In-in (Schwinger-Keldysh) formalism in cosmology

Question

I'm working on some application of the Schwinger-Keldysh formalism in cosmology (studying correlation function during inflation).

1. I saw both the operator and the path integral approach, maybe something that connects the two?

2. Moreover, do you have an intuitive idea of why this is the correct approach to non-equilibrium QFT and to QFT in curved background?

3. From the in-in formula: \begin{equation}\langle\mathcal{O}(t)\rangle=\langle 0|\left(\mathrm{T} e^{-i \int_{\mathrm{tin}}^{t} d \tau \hat{H}_{I}(\tau)}\right)^{\dagger}\mathcal{O}(t)\left(\mathrm{T} e^{-i \int_{t_{\mathrm{in}}}^{t} d \tau \hat{H}_{I}(\tau)}\right)| 0\rangle \end{equation} how do you proceed in perturbation theory? Just expanding both exponential? In standard QFT we were expanding the exponential in $U(-\infty, +\infty)$.

4. How come we use two different set of fields $\phi_+$ and $\phi_-$ to do computation then?

Answer 1

I reproduce here with adjustments my answer to a somewhat different question, and with admittedly condensed-matter bias. Still, it might be useful.

• In principle, AGD combined with the Keldysh's original paper are quite sufficient for serious work.
• The introduction of the review article by Rammer & Smith is quite good as a short reference.
• Haug and Jauho's book is quite popular, however it mainly reproduces the results of the series of (very readable) papers by Meir, Wingreen and Lee, Meir and Wingreen, and Meir, Wingreen and Jauho.
• This review by Kamenev is a good entry point into the Keldysh formalism in terms of the path integral formulation.
• Finally, the book by Kadanoff & Baym is a classic, that is worth mentioning, because they developed the formalism simultaneously with Keldysh.

You can find more literature here: Good reading on the Keldysh formalism