From Monge to Higgs: a survey of distance computations in noncommutative geometry. (English) Zbl 1361.81077
Martinetti, Pierre (ed.) et al., Noncommutative geometry and optimal transport. Workshop on noncommutative geometry and optimal transport, Besançon, France, November 27, 2014. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2297-4/pbk; 978-1-4704-3560-8/ebook). Contemporary Mathematics 676, 1-46 (2016).
Summary: This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, and the comparison of Connes distance with the minimal length that emerges in various models of quantum spacetime. Links with other areas of mathematics are studied, in particular the horizontal distance in sub-Riemannian geometry. The interpretation of Connes distance as a noncommutative version of the Monge-Kantorovich metric in optimal transport is also discussed.
For the entire collection see [Zbl 1353.46001].
For the entire collection see [Zbl 1353.46001].
MSC:
| 81R60 | Noncommutative geometry in quantum theory |
| 58B34 | Noncommutative geometry (à la Connes) |
| 81V25 | Other elementary particle theory in quantum theory |
| 83C65 | Methods of noncommutative geometry in general relativity |
| 83F05 | Relativistic cosmology |
| 81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |
References:
| [1] | Amelino-Camelia, Giovanni; Gubitosi, Giulia; Mercati, Flavio, Discreteness of area in noncommutative space, Phys. Lett. B, 676, 4-5, 180-183 (2009) · doi:10.1016/j.physletb.2009.04.045 |
| [2] | Bahns, D.; Doplicher, S.; Fredenhagen, K.; Piacitelli, G., Quantum geometry on quantum spacetime: distance, area and volume operators, Comm. Math. Phys., 308, 3, 567-589 (2011) · Zbl 1269.83027 · doi:10.1007/s00220-011-1358-y |
| [3] | J. V. Bellissard, M. Marcolli, and K. Reihani, Dynamical systems on spectral metric spaces, arXiv:1008.4617v1 [math.OA] (2010). |
| [4] | Bimonte, G.; Lizzi, F.; Sparano, G., Distances on a lattice from non-commutative geometry, Phys. Lett. B, 341, 2, 139-146 (1994) · doi:10.1016/0370-2693(94)90302-6 |
| [5] | Bratteli, Ola; Robinson, Derek W., Operator algebras and quantum statistical mechanics. 1, Texts and Monographs in Physics, xiv+505 pp. (1987), Springer-Verlag, New York · Zbl 0905.46046 · doi:10.1007/978-3-662-02520-8 |
| [6] | Cagnache, Eric; D’Andrea, Francesco; Martinetti, Pierre; Wallet, Jean-Christophe, The spectral distance in the Moyal plane, J. Geom. Phys., 61, 10, 1881-1897 (2011) · Zbl 1226.81095 · doi:10.1016/j.geomphys.2011.04.021 |
| [7] | Cagnache, Eric; Wallet, Jean-Christophe, Spectral distances: results for Moyal plane and noncommutative torus, SIGMA Symmetry Integrability Geom. Methods Appl., 6, Paper 026, 17 pp. (2010) · Zbl 1190.58010 · doi:10.3842/SIGMA.2010.026 |
| [8] | Chamseddine, Ali H.; Connes, Alain, The spectral action principle, Comm. Math. Phys., 186, 3, 731-750 (1997) · Zbl 0894.58007 · doi:10.1007/s002200050126 |
| [9] | Chamseddine, Ali H.; Connes, Alain; Marcolli, Matilde, Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys., 11, 6, 991-1089 (2007) · Zbl 1140.81022 |
| [10] | A. H. Chamseddine, A. Connes, and Walter van Suijlekom, Beyond the spectral standard model: emergence of Pati-Salam unification, JHEP 11 (2013), 132. · Zbl 1388.81965 |
| [11] | Christensen, Erik; Ivan, Cristina, Spectral triples for AF \(C^*\)-algebras and metrics on the Cantor set, J. Operator Theory, 56, 1, 17-46 (2006) · Zbl 1111.46052 |
| [12] | Christensen, Erik; Ivan, Cristina; Schrohe, Elmar, Spectral triples and the geometry of fractals, J. Noncommut. Geom., 6, 2, 249-274 (2012) · Zbl 1244.28010 · doi:10.4171/JNCG/91 |
| [13] | Connes, A., Compact metric spaces, Fredholm modules, and hyperfiniteness, Ergodic Theory Dynam. Systems, 9, 2, 207-220 (1989) · Zbl 0718.46051 · doi:10.1017/S0143385700004934 |
| [14] | Connes, Alain, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys., 182, 1, 155-176 (1996) · Zbl 0881.58009 |
| [15] | Connes, Alain; Lott, John, The metric aspect of noncommutative geometry. New symmetry principles in quantum field theory (Carg\`“ese, 1991), NATO Adv. Sci. Inst. Ser. B Phys. 295, 53-93 (1992), Plenum, New York · Zbl 1221.58007 |
