In non-equilibrium physics, the Keldysh formalism or Keldysh–Schwinger 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[1] and, separately, Leo Kadanoff and Gordon Baym.
[5] 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.
Consider a general quantum mechanical system.
If we now add a time-dependent perturbation to this Hamiltonian, say
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.
In the Heisenberg picture of quantum mechanics, this operator is time-dependent and the state is not.
is given by where, due to time evolution of operators in the Heisenberg picture,
we have Since the time-evolution unitary operators satisfy
replaced by any time value greater than
We can write the above expression more succinctly by, purely formally, replacing each operator
parametrizes the contour path on the time axis starting at
Then we can introduce notation of path ordering on this contour, by defining
, 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 Where
on the forward branch of the Keldysh contour, and the integral over
For the rest of this article, as is conventional, we will usually simply use the notation
is on the forward or reverse branch is inferred from context.
We can expand the exponential as a Taylor series to obtain the perturbation series 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.
is a polynomial or series as a function of the elementary fields
, 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.
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
Then each (unrenormalized) edge directed from a vertex
is the number of internal vertices) are all added up to find the total value of the diagram.