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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 and, separately, Leo Kadanoff and Gordon Baym. It was further developed by later contributors such as O. V. Konstantinov and V. I. Perel. Extensions to driven-dissipative open quantum systems is given not only for bosonic systems, but also for fermionic systems. 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 H 0 {\displaystyle H_{0}} . Let the initial state of the system be the pure state | n ⟩ {\displaystyle |n\rangle } . If we now add a time-dependent perturbation to this Hamiltonian, say H ′ ( t ) {\displaystyle H'(t)} , the full Hamiltonian is H ( t ) = H 0 + H ′ ( t ) {\displaystyle H(t)=H_{0}+H'(t)} 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 O {\displaystyle {\mathcal {O}}} . In the Heisenberg picture of quantum mechanics, this operator is time-dependent and the state is not. The expectation value of the operator O ( t ) {\displaystyle {\mathcal {O}}(t)} is given by ⟨ O ( t ) ⟩ = ⟨ n | U † ( t , 0 ) O ( 0 ) U ( t , 0 ) | n ⟩ {\displaystyle {\begin{aligned}\langle {\mathcal {O}}(t)\rangle &=\langle n|{U}^{\dagger }(t,0)\,{\mathcal {O}}(0)\,U(t,0)|n\rangle \\\end{aligned}}} where, due to time evolution of operators in the Heisenberg picture, O ( t ) = U † ( t , 0 ) O ( 0 ) U ( t , 0 ) {\displaystyle {\mathcal {O}}(t)=U^{\dagger }(t,0){\mathcal {O}}(0)U(t,0)} . The time-evolution unitary operator U ( t 2 , t 1 ) {\displaystyle U(t_{2},t_{1})} is the time-ordered exponential of an integral, U ( t 2 , t 1 ) = T ( e − i ∫ t 1 t 2 H ( t ′ ) d t ′ ) . {\displaystyle U(t_{2},t_{1})=T(e^{-i\int _{t_{1}}^{t_{2}}H(t')dt'}).} (Note that if the Hamiltonian at one time commutes with the Hamiltonian at different times, then this can be simplified to U ( t 2 , t 1 ) = e − i ∫ t 1 t 2 H ( t ′ ) d t ′ {\displaystyle U(t_{2},t_{1})=e^{-i\int _{t_{1}}^{t_{2}}H(t')dt'}} .) For perturbative quantum mechanics and quantum field theory, it is often more convenient to use the interaction picture. The interaction picture operator is O .... Discover the Erik Dahlen popular books. Find the top 100 most popular Erik Dahlen books.