| [16] | Connes, Alain, On the spectral characterization of manifolds, J. Noncommut. Geom., 7, 1, 1-82 (2013) · Zbl 1287.58004 · doi:10.4171/JNCG/108 |
| [17] | Kastler, Daniel; Testard, Daniel, Quantum forms of tensor products, Comm. Math. Phys., 155, 1, 135-142 (1993) · Zbl 0786.58007 |
| [18] | Dai, Jian; Song, Xing-Chang, Pythagoras’s theorem on a two-dimensional lattice from a “natural” Dirac operator and Connes’s distance formula, J. Phys. A, 34, 27, 5571-5581 (2001) · Zbl 0985.46043 · doi:10.1088/0305-4470/34/27/307 |
| [19] | F. D’Andrea, Pythagoras theorem in noncommutative geometry, to be published in Comtemp. Math. arXiv 1507.08773 [math-ph] arXiv: (2015). |
| [20] | D’Andrea, Francesco; Lizzi, Fedele; Martinetti, Pierre, Spectral geometry with a cut-off: topological and metric aspects, J. Geom. Phys., 82, 18-45 (2014) · Zbl 1294.58002 · doi:10.1016/j.geomphys.2014.03.014 |
| [21] | D’Andrea, Francesco; Martinetti, Pierre, On Pythagoras theorem for products of spectral triples, Lett. Math. Phys., 103, 5, 469-492 (2013) · Zbl 1348.58004 · doi:10.1007/s11005-012-0598-x |
| [22] | D’Andrea, Francesco; Martinetti, Pierre, A view on optimal transport from noncommutative geometry, SIGMA Symmetry Integrability Geom. Methods Appl., 6, Paper 057, 24 pp. (2010) · Zbl 1227.82066 · doi:10.3842/SIGMA.2010.057 |
| [23] | A. Devastato, Noncommutative geometry, grand symmetry and twisted spectral triple, J. Phys.: Conf. Ser. 634 (2015), 012008. |
| [24] | A. Devastato, F. Lizzi, and P. Martinetti, Grand Symmetry, Spectral Action and the Higgs mass, JHEP 01 (2014), 042. · Zbl 1338.81395 |
| [25] | S. Doplicher, Spacetime and fields, a quantum texture, Proceedings 37th Karpacz Winter School of Theo. Physics (2001), 204-213. |
| [26] | \bysame, Quantum field theory on quantum spacetime, J. Phys.: Conf. Ser. 53 (2006), 793-798. |
| [27] | Doplicher, Sergio; Fredenhagen, Klaus; Roberts, John E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys., 172, 1, 187-220 (1995) · Zbl 0847.53051 |
| [28] | Dubois-Violette, M.; Kerner, R.; Madore, J., Classical bosons in a noncommutative geometry, Classical Quantum Gravity, 6, 11, 1709-1724 (1989) · Zbl 0675.53071 |
| [29] | Franco, Nicolas, Global eikonal condition for Lorentzian distance function in noncommutative geometry, SIGMA Symmetry Integrability Geom. Methods Appl., 6, Paper 064, 11 pp. (2010) · Zbl 1217.58006 · doi:10.3842/SIGMA.2010.064 |
| [30] | Gayral, V.; Gracia-Bond{\'{\i }}a, J. M.; Iochum, B.; Sch{\`“u}cker, T.; V{\'”a}rilly, J. C., Moyal planes are spectral triples, Comm. Math. Phys., 246, 3, 569-623 (2004) · Zbl 1084.58008 · doi:10.1007/s00220-004-1057-z |
| [31] | Goodearl, K. R., Notes on real and complex \(C^{\ast } \)-algebras, Shiva Mathematics Series 5, iv+211 pp. (1982), Shiva Publishing Ltd., Nantwich · Zbl 0495.46039 |
| [32] | Grosse, Harald; Wulkenhaar, Raimar, 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory, J. Geom. Phys., 62, 7, 1583-1599 (2012) · Zbl 1243.58005 · doi:10.1016/j.geomphys.2012.03.005 |
| [33] | S. Hasselmann, Spectral triples on Carnot manifolds, arXiv 1404.5494 (2014). |
| [34] | Iochum, Bruno; Krajewski, Thomas; Martinetti, Pierre, Distances in finite spaces from noncommutative geometry, J. Geom. Phys., 37, 1-2, 100-125 (2001) · Zbl 0983.46053 · doi:10.1016/S0393-0440(00)00044-9 |
| [35] | Kadison, Richard V.; Ringrose, John R., Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics 100, xv+398 pp. (1983), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York · Zbl 0888.46039 |
| [36] | L. V. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk. SSSR 37 (1942), 227-229. |
| [37] | F. Latremoliere, Quantum metric space and the Gromov-Hausdorff propinquity, to be published in Contemp. Maths. arXiv 1506.04341v2 (2015). |
| [38] | Lisini, Stefano, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations, 28, 1, 85-120 (2007) · Zbl 1132.60004 · doi:10.1007/s00526-006-0032-2 |
