Thermal boundary conditions as constraints

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.