Thermal Time and Irreversibility from Non-Commuting Observables in Accelerated Quantum Systems

TL;DR

This paper explores how temporal ordering becomes meaningful in relativistic quantum systems, showing that non-commuting observables and thermal conditions induce irreversibility and distinguishability in accelerated detectors.

Contribution

It demonstrates that sequential interactions through non-commuting observables in accelerated quantum detectors lead to observable irreversibility and temporal asymmetry.

Findings

Sequential couplings depend on order at second order in interaction.

The reduced detector state shows asymmetry controlled by the KMS parameter.

States form a family of non-commuting Gibbs states with identical spectra.

Abstract

We investigate when temporal ordering becomes operationally meaningful in relativistic quantum field theory using localized detector models. A time parameter alone does not ensure that different sequences of operations are physically distinguishable. We show that distinguishability arises when the state satisfies the Kubo--Martin--Schwinger (KMS) condition and the detector couples through non-commuting observables. We consider uniformly accelerated two-level detectors interacting with a quantum field in the Minkowski vacuum. The restriction of the vacuum to the detector trajectory induces a thermal response characterized by the Unruh temperature and the Tolman profile. For sequential couplings through distinct observables, the reduced detector state depends on the ordering of interactions already at second order, with a dependence controlled by the KMS parameter. This asymmetry is quantified using quantum relative entropy. In a minimal model, the relevant states form a family of non-commuting Gibbs states with identical spectra and different generators, yielding a closed-form expression depending only on the dimensionless combination of temperature and detector energy scale.