Derivation of the Keldysh Path Integral from the Master Equation of a Harmonic Oscillator in a Thermal Bath Via Supercoherent States

Abstract

We consider a harmonic oscillator coupled to a Markovian environment, i.e. a thermal reservoir. We derive the Keldysh path integral and the corresponding lesser, greater, time-ordered, antitime-ordered, retarded, advanced, and Keldysh Green’s functions for this system. In order to give a conceptually clear derivation, we use the superoperator formalism and also define supercoherent states. The supercoherent states and the creation and annihilation superoperators provide a natural way to rewrite the master equation into a path integral.