Thermal boundary conditions as constraints

TL;DR

This paper introduces a novel way to implement thermal boundary conditions as constraints in the path integral formalism, resulting in a $d$-dimensional, time-independent effective representation of the finite-temperature partition function.

Contribution

It develops a new formalism using Lagrange multipliers to impose boundary conditions, simplifying the finite-temperature partition function to a $d$-dimensional integral.

Findings

Effective representation depends only on static fields

Formalism applied successfully to scalar and Dirac fields

Provides a non-local classical-like partition function

Abstract

We introduce the boundary conditions corresponding to the imaginary-time (Matsubara) formalism for the finite-temperature partition function in $d+1$ dimensions as {\em constraints} in the path integral for the vacuum amplitude (the zero-temperature partition function). We implement those constraints by using Lagrange multipliers, which are static fields, two of them associated to each physical degree of freedom. After integrating out the original, physical fields, we obtain an effective representation for the partition function, depending only on the Lagrange multipliers. The resulting functional integral has the appealing property of involving only $d$-dimensional, {\em time independent} fields, looking like a non local version of the classical partition function. We analyze the main properties of this novel representation for the partition function, developing the formalism within the context of two concrete examples: the real scalar and Dirac fields.