochastic dynamics for the $\varphi$, $\gamma$ fields

\_[,AC]{} \^C+\_[,A(CD)]{}\^[CD]{}=-\_A \[2pilan1\] Observe that a possible term $\kappa_{AB}\phi^B$ is absent,

\_[,(AB)C]{} \^C+\_[,(AB)(CD)]{} \^[CD]{}=2\_[AB]{} \[2pilan2\] provided

\_A\^E=-i\_A\^E

\_A\^[CD]{}=0

\_[AB]{}\^E=0

\_[AB]{}\^[EF]{}=-i{\_A\^E\_B\^F+\_A\^F\_B\^E} Multiplying the Langevin equations by the sources and using these expectation values, we get

\_A\_B=i\_[,AB]{}

\_A\_[CD]{}=2i\_[,A(CD)]{}

\_[CD]{}\_[AB]{}=4i\_[,(AB)(CD)]{}

Recovery of the 1PI stochastic theory from the 2PI one
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Let us check that the 2PI and 1PI theories agree as far as the mean field fluctuations are concerned. This mu