Quantum statistical correlations in thermal field theories: boundary effective theory

TL;DR

This paper develops a boundary effective theory approach for thermal scalar fields, showing that the one-loop effective action at finite temperature can be expressed similarly to zero temperature, leading to a dimensionally-reduced description.

Contribution

It introduces a novel boundary effective theory framework for thermal field systems, connecting finite-temperature effects to boundary correlations and saddle-point expansions.

Findings

Finite-temperature effective action matches zero-temperature form when expressed in classical boundary fields.

The method yields a dimensionally-reduced effective theory for thermal systems.

Calculated two-point correlation exemplifies the approach.

Abstract

We show that the one-loop effective action at finite temperature for a scalar field with quartic interaction has the same renormalized expression as at zero temperature if written in terms of a certain classical field $\phi_c$, and if we trade free propagators at zero temperature for their finite-temperature counterparts. The result follows if we write the partition function as an integral over field eigenstates (boundary fields) of the density matrix element in the functional Schr\"{o}dinger field-representation, and perform a semiclassical expansion in two steps: first, we integrate around the saddle-point for fixed boundary fields, which is the classical field $\phi_c$, a functional of the boundary fields; then, we perform a saddle-point integration over the boundary fields, whose correlations characterize the thermal properties of the system. This procedure provides a dimensionally-reduced effective theory for the thermal system. We calculate the two-point correlation as an example.