| [39] | Madore, J., An introduction to noncommutative differential geometry and its physical applications, London Mathematical Society Lecture Note Series 206, iv+200 pp. (1995), Cambridge University Press, Cambridge · Zbl 0842.58002 |
| [40] | P. Martinetti, Distances en geometrie non-commutative, PhD thesis (2001), arXiv:math-ph/0112038v1. |
| [41] | \bysame, Towards a Monge-Kantorovich distance in noncommutative geometry, Zap. Nauch. Semin. POMI 411 (2013). |
| [42] | Martinetti, Pierre; Mercati, Flavio; Tomassini, Luca, Minimal length in quantum space and integrations of the line element in noncommutative geometry, Rev. Math. Phys., 24, 5, 1250010, 36 pp. (2012) · Zbl 1253.81080 · doi:10.1142/S0129055X12500109 |
| [43] | Martinetti, Pierre; Tomassini, Luca, Noncommutative geometry of the Moyal plane: translation isometries, Connes’ distance on coherent states, Pythagoras equality, Comm. Math. Phys., 323, 1, 107-141 (2013) · Zbl 1283.81092 · doi:10.1007/s00220-013-1760-8 |
| [44] | Martinetti, Pierre, Carnot-Carath\'eodory metric and gauge fluctuation in noncommutative geometry, Comm. Math. Phys., 265, 3, 585-616 (2006) · Zbl 1107.58011 · doi:10.1007/s00220-006-0001-9 |
| [45] | Martinetti, Pierre, Smoother than a circle. Modern trends in geometry and topology, 283-293 (2006), Cluj Univ. Press, Cluj-Napoca · Zbl 1138.46043 |
| [46] | Martinetti, Pierre, Spectral distance on the circle, J. Funct. Anal., 255, 7, 1575-1612 (2008) · Zbl 1151.58004 · doi:10.1016/j.jfa.2008.07.018 |
| [47] | Pierre Martinetti and Luca Tomassini, Length and distance on a quantum space, Proc. of Sciences 042 (2011). · Zbl 1283.81092 |
| [48] | Martinetti, Pierre; Wulkenhaar, Raimar, Discrete Kaluza-Klein from scalar fluctuations in noncommutative geometry, J. Math. Phys., 43, 1, 182-204 (2002) · Zbl 1059.58021 · doi:10.1063/1.1418012 |
| [49] | B. Mesland and A. Rennie, Nonunital spectral triples and metric completness in unbounded kk-theory, (2015). · Zbl 1345.19003 |
| [50] | Montgomery, Richard, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs 91, xx+259 pp. (2002), American Mathematical Society, Providence, RI · Zbl 1044.53022 |
| [51] | Moretti, Valter, Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes, Rev. Math. Phys., 15, 10, 1171-1217 (2003) · Zbl 1055.46048 · doi:10.1142/S0129055X03001886 |
| [52] | Nestruev, Jet, Smooth manifolds and observables, Graduate Texts in Mathematics 220, xiv+222 pp. (2003), Springer-Verlag, New York · Zbl 1021.58001 |
| [53] | Paolini, Emanuele; Stepanov, Eugene, Decomposition of acyclic normal currents in a metric space, J. Funct. Anal., 263, 11, 3358-3390 (2012) · Zbl 1254.49027 · doi:10.1016/j.jfa.2012.08.009 |
| [54] | Rieffel, Marc A., Compact quantum metric spaces. Operator algebras, quantization, and noncommutative geometry, Contemp. Math. 365, 315-330 (2004), Amer. Math. Soc., Providence, RI · Zbl 1084.46060 · doi:10.1090/conm/365/06709 |
| [55] | Rieffel, Marc A., Metrics on states from actions of compact groups, Doc. Math., 3, 215-229 (electronic) (1998) · Zbl 0993.46043 |
| [56] | Rieffel, Marc A., Metrics on state spaces, Doc. Math., 4, 559-600 (electronic) (1999) · Zbl 0945.46052 |
| [57] | Carlo Rovelli, Lorentzian Connes distance, spectral graph distance and loop gravity, arXiv 1408.3260 (2014). |
| [58] | van Suijlekom, Walter D., Noncommutative geometry and particle physics, Mathematical Physics Studies, xvi+237 pp. (2015), Springer, Dordrecht · Zbl 1305.81007 · doi:10.1007/978-94-017-9162-5 |
| [59] | Villani, C{\'e}dric, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338, xxii+973 pp. (2009), Springer-Verlag, Berlin · Zbl 1156.53003 · doi:10.1007/978-3-540-71050-9 |
| [60] | Wallet, Jean-Christophe, Connes distance by examples: homothetic spectral metric spaces, Rev. Math. Phys., 24, 9, 1250027, 26 pp. (2012) · Zbl 1256.46040 · doi:10.1142/S0129055X12500274 |
| [61] | Weaver, Nik, Lipschitz algebras, xiv+223 pp. (1999), World Scientific Publishing Co., Inc., River Edge, NJ · Zbl 0936.46002 · doi:10.1142/4100 |
| [62] | D. Zaev, Lp-Wasserstein