Noncommutative Spectral Invariants and Black Hole Entropy

TL;DR

This paper introduces noncommutative geometric invariants derived from conformal nets to analyze quantum entropy, proposing a link between noncommutative area and black hole entropy in quantum gravity.

Contribution

It develops a method to define noncommutative spectral invariants and entropy, connecting them to conformal field theory parameters and black hole entropy.

Findings

Leading noncommutative area proportional to central charge c

First order correction proportional to log of the global index μ_A

Proposed link between noncommutative area and classical black hole entropy

Abstract

We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the ``heat kernel'' semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system with infinitely many degrees of freedom, we regard these coefficients as noncommutative geometric invariants. Under a natural modularity assumption, the leading term of the entropy (noncommutative area) is proportional to the central charge c, the first order correction (noncommutative Euler characteristic) is proportional to log\mu_A, where \mu_A is the global index of A, and the second spectral invariant is again proportional to c. We give a further general method to define a mean entropy by considering conformal symmetries that preserve a discretization of S^1 and we get the same value proportional to c. We then make the corresponding analysis with the proper Hamiltonian associated to an interval. We find here, in complete generality, a proper mean entropy proportional to log\mu_A with a first order correction defined by means of the relative entropy associated with canonical states. By considering a class of black holes with an associated conformal quantum field theory on the horizon, and relying on arguments in the literature, we indicate a possible way to link the noncommutative area with the Bekenstein-Hawking classical area description of entropy.