Relative state-counting for semiclassical black holes

TL;DR

This paper develops a framework for interpreting entropy differences in perturbative quantum gravity black hole states as relative state counts, using type II von Neumann algebras to give physical meaning to these differences.

Contribution

It introduces a method to interpret entropy differences in black hole states as relative state counts via type II algebra constructions, clarifying their physical significance.

Findings

Type II entropy differences correspond to the logarithm of the dimension of an auxiliary Hilbert space.

Coupling mass fluctuations to quantum matter embeds the algebra into a type II factor, enabling meaningful entropy differences.

The approach applies to microcanonical wavefunctions and suggests broader relevance for one-shot entropy differences.

Abstract

It has been shown that entropy differences between certain states of perturbative quantum gravity can be computed without specifying an ultraviolet completion. This is analogous to the situation in classical statistical mechanics, where entropy differences are defined but absolute entropy is not. Unlike in classical statistical mechanics, however, the entropy differences computed in perturbative quantum gravity do not have a clear physical interpretation. Here we construct a family of perturbative black hole states for which the entropy difference can be interpreted as a relative counting of states. Conceptually, this paper begins with the algebra of mass fluctuations around a fixed black hole background, and points out that while this is a type I algebra, it is not a factor and therefore has no canonical definition of entropy. As in previous work, coupling the mass fluctuations to quantum matter embeds the mass algebra within a type II factor, in which entropy differences (but not absolute entropies) are well defined. It is then shown that for microcanonical wavefunctions of mass fluctuation, the type II entropy difference equals the logarithm of the dimension of the extra Hilbert space that is needed to map one microcanonical window to another using gauge-invariant unitaries. The paper closes with comments on type II entropy difference in a more general class of states, where the von Neumann entropy difference does not have a physical interpretation, but "one-shot" entropy differences do.