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Noncommutative Spectral Invariants and Black Hole Entropy

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Abstract.

We consider an intrinsic entropy associated with a local conformal net 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 , where is the global index of , 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 S1 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 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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Yasuyuki Kawahigashi

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Roma, Italy

    Roberto Longo

Authors
  1. Yasuyuki Kawahigashi
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  2. Roberto Longo
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Correspondence to Yasuyuki Kawahigashi.

Additional information

Communicated by A. Connes

Supported in part by GNAMPA and MIUR.

Supported in part by JSPS.

Dedicated to Richard V. Kadison on the occasion of his eightieth birthday

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Kawahigashi, Y., Longo, R. Noncommutative Spectral Invariants and Black Hole Entropy. Commun. Math. Phys. 257, 193–225 (2005). https://doi.org/10.1007/s00220-005-1322-9

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