Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 057, 24 pages      arXiv:0906.1267
Contribution to the Special Issue “Noncommutative Spaces and Fields”

A View on Optimal Transport from Noncommutative Geometry

Francesco D'Andrea a and Pierre Martinetti b
a) Ecole de Mathématique, Univ. Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-La-Neuve, Belgium
b) Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

Received April 14, 2010, in final form July 08, 2010; Published online July 20, 2010

We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space Rn, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation.

Key words: noncommutative geometry; spectral triples; transport theory.

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  1. Ambrosio L., Lecture notes on optimal transport problems, in Mathematical Aspects of Evolving Interfaces (Funchal, 2000), Lecture Notes in Math., Vol. 1812, Springer, Berlin, 2003, 1-52.
  2. Azagra D., Ferrera J., López-Mesas F., Rangel Y., Smooth approximation of Lipschitz functions on Riemannian manifolds, J. Math. Anal. Appl. 326 (2007), 1370-1378, math.DG/0602051.
  3. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. 1. C*- and W*-algebras, symmetry groups, decomposition of states, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987.
  4. Biane P., Voiculescu D., A free probability analogue of the Wasserstein metric on the trace-state space, Geom. Funct. Anal. 11 (2001), 1125-1138, math.OA/0006044.
  5. Bimonte G., Lizzi F., Sparano G., Distances on a lattice from non-commutative geometry, Phys. Lett. B 341 (1994), 139-146, hep-lat/9404007.
  6. Brenier Y., Extended Monge-Kantorovich theory, in Optimal Transportation and Applications (Martina Franca, 2001), Lecture Notes in Math., Vol. 1813, Springer, Berlin, 2003, 91-121.
  7. Cagnache E., D'Andrea F., Martinetti P., Wallet J.C., The Spectral distance in the Moyal plane, arXiv:0912.0906.
  8. Chamseddine A.H., Connes A., Marcolli M., Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007), 991-1089, hep-th/0610241.
  9. Chavel I., Riemannian geometry - a modern introduction, Cambridge Tracts in Mathematics, Vol. 108, Cambridge University Press, Cambridge, 1993.
  10. Connes A., Compact metric spaces, Fredholm modules, and hyperfiniteness, Ergodic Theory Dynam. Systems 9 (1989), 207-220.
  11. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  12. Connes A., Noncommutative geometry and reality, J. Math. Phys. 36 (1995), 6194-6231.
  13. Connes A., Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), 155-176.
  14. Connes A., A unitary invariant in Riemannian geometry, Int. J. Geom. Methods Mod. Phys. 5 (2008), 1215-1242, arXiv:0810.2091.
  15. Connes A., Lott J., The metric aspect of noncommutative geometry, in New symmetry principles in quantum field theory (Cargèse, 1991), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 295, Plenum, New York, 1992, 53-93.
  16. Connes A., Marcolli M., Noncommutative geometry, quantum fields and motives, American Mathematical Society Colloquium Publications, Vol. 55, American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008.
  17. Doplicher S., Fredenhagen K., Roberts J.E., Spacetime quantization induced by classical gravity, Phys. Lett. B 331 (1994), 39-44.
  18. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, hep-th/0303037.
  19. Evans L.C., Gangbo W., Differential equations methods for the Monge-Kantorevich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653.
  20. Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Varilly J.C., Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), 569-623, hep-th/0307241.
  21. Givens C.R., Shortt R.M., A class of Wasserstein metrics for probability distributions, Michigan Math. J. 31 (1984), 231-240.
  22. Goodearl K.R., Notes on real and complex C*-algebras, Shiva Mathematics Series, Vol. 5, Shiva Publishing Ltd., Nantwich, 1982.
  23. Gracia-Bondía J.M., Varilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001.
  24. Greene R.E., Wu H., C approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), 47-84.
  25. Gromov M., Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, Vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999.
  26. Iochum B., Krajewski T., Martinetti P., Distances in finite spaces from noncommutative geometry, J. Geom. Phys. 37 (2001), 100-125, hep-th/9912217.
  27. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras. Vol. II. Advanced theory, Pure and Applied Mathematics, Vol. 100, Academic Press, Inc., Orlando, FL, 1986.
  28. Kantorovich L.V., On the transfer of masses, Dokl. Akad. Nauk. SSSR 37 (1942), 227-229.
  29. Kantorovich L.V., Rubinstein G.S., On a space of totally additive functions, Vestnik Leningrad. Univ. 13 (1958), no. 7, 52-59.
  30. Latrémolière F., Bounded-Lipschitz distances on the state space of a C*-algebra, Taiwanese J. Math. 11 (2007), 447-469.
  31. Martinetti P., Distances en géométrie non-commutative, PhD Thesis, math-ph/0112038.
  32. Martinetti P., Carnot-Carathéodory metric and gauge fluctuations in noncommutative geometry, Comm. Math. Phys. 265 (2006), 585-616, hep-th/0506147.
  33. Martinetti P., Spectral distance on the circle, J. Funct. Anal. 255 (2008), 1575-1612, math.OA/0703586.
  34. Martinetti P., Smoother than a circle or how noncommutative geometry provides the torus with an egocentred metric, in Modern Trends in Geometry and Topology (Deva, 2005), Cluj Univ. Press, Cluj-Napoca, 2006, 283-293, hep-th/0603051.
  35. Martinetti P., Wulkenhaar R., Discrete Kaluza-Klein from scalar fluctuations in noncommutative geometry, J. Math. Phys. 43 (2002), 182-204, hep-th/0104108.
  36. Monge G., Mémoire sur la Théorie des Déblais et des Remblais, Histoire de l'Acad. des Sciences de Paris, 1781.
  37. Nash J., C1-isometric imbeddings, Ann. of Math. (2) 60 (1954), 383-396.
  38. Nash J., The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63.
  39. Rieffel M.A., Metric on state spaces, Doc. Math. 4 (1999), 559-600, math.OA/9906151.
  40. Rieffel M.A., Compact quantum metric spaces, in Operator Algebras, Quantization, and Noncommutative Geometry, Contemp. Math., Vol. 365, Amer. Math. Soc., Providence, RI, 2004, 315-330, math.OA/0308207.
  41. Rieffel M.A., Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), no. 796, 1-65, math.OA/0011063.
  42. Roe J., Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, Vol. 90, American Mathematical Society, Providence, RI, 1996.
  43. Villani C., Topics in optimal transportation, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003.
  44. Villani C., Optimal transport. Old and new, Grundlehren der Mathematischen Wissenschaften, Vol. 338, Springer-Verlag, Berlin, 2009.

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