Spacetime Dependence of Local Temperature in Relativistic Quantum Field Theory

TL;DR

This paper investigates how local temperature varies in spacetime for quantum Klein-Gordon fields, revealing conditions under which states resemble thermal equilibrium or local thermal states, with implications for relativistic thermodynamics.

Contribution

It characterizes the spacetime dependence of inverse temperature vectors in quantum fields satisfying the local KMS condition, extending previous LTE results and highlighting discrepancies in massless cases.

Findings

States can be extended to LKMS states with linearly dependent temperature vectors

States can be extended to global KMS states with constant temperature

Results reveal differences with classical relativistic thermodynamics in massless cases

Abstract

The spacetime dependence of the inverse temperature four-vector $\boldsymbol{\beta}$ for certain states of the quantized Klein-Gordon field on (parts of) Minkowski spacetime is discussed. These states fulfill a recently proposed version of the Kubo-Martin-Schwinger (KMS) boundary value condition, the so-called "local KMS (LKMS) condition". It turns out that, depending on the mass parameter $m\geq 0$, any such state can be extended either (i) to a LKMS state on some forward or backward lightcone, with $\boldsymbol{\beta}$ depending linearily on spacetime, or (ii) to a thermal equilibrium (KMS) state on all of Minkowski space with constant $\boldsymbol{\beta}$. This parallels previously known results for local thermal equilibrium (LTE) states of the quantized Klein-Gordon field. Furthermore, in the case of a massless field our results point to a discrepancy with some classic results in general approaches to (non-quantum) relativistic thermodynamics.