An Analogue of the Kac-Wakimoto Formula and Black Hole Conditional Entropy

TL;DR

This paper derives a local formula for superselection sector dimensions in Quantum Field Theory, linking black hole thermodynamics with algebraic entropy, and showing quantization of free energy differences related to Hawking temperature.

Contribution

It introduces a model-independent local formula for superselection sectors and connects it to black hole entropy and thermodynamics using algebraic methods.

Findings

The relative free energy between black hole states is proportional to the variation of conditional entropy.

The free energy difference is quantized and related to the logarithm of rational numbers.

The formula applies broadly in Quantum Field Theory and provides insights into black hole thermodynamics.

Abstract

A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes-Stoermer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed.