Keldysh formalism

In non-equilibrium physics, the Keldysh formalism is a general framework for describing the quantum mechanical evolution of a system in a non-equilibrium state or systems subject to time varying external fields (electrical field, magnetic field etc.). Historically, it was foreshadowed by the work of Julian Schwinger and proposed almost simultaneously by Leonid Keldysh and, separately, Leo Kadanoff and Gordon Baym. It was further developed by later contributors such as O. V. Konstantinov and V. I. Perel.
Extension to driven-dissipative open quantum systems is given in 
The Keldysh formalism provides a systematic way to study non-equilibrium systems, usually based on the two-point functions corresponding to excitations in the system. The main mathematical object in the Keldysh formalism is the non-equilibrium Green's function (NEGF), which is a two-point function of particle fields. In this way, it resembles the Matsubara formalism, which is based on equilibrium Green functions in imaginary-time and treats only equilibrium systems.
Time evolution of a quantum system.
Consider a general quantum mechanical system. This system has the Hamiltonian formula_1. Let the initial state of the system be formula_2, which can either be a pure state or a mixed state. If we now add a time-dependent perturbation to this Hamiltonian, say formula_3, the full Hamiltonian is formula_4 and hence the system will evolve in time under the full Hamiltonian. In this section, we will see how time evolution actually works in quantum mechanics.
Consider a Hermitian operator formula_5. In the Heisenberg Picture of quantum mechanics, this operator is time-dependent and the state is not. The expectation value of the operator formula_6 is given by
formula_7
Where, due to time evolution of operators in the Heisenberg Picture, formula_8. The time-evolution unitary operator formula_9 is the time-ordered exponential of an integral formula_10. (Note that if the Hamiltonian at one time commutes with the Hamiltonian at different times, then this can be simplified to formula_11.)
For perturbative quantum mechanics and quantum field theory, it is often more convenient to use the interaction picture. The interaction picture operator is
formula_12
Where formula_13. Then defining formula_14 we have
formula_15
Since the time-evolution unitary operators satisfy formula_16, the above expression can be rewritten as
formula_17
or with formula_18 replaced by any time value greater than formula_19.
Path ordering on the Keldysh contour.
We can write the above expression more succinctly by, purely formally, replacing each operator formula_20 with a contour-ordered operator formula_21 , such that formula_22 parametrizes the contour path on the time axis starting at formula_23, proceeding to formula_24, and then returning to formula_23. This path is known as the Keldysh contour. formula_26 has the same operator action as formula_20 (where formula_19 is the time value corresponding to formula_22) but also has the additional information of formula_22 (that is, strictly speaking formula_31 if formula_32, even if for the corresponding times formula_33).
Then we can introduce notation of path ordering on this contour, by defining formula_34, where formula_35 is a permutation such that formula_36, and the plus and minus signs are for bosonic and fermionic operators respectively. Note that this is a generalization of time ordering.
With this notation, the above time evolution is written as
formula_37
Where formula_22 corresponds to the time formula_19 on the forward branch of the Keldysh contour, and the integral over formula_40 goes over the entire Keldysh contour. For the rest of this article, as is conventional, we will usually simply use the notation formula_20 for formula_26 where formula_19 is the time corresponding to formula_22, and whether formula_22 is on the forward or reverse branch is inferred from context.
Keldysh diagrammatic technique for Green's functions.
The non-equilibrium Green's function is defined as formula_46.
Or in the interaction picture formula_47 . We can expand the exponential as a Taylor series to obtain the perturbation series formula_48. This is the same procedure as in equilibrium diagrammatic perturbation theory, but with the important difference that both forward and reverse contour branches are included.
If, as is often the case, formula_49 is a polynomial or series as a function of the elementary fields formula_50, we can organize this perturbation series into monomial terms and apply all possible Wick pairings to the fields in each monomial, obtaining a summation of Feynman diagrams. However, the edges of the Feynman diagram correspond to different propagators depending on whether the paired operators come from the forward or reverse branches. Namely,
where the anti-time ordering formula_55 orders operators in the opposite way as time ordering and the formula_56 sign in formula_57 is for bosonic or fermionic fields. Note that formula_58 is the propagator used in ordinary ground state theory.
Thus, Feynman diagrams for correlation functions can be drawn and their values computed the same way as in ground state theory, except with the following modifications to the Feynman rules: Each internal vertex of the diagram is labeled with either formula_59 or formula_60, while external vertices are labelled with formula_61. Then each (unrenormalized) edge directed from a vertex formula_62 (with position formula_63, time formula_64 and sign formula_65) to a vertex formula_66 (with position formula_67, time formula_68 and sign formula_69) corresponds to the propagator formula_70. Then the diagram values for each choice of formula_56 signs (there are formula_72 such choices, where formula_73 is the number of internal vertices) are all added up to find the total value of the diagram.