Abstract
We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann zeta function. It manifests itself in particular by the values of the coefficients {c(d)} by which it multiplies the d dimensional terms in the heat expansion of the spectral triple. We find that {c(d)} is the product of the Riemann xi function evaluated at {-d} by an elementary expression. In particular {c(4)} is a rational multiple of {zeta(5)} and {c(2)} a rational multiple of {zeta(3)}. The functional equation gives a duality between the coefficients in positive dimension, which govern the high energy expansion, and the coefficients in negative dimension, exchanging even dimension with odd dimension.
Highlights
Is an additive functional of the spectral triple (A, H, D), i.e. when evaluated on the direct sum of two spectral triples it behaves additively
Motivated by the formalism of Quantum field theory, one gets another natural way to obtain an additive functional of spectral triples: given (A, H, D) one performs the fermionic second quantization using the Clifford algebra of the underlying real Hilbert space HR and evaluating the von Neumann information theoretic entropy of the unique state satisfying the KMS condition at inverse temperature β with respect to the time evolution of the Clifford algebra induced by the operator D
The second quantization transforms direct sums into tensor products and the von Neumann entropy of the tensor product of two states is the sum of the respective entropies
Summary
Is an additive functional of the spectral triple (A, H, D), i.e. when evaluated on the direct sum of two spectral triples it behaves additively. Let C := CliffC(HR) be the complexified Clifford algebra of the underlying real Hilbert space HR and σt ∈ Aut(C) be the one-parameter group of automorphisms associated to exp(it D) ∈ Aut(HR). One applies the existence and uniqueness of KMS states on a matrix algebra to the subalgebra of C = CliffC(HR) associated to the subspace corresponding to a finite dimensional spectral projection of D.
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