# 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, e.g. in the presence of time varying fields (electrical field, magnetic field etc.). Keldysh formalism is named after Leonid Keldysh. It is sometimes called Schwinger-Keldysh formalism, referring to Julian Schwinger. However, many physicists, like Leo Kadanoff, Gordon Baym, O.V. Konstantinov and V. I. Perel,[1] made significant contributions to developing the method that is now called Keldysh formalism.

To study non-equilibrium systems, one is interested in one-point functions or average values of quantum operators, two-point functions and so on. These quantities are calculated using Keldysh formalism. The main mathematical object in the Keldysh formalism is the non-equilibrium Green's function (NEGF), which is related to the two-point function of operators in the system we are looking at.

## Time evolution of a quantum system

Consider a general quantum mechanical system. This system has the Hamiltonian ${\displaystyle H_{0}}$. Let the ground state of the system with respect to this Hamiltonian be ${\displaystyle |0\rangle }$. If we now add a time-dependent perturbation to this Hamiltonian, say ${\displaystyle H'(t)}$, the state will no more be the ground state for the full Hamiltonian ${\displaystyle H(t)=H_{0}+H'(t)}$ and hence the system will evolve in time towards an equilibrium state under the full Hamiltonian. In this section, we will see how time evolution actually works in quantum mechanics.

Let us consider a Hermitian operator ${\displaystyle {\mathcal {O}}}$. In the Heisenberg Picture of quantum mechanics, this operator is time-dependent and the state is not. We are interested in the average ${\displaystyle \langle {\mathcal {O}}(t)\rangle }$. In the natural units, if we define the time-evolution operator as ${\displaystyle U(t,0)=e^{-i\int _{0}^{t}H(t')dt'}}$, then the average of the operator ${\displaystyle {\mathcal {O}}(t)}$ is given by

{\displaystyle {\begin{aligned}\langle {\mathcal {O}}(t)\rangle &=\langle 0|U^{\dagger }(t,0)\,{\mathcal {O}}\,U(t,0)|0\rangle \\\end{aligned}}}

As can be seen, we need to use both the forward time evolution operator ${\displaystyle U(t,0)}$ as well as the backward time evolution operator ${\displaystyle U^{\dagger }(t,0)}$. But usually, only the forward time evolution is considered. This is done by assuming that ${\displaystyle H'(t)}$ obeys the adiabatic theorem. This is basically saying that the perturbation ${\displaystyle H'(t)}$ is turned on slowly as ${\displaystyle t}$ increases from ${\displaystyle 0}$ and it is also turned off slowly as ${\displaystyle t}$ approaches ${\displaystyle \infty }$.

This forward as well as backward time evolution is a characteristic feature of Keldysh formalism.

### Equilibrium case

We started with a system in its ground state ${\displaystyle |0\rangle }$. In this case, Keldysh formalism becomes simpler. It is then also called Matsubara formalism.