Dissipative noninteracting scalar field theory: A covariant formulation

TL;DR

This paper develops a covariant formulation of dissipative scalar field theory by generalizing the Caldeira-Leggett reservoir model to a continuum of Klein-Gordon fields, deriving a quantum Langevin equation and analyzing medium susceptibility.

Contribution

It introduces a covariant approach to dissipative scalar fields using a continuum reservoir model and establishes the relation between coupling functions and medium susceptibility.

Findings

Derived a quantum Langevin equation for scalar fields.

Established the relation between coupling function and susceptibility via Hankel transform.

Obtained explicit forms of the scalar field and its large-time behavior in homogeneous media.

Abstract

Caldeira-Leggett model of reservoir is generalized to a reservoir modeled by a continuum of real Klein-Gordon fields, instead of harmonic oscillators. A quantum Langevin type dissipative equation is obtained for the scalar field. The susceptibility of the medium is defined in terms of the reservoir Green's function and the coupling function satisfying causality condition. The connection between the coupling function and the susceptibility of the medium is found to be a Hankel transform from which the coupling function can be determined in terms of the susceptibility of the medium. Noise currents and their fluctuation-dissipation relation are obtained. In a homogeneous medium or reservoir, explicit form of the quantum scalar field, and its large-time limit, are found.