Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories

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 generally 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 us 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, such as the Unruh temperature and the Hawking radiation. We point out the existence of a state-independent notion of `time', given by the canonical one-parameter subgroup of outer automorphisms provided by the co-cycle Radon--Nikodym theorem.