Statistical mechanics of generally covariant quantum theories: A Boltzmann-like approach

TL;DR

This paper explores how to apply statistical mechanics to generally covariant quantum theories lacking a traditional Hamiltonian, using a Boltzmann-like approach to define thermodynamics without external time.

Contribution

It introduces a framework for statistical mechanics in covariant quantum theories, demonstrating its validity and applying it to a model resembling general relativity.

Findings

Statistical mechanics can be formulated without a notion of energy or external time.

Thermodynamic parameters are derived from thermalizing interactions.

Application to a model with a single degree of freedom mimicking general relativity's algebra.

Abstract

We study the possibility of applying statistical mechanics to generally covariant quantum theories with a vanishing Hamiltonian. We show that (under certain appropiate conditions) this makes sense, in spite of the absence of a notion of energy and external time. We consider a composite system formed by a large number of identical components, and apply Boltzmann's ideas and the fundamental postulates of ordinary statistical physics. The thermodynamical parameters are determined by the properties of the thermalizing interaction. We apply these ideas to a simple example, in which the component system has one physical degree of freedom and mimics the constraint algebra of general relativity.