I understand that $$G^M(0,0^+) = \operatorname{tr}\{\rho O_2 O_1\}$$ (I am not putting hats on the operators here because they don't render in the correct position) is simply the expectation value of the operator product. E.g. for $O_1 = \psi$ and $O_2 = \psi^\dagger$ gives $G^M(0,0^+) = \operatorname{tr}\{\rho n\}$. So they have a physical meaning. But what about $G^M(\tau)$ for $\tau\neq 0$ (where $\tau = \tau_1-\tau_2$)?
Is the only 'meaning' of this that it is an analytic continuation of the propagator for real times? Is there maybe any physical intuition for Matsubara frequency components $G^M (\omega_m)$ if not for imaginary time directly?
I have read quite a few books and papers on (non-) equilibrium field theory, but have yet to see anyone give physical meaning to these. Possibly there just is none, and it's only a tool for calculation, but if there is I would like to understand it.
As a bonus I would also like some intuition for the left and right propagators in the general time contour (Schwinger-Keldysh/Kadanoff-Baym etc.) formalism. I.e. the propagators with one real and one imaginary time argument.
