Quantum stochastic differential equations for boson and fermion systems -- Method of Non-Equilibrium Thermo Field Dynamics

TL;DR

This paper develops a unified operator formalism for quantum stochastic differential equations, including Liouville and Langevin equations, within Non-Equilibrium Thermo Field Dynamics, applicable to bosonic and fermionic systems.

Contribution

It introduces a general framework for quantum stochastic equations in NETFD, incorporating fermionic and bosonic Brownian motions and generalizing thermal state conditions.

Findings

Unified formalism for quantum stochastic equations in NETFD

Inclusion of fermionic and bosonic Brownian motions

Generalization of thermal state conditions for fermions

Abstract

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the It\^o and the Stratonovich types, is presented within the framework of Non-Equilibrium Thermo Field Dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schr\"odinger and the Heisenberg representations with fermionic and bosonic Brownian motions. In order to decide the double tilde conjugation rule and the thermal state conditions for fermions, a generalization of the system consisting of a vector field and Faddeev-Popov ghosts to dissipative open situations is carried out within NETFD.