Schwinger-Keldysh propagator

The closed time contour of the in-in formalism gives four bulk propagators: time-ordered $G_{RR}$, anti-time-ordered $G_{LL}$, and the two Wightman functions $G_{LR}$ and $G_{RL}$. Vertices on the two contour branches carry opposite factors of $i$.

For a free mode function $f(k,\eta )$ and a correlator evaluated at ${\eta }_0$, the two cosmological bulk-to-boundary propagators are $K_R=f(k,\eta )f^{\ast }(k,{\eta }_0)$ and $K_L=f^{\ast }(k,\eta )f(k,{\eta }_0)$, up to the chosen contour-label convention.

For a free mode $f(k,\eta )$, the right-right bulk propagator is time ordered,whereas the mixed propagators are Wightman functions.

For real couplings and real external data, reflection of every contour label complex conjugates an in-in diagram: $B_{LL}=B_{RR}^{\ast }$ and $B_{RL}=B_{LR}^{\ast }$.