Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories

TL;DR

This paper explores how the flow of time in covariant quantum theories can be derived from thermodynamical states using von Neumann algebra automorphisms, linking quantum, gravitational, and thermodynamic concepts.

Contribution

It introduces a framework based on the Tomita-Takesaki theorem to derive a physical time-flow from thermal states in covariant quantum theories, unifying several fundamental ideas.

Findings

Time-flow can be derived from thermal states using von Neumann algebra automorphisms.

The classical limit of this time-flow relates to known phenomena like Unruh temperature and Hawking radiation.

A state-independent notion of time is identified via outer automorphisms.

Abstract

We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the "timelessness" of the hypothetical fundamental general covariant quantum field theory with the "evidence" of the flow of time. By using the algebraic formulation of quantum theory, we propose a unifying perspective on these problems, based on the hypothesis that in a generally covariant quantum theory the physical time-flow is not a universal property of the mechanical theory, but rather it is determined by the thermodynamical state of the system ("thermal time hypothesis"). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita-Takesaki theorem, which allows to derive a time-flow, namely a one-parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time-flow, its classical limit, and we relate it to various characteristic theoretical facts, as the Unruh temperature and the Hawking radiation. We also point out the existence of a state-independent notion of "time", given by the canonical one-parameter subgroup of outer automorphisms provided by the Cocycle Radon-Nikodym theorem.