Properties of the Boltzmann equation in the classical approximation

TL;DR

This paper investigates how classical approximations of the Boltzmann equation with scalar interactions depend on ultraviolet cutoffs, highlighting issues related to non-renormalizability in quantum field theory.

Contribution

It compares classical approximations of the Boltzmann equation with the full quantum version, analyzing cutoff dependence and non-renormalizability effects.

Findings

Classical approximations show cutoff dependence related to non-renormalizability.

Numerical solutions of the unapproximated Boltzmann equation are straightforward.

The setup simplifies studying ultraviolet cutoff effects in kinetic theory.

Abstract

We study the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since one has also access to the non-approximated result for comparison.