Relative State Counting for Semiclassical Black Holes

Area of Science:

• Theoretical physics
• Quantum gravity
• Black hole thermodynamics

Background:

• Entropy differences in perturbative quantum gravity lack clear physical interpretation.
• Classical statistical mechanics defines entropy differences but not absolute entropy.
• Perturbative quantum gravity calculations often require specifying an ultraviolet completion.

Purpose of the Study:

• To interpret entropy differences in perturbative quantum gravity.
• To construct perturbative black hole states with interpretable entropy differences.
• To explore the role of algebra types in entropy definition.

Main Methods:

• Analysis of mass fluctuation algebra around black hole backgrounds.
• Coupling mass fluctuations to quantum matter to embed the algebra within a type II factor.
• Calculation of type II entropy differences for microcanonical wave functions.

Main Results:

• The algebra of mass fluctuations is type I but not a factor, lacking canonical entropy.
• Coupling to quantum matter yields a type II factor where entropy differences are defined.
• Type II entropy difference for microcanonical states equals the log dimension of an extra Hilbert space.

Conclusions:

• Entropy differences in perturbative quantum gravity can be physically interpreted as state counting.
• The framework allows for well-defined entropy differences within type II factors.
• One-shot entropy differences offer interpretation where von Neumann entropy differences do not.