Causal signal transmission by quantum fields. IV. The causal Wick theorem

Wick’s theorem in the Schwinger-Perel-Keldysh closed-time-loop formalism is written in a form where the place of contractions is taken by the linear response function of the field. This result demonstrates that the physical information supplied by Wick’s theorem for operators is propagation of the free field in space and time.

Figure 1

The $C$-contour.

Figure 2

The three possible ways a pair of displacement operators (bold dots) may be placed on the $C$-contour; $\widehat{Q}=\widehat{Q}(t)$, ${\widehat{Q}}^{\prime }=\widehat{Q}(t^{\prime })$. Dashed lines signify the corresponding contractions.

Figure 3

The 10 possible ways a pair of channel operators (bold dots) may be placed on the $C$-contour; $\widehat{\psi }=\widehat{\psi }(x,t)$, ${\widehat{\psi }}^{\prime }=\widehat{\psi }(x^{\prime },t^{\prime })$, and similarly for the tilded operators. Dashed lines signify the corresponding contractions. Only those on the top row are nonzero.