A classical model for semiclassical state-counting

TL;DR

This paper explores the concept of state-counting in semiclassical gravity using an analogy with classical phase space, clarifying how classical structures emerge from quantum theory in the semiclassical limit.

Contribution

It provides a classical analogy to explain features of relative state-counting in semiclassical gravity, linking quantum gauge-invariant operators to classical symplectomorphisms.

Findings

Entropy differences relate to phase space volume ratios.

Classical symplectomorphisms emerge from quantum gauge-invariant operators.

The analogy clarifies how classical structures are inherited from quantum theory.

Abstract

In the type II von Neumann algebras that appear in semiclassical gravity, all states have infinite entropy, but entropy differences are uniquely defined. Akers and I have shown that the entropy difference of microcanonical states has a relative state-counting interpretation in terms of the additional (finite) number of degrees of freedom that are needed to represent the "larger-entropy" state supposing that one already has a representation of the "smaller-entropy" state, and supposing that one is restricted to act with gauge-invariant operators. This short paper explains some of the curious features of relative state-counting by analogy to the classical limit of quantum statistical mechanics. In this analogy the preferred family of renormalized traces becomes the preferred family of symplectic measures on phase space; the trace-index of infinite-dimensional subspaces becomes the ratio of phase space volumes; and the restriction that one must act with gauge-invariant operators becomes the restriction that one must act with symplectomorphisms. Because in the phase-space analogy one has exact control over the quantum deformation away from the classical theory, one can see precisely how the relevant aspects of the classical structure are inherited from the quantum theory -- though even in this simple setting, it is a nontrivial technical task to show how classical symplectomorphisms emerge from the underlying quantum theory in the $\hbar \to 0$ limit.