Minding one's P's and Q's : from the one loop effective action in quantum field theory to classical transport theory

TL;DR

This paper develops a real-time world line formalism for one loop effective actions in quantum field theory, deriving classical transport equations for hot QCD and connecting quantum and classical descriptions.

Contribution

It introduces a real-time, many-body world line formalism for the one loop effective action and derives classical transport equations from quantum field theory.

Findings

Derived classical transport equations for hot QCD.

Connected quantum effective actions to classical kinetic theory.

Discussed imaginary time formalism for hot  theory.

Abstract

The one loop effective action in quantum field theory can be expressed as a quantum mechanical path integral over world lines, with internal symmetries represented by Grassmanian variables. In this paper, we develop a real time, many body, world line formalism for the one loop effective action. In particular, we study hot QCD and obtain the classical transport equations which, as Litim and Manuel have shown, reduce in the appropriate limit to the non-Abelian Boltzmann-Langevin equation first obtained by B\"{o}deker. In the Vlasov limit, the classical kinetic equations are those that correspond to the hard thermal loop effective action. We also discuss the imaginary time world line formalism for a hot $\phi^4$ theory, and elucidate its relation to classical transport theory.