Temperature-reflection I: field theory, ensembles, and interactions

TL;DR

This paper investigates the invariance of partition functions under temperature reflection (T-reflection) in quantum field theories, clarifying which functions should be invariant and exploring implications for various systems and interactions.

Contribution

It establishes that finite-temperature path integrals for QFTs are T-reflection invariant and analyzes conditions under which partition functions exhibit this symmetry, including non-unitary and interacting systems.

Findings

Multi-particle partition functions are T-reflection invariant

Single-particle partition functions may not be invariant without extension

T-reflection invariance is unrelated to time-reversal symmetry

Abstract

In this paper, we revisit the claim that many partition functions are invariant under reflecting temperatures to negative values (T-reflection). The goal of this paper is to demarcate which partition functions should be invariant under T-reflection, and why. Our main claim is that finite-temperature path integrals for quantum field theories (QFTs) should be T-reflection invariant. Because multi-particle partition functions are equal to Euclidean path integrals for QFTs, we expect them to be T-reflection invariant. Single-particle partition functions though are often not invariant under T-reflection. Several exactly solvable systems are non-invariant under naive T-reflection, but are likely invariant under an extended T-reflection. We give example systems that are T-reflection invariant but are (1) non-unitary, (2) chiral, (3) interacting, (4) non-supersymmetric, or (5) non-conformal, and (6) argue that T-reflection is unrelated to time-reversal. Finally, we study the interplay between T-reflection and perturbation theory in the anharmonic harmonic oscillator in quantum mechanics and in Yang-Mills in four-dimensions. This is the first in a series of papers on temperature-reflections.