Axiomatic construction of quantum Langevin equations

TL;DR

This paper develops a rigorous axiomatic framework for quantum Langevin equations, clarifying the conditions under which Markovian and non-Markovian quantum dynamics are valid, with applications to quantum models.

Contribution

It introduces an axiomatic approach to derive quantum Langevin equations based on fundamental physical principles, distinguishing between semi-classical and full quantum regimes.

Findings

Markovian semi-classical approach is distinguished from non-Markovian quantum approach.

Quantum fluctuation-dissipation theorem is incompatible with Markovian dynamics.

Application to quantum spherical model discussed.

Abstract

A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation-dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a markovian semi-classical approach, due to Bedeaux and Mazur, from a non-markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation-dissipation theorem is seen to be incompatible with a markovian dynamics. Possible applications to the quantum spherical model are discussed.