Noncommutative spectral invariants and black hole entropy

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.