Journal of Mathematical Sciences

, Volume 196, Issue 2, pp 165–174 | Cite as

Towards a Monge − Kantorovich Metric in Noncommutative Geometry

Article

We investigate whether the identification between Connes’s spectral distance in noncommutative geometry and the Monge–Kantorovich distance of order 1 in the theory of optimal transport which has been pointed out by Rieffel in the commutative case still makes sense in a noncommutative framework. To this aim, given a spectral triple (\( \mathcal{A} \), \( \mathcal{H} \), \( \mathcal{D} \)) with noncommutative \( \mathcal{A} \), we introduce a "Monge–Kantorovich"-like distance WD on the space of states of \( \mathcal{A} \), taking as a cost function the spectral distance dD between pure states. We show in full generality that dD ≤ WD, and exhibit several examples where thee quality actually holds true, in particular, on the unit two-ball viewed as the state space of M2(ℂ). We also discuss WD in a two-sheet model (the product of a manifold and ℂ2), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish on the diagonal.

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References

  1. 1.
    P. Biane and D. Voiulesu, "A free probability analogue of the Wasserstein metric in the trace state space," Geom. Funct. Anal., 11, No.6, 1125–1138 (2001).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics.1, Springer, New York (1987).CrossRefMATHGoogle Scholar
  3. 3.
    E. Cagnahe, F. d'Andrea, P. Martinetti, and J.-C. Wallet, "The spectral distance on Moyal plane," J. Geom. Phys., 61, 1881–1897 (2011).Google Scholar
  4. 4.
    A. H. Chamseddine, A. Connes, and M. Marolli, "Gravity and the standard model with neutrino mixing," Adv. Theor. Math. Phys., 11, 1881–1897 (2007).CrossRefGoogle Scholar
  5. 5.
    The Atlas Collaboration, "Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC" (2012).Google Scholar
  6. 6.
    A. Connes, "Compact metric spaces, Fredholm modules, and hyperfiniteness, "Ergod. Theory Dynam. Systems, 9, 207–220(1989).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    A. Connes, Noncommutative Geometry, Academic Press, San Diego (1994).MATHGoogle Scholar
  8. 8.
    A. Connes, "Gravity coupled with matter and the foundations of noncommutative geometry," Comm. Math. Phys., 182,155–176(1996).CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    A. Connes, "On the spectral characterization of manifolds," arXiv: 0810.2088 (2008).Google Scholar
  10. 10.
    A. Connes and J. Lott, "The metric aspect of noncommutative geometry," NATO Adv. Study Inst., Ser. B Phys., 295, 53–93(1992).Google Scholar
  11. 11.
    F. D'Andrea, F. Lizzi, and P. Martinetti, "Spectral geometry with a cutoff: topological and metric properties," arXiv: 1305.2605.Google Scholar
  12. 12.
    F. D'Andrea and P. Martinetti, "A view on optimal transport from noncommutative geometry," SIGMA, 6, No.057 (2010).Google Scholar
  13. 13.
    B. Iohum, T. Krajewski, and P. Martinetti, "Distances in finite spaces from noncommutative geometry,"J. Geom. Phys., 31, 100–125(2001).CrossRefGoogle Scholar
  14. 14.
    L. V. Kantorovih, "On the translocation of masses," Dokl. Akad. Nauk SSSR, 37, 227–229(1942).Google Scholar
  15. 15.
    L. V. Kantorovih and G. S. Rubinstein, "On a space of totally addictive functions," Vestn. Leningrad. Univ., 13, 52–58(1958).Google Scholar
  16. 16.
    P. Martinetti, "Distances en geometric noncommutative," PhD Thesis (2001); arXiv: math-ph/0112038v1.Google Scholar
  17. 17.
    P. Martinetti, "Spectral distance on the circle," J. Funct. Anal., 255, 15751612(2008).CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    P. Martinetti and R. Wulkenhaar, "Discrete Kaluza–Klein from scalar fluctuations in noncommutative geometry," J. Math. Phys., 43, No.1, 182–204 (2002).Google Scholar
  19. 19.
    M. A. Rieffel, "Metric on state spaces," Doc. Math., 4, 559–600 (1999).MATHMathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.CMTP & Dipartimento di MatematiaUniversità di Roma Tor Vergata, Università di Napoli Federico IIRomeItaly
  2. 2.CMTP & Dipartimento di MatematiaUniversità di Napoli Federico IINapoliItaly

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