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I am working on out-of-equilibrium problems of strongly correlated materials, so I am interested in the Keldysh formalism. I just started reading about the subject, and I don't understand quite well the interest of the formalism. Since one can use the Kubo and Matsubara formalism to treat a time-dependent Hamiltonian.

I can see that the formalism allows for a diagrammatic approach in a more compact and systematic way. Generalization of the self energy to the time-dependent $H$.

What can the Keldysh formalism do that the Kubo formula along with the Matsubara formalism can't?

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Matsubara formalism works for statistical systems in equilibrium, and Kubo formula corresponds to the linear response approximation when the system is weakly perturbed by a time-dependent interaction. If you want to go beyond these two approximations (equilibrium or weakly perturbed systems), you need Keldysh method or some other methods of your preference, as e.g. to work directly with the density matrix and not Green's functions. For instance, if you want to treat far-from-equilibrium systems you need better things than Kubo or Matsubara.

Also, at the heart of the Green's function methods is the hypothesis of adiabatically switching on the interaction, using a theorem of Gell-Mann and Low [Bound states in quantum field theory. Physical Review, 84, 350–354, (1951).]. If you want to go beyond this approximation, you also need Keldysh method. Apparently, this is a problem which has only recently been seriously discussed, so I will not comment much about it.

Perhaps you mistakenly said Matsubara for the analytical continuation of the statistical equilibrium methods (i.e. the true Matsubara method) to time-dependent problems of statistical mechanics. Then the two methods are equivalent. It is just a matter of convention and preference which one you prefer to use (either Keldysh or analytic continuation of imaginary time). The reason is that in Matsubara method of equilibrium systems you suppose the time to be pure imaginary, so taking both real-time (i.e. evolution) and statistics means the time variable goes smoothly to the complex plane $t\rightarrow t+i/k_{B}T$ (sketchilly speaking, more details can be found in this question).

Also, time-dependent quantum field problems can be attacked using Keldysh methods (I mean, problem discussing only one particle, not statistical mechanics). But this is completely unrelated to Kubo or Matsubara approximations, so I guess your question was not about that possibility.

Finally, (and I'm less certain about that) I feel that quantum statistics can be discussed only in the Keldysh formalism. The reason is that the Green's function approach uses the classical statistics (I call classical both the fermionic and bosonic gases), i.e. the thermal equilibrium. This is the condition under which the replacement $e^{iHt}\rightarrow e^{H/k_{B}T}$ between quantum field and statistical fields works. Beyond this Boltzmann statistical weighting, I fear you need Keldysh or more powerful methods.

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What can the Keldysh formalism do that the Kubo formula along with the Matsubara formalism can't?

The Fermi and Bose distribution functions, with which you must be familiar, hold strictly at equilibrium. The Keldysh formalism allows you to capture nonequilibrium distribution functions which cannot be captured within the Matsubara formulation, by its very construction.

This is relevant when one tries to calculate the response of a system is driven strongly out of equilibrium. For systems which are weakly driven out of equilibrium, there exists the fluctuation-dissipation theorem which relates the experimentally observable quantities to some response function that depends only on the equilibrium state, and not on the strength of the perturbation. This is the gist of linear response theory and the response function can be calculated using a Kubo formula. However, the crucial assumption entering here is that the distribution function doesn't change upon applying the perturbation, which breaks down in case of strong perturbations.

The Matsubara theory can be used to calculate imaginary time ordered correlation functions, which are then analytically continued to obtain causal real time response functions. However, this procedure isn't very useful beyond linear response regime because in order to make connection with experiments one must also obtain the renormalised distribution function; something beyond the scope of Matsubara theory.

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