On the derivation of the Boltzmann equation in quantum field theory: Flat spacetime

TL;DR

This paper rigorously derives a quantum field theoretic version of the Boltzmann equation in flat spacetime, highlighting the importance of rescattering terms and loop corrections for accurate physical modeling.

Contribution

It introduces a non-perturbative pre-Boltzmann equation and analyzes the effects of rescattering and loop corrections within quantum field theory in flat spacetime.

Findings

Derived a non-perturbative integro-differential equation for number densities.

Identified additional rescattering terms in the long-time limit of the equation.

Showed that loop corrections necessitate including rescattering terms for consistency.

Abstract

In this paper, we analyze in a mathematically rigorous fashion the validity of the Boltzmann transport equation within quantum field theory. We work within the specific model of a hermitian, scalar field with polynomial self-interaction in two-dimensional Minkowski space. Our main results are as follows: Firstly, that one can obtain a non-perturbative, exact integro-differential equation for the number densities, which we called the pre-Boltzmann equation. We secondly take the long-time-dilute-medium limit of this equation, to obtain a simpler equation. This limiting equation is qualitatively similar to the Boltzmann equation, but it involves additional re-scattering terms. These terms disappear if perform a perturbation expansion in the coupling constant and ignore the loop corrections (Born approximation). If loop corrections are included, then we argue that for consistency, one must also keep corresponding rescattering terms which are normally ignored. Our analysis is hence of potential relevance for physical applications of the Boltzmann equation wherein loop effects are essential, such as in the standard scenario of baryogensis in the Early Universe. Our analysis is performed in the context of flat spacetime, but in such a way that the main ingredients can be transferred, straightforwardly to the case of a curved spacetime of Robertson-Walker-type. Our main technical tools are methods from constructive quantum field theory, as well as a general method called "projection technique". This turns out to give convergent expansions, and a rather elegant way of organizing the combinatorics of the various quantum field theoretic expansions in the analysis.