Quantum Transport Equations for a Scalar Field

TL;DR

This paper derives quantum transport equations for a scalar field, revealing that higher-order gradients encode quantum coherence, which simplifies to classical Boltzmann equations under frequent scattering conditions.

Contribution

It introduces a gradient expansion approach that incorporates quantum coherence effects into scalar field transport equations, extending beyond traditional Boltzmann descriptions.

Findings

Higher-order gradients encode quantum coherence.

Quantum effects are suppressed with frequent scatterings.

Transport equations reduce to classical Boltzmann form in certain limits.

Abstract

We derive quantum Boltzmann equations from Schwinger-Dyson equations in gradient expansion for a weakly coupled scalar field theory with a spatially varying mass. We find that at higher order in gradients a full description of the system requires specifying not only an on shell distribution function but also a finite number of its derivatives, or equivalently its higher moments. These derivatives describe quantum coherence arising as a consequence of localization in position space. We then show that in the limit of frequent scatterings coherent quantum effects are suppressed, and the transport equations reduce to the single Boltzmann equation for particle density, in which particles flow along modified semiclassical trajectories in phase space.