Main idea behind this paper on Closed-time-path functional formalism I tried to understand following paper: Closed-time-path functional formalism in curved spacetime: Application to cosmological back-reaction problems but I can't understand what is going on because I don't understand how putting one additional term $J^-$ in lagrangian density makes one go backwards in time see $(1.3)$ maybe it's because I don't understand Schwinger Keldysh formalism.
Also I'm confusing it with Feynman path integral (fpi) formalism as if it's same as fpi but we are additional including contribution from paths going backwards in time since these paths are not included in fpi.
So can anyone please give me the main idea behind this paper.
 A: My PhD thesis (section 2.2) has an introduction to the Schwinger-Keldysh "closed-time-path" (CTP) formalism.
https://arxiv.org/abs/gr-qc/0209010
CTP is a way of computing dynamical equations for expectation values of operators for quantum fields that are in disequilibrium. The basic idea is that the usual way of defining the amplitude for a quantum field's response to an external source J in terms of a "path" integral over all field configurations will give you amplitudes that are not expectation values in a single vacuum state at $t=-\infty$ (the "in" vacuum state) but instead is an amplitude between the "in" vacuum state (propagated to $t=+\infty$ under the dynamics of the action of the field Lagrangian and responding to the source J) and the "out" vacuum state, which is the vacuum state at $t=+\infty$.
By defining two different fields on two different time branches with a boundary condition such that the fields are equal at some specific time point (far to the future of any dynamics that we wish to study), the CTP formalism is able to derive--instead of "in-out" amplitudes--"in-in" expectation values for any kind of operator. In particular, using CTP one can derive dynamical equations for expectation values of the field operator, the field operator squared, etc.
