Thermal time as an unsharp observable

TL;DR

This paper demonstrates that the thermal time concept for quantum systems can be represented as an unsharp observable using positive operator valued measures, with extensions to relativistic particles and noncommutative systems.

Contribution

It introduces a novel representation of thermal time as an unsharp observable and extends this framework to other physical systems.

Findings

Thermal time for quantum harmonic oscillator can be described as a positive operator valued measure.

Extensions to free massless relativistic particles are established.

Results apply to systems with equilibrium states defined by noncommutative integrals.

Abstract

We show that the Connes-Rovelli thermal time associated with the quantum harmonic oscillator can be described as an (unsharp) observable, that is, as a positive operator valued measure. We furthermore present extensions of this result to the free massless relativistic particle in one dimension and to a hypothetical physical system whose equilibrium state is given by the noncommutative integral.