Coherent State Path Integral Reveals Unexpected Vacuum Structure in Thermal Field Theory

TL;DR

This paper develops a coherent state path integral approach for thermal scalar fields, revealing overlooked vacuum contributions and novel vacuum expectation value effects in both free and interacting theories.

Contribution

It introduces a new mapping between field-basis and ladder-basis coherent states in quantum field theory and uncovers previously missed vacuum energy terms and vacuum expectation value couplings.

Findings

Identifies vacuum energy contributions in the path integral formulation.

Extends the method to interacting $$ theory revealing new vacuum effects.

Provides a novel mapping between different coherent state bases.

Abstract

We construct the path integral formulation of the partition function for a free scalar thermal field theory using coherent states, first in the ladder operator basis and then in the field operator basis. In so doing, we provide for the first time a mapping in quantum field theory between the field-basis coherent states and ladder-basis coherent states. Using either basis, one finds terms missed in the usual path integral derivation, which we identify as the vacuum energy contribution to the partition function. We then extend the field-basis coherent state method to the interacting $\phi^4$ theory. In addition to the vacuum energy contribution, one finds a coupling of a vacuum expectation value to the mass term that is absent in the existing literature.