Hamiltonian finite-temperature quantum field theory from its vacuum on partially compactified space

TL;DR

This paper introduces a novel Hamiltonian approach to finite-temperature quantum field theory by relating the partition function to vacuum energy on a partially compactified space, simplifying calculations and avoiding thermal averages.

Contribution

It presents a new method to analyze finite-temperature quantum field theories using vacuum wave functionals on compactified spaces, applicable to both free gases and interacting gauge theories.

Findings

Reproduces known results for Bose and Fermi gases.

Calculates Yang-Mills pressure as a function of temperature.

Provides a mathematically well-defined framework for finite-temperature QFT.

Abstract

The partition function of a relativistic invariant quantum field theory is expressed by its vacuum energy calculated on a spatial manifold with one dimension compactified to a 1-sphere $S^1 (\beta)$, whose circumference $\beta$ represents the inverse temperature. Explicit expressions for the usual energy density and pressure in terms of the energy density on the partially compactified spatial manifold $\mathbb{R}^2 \times S^1 (\beta)$ are derived. To make the resulting expressions mathematically well-defined a Poisson resummation of the Matsubara sums as well as an analytic continuation in the chemical potential are required. The new approach to finite-temperature quantum field theories is advantageous in a Hamilton formulation since it does not require the usual thermal averages with the density operator. Instead, the whole finite-temperature behaviour is encoded in the vacuum wave functional on the spatial manifold $\mathbb{R}^2 \times S^1 (\beta)$. We illustrate this approach by calculating the pressure of a relativistic Bose and Fermi gas and reproduce the known results obtained from the usual grand canonical ensemble. As a first non-trivial application we calculate the pressure of Yang-Mills theory as function of the temperature in a quasi-particle approximation motivated by variational calculations in Coulomb gauge.