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Entropy and the Spectral Action

Communications in Mathematical Physics volume 373, pages 457–471 (2020)Cite this article

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.

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Acknowledgements

The work of A. H. C is supported in part by the National Science Foundation Grant No. Phys-1518371. A. H. C would also like to thank Institute forMathematics, Astrophysics and Particle Physics, Radboud University Nijmegen for hospitality where part of this work was done. The involved research has partly been enabled by a Radboud Excellence Professorship awarded to Prof. Chamseddine. WvS would like to thank Sijbrand de Jong for stimulating discussions on entropy and the spectral action. WvS thanks IHÉS for hospitality during a visit in early 2018, as well as NWO for support via VIDI-Grant 016.133.326.

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Affiliations

  1. Physics Department, American University of Beirut, Beirut, Lebanon

    Ali H. Chamseddine

  2. College de France, 3 rue Ulm, 75005, Paris, France

    Alain Connes

  3. I.H.E.S., 91440, Bures-sur-Yvette, France

    Ali H. Chamseddine & Alain Connes

  4. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Alain Connes

  5. Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

    Ali H. Chamseddine & Walter D. van Suijlekom

Authors
  1. Ali H. Chamseddine
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  2. Alain Connes
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  3. Walter D. van Suijlekom
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Corresponding author

Correspondence to Walter D. van Suijlekom.

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Communicated by Y. Kawahigashi

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Chamseddine, A.H., Connes, A. & van Suijlekom, W.D. Entropy and the Spectral Action. Commun. Math. Phys. 373, 457–471 (2020). https://doi.org/10.1007/s00220-019-03297-8

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  • DOI: https://doi.org/10.1007/s00220-019-03297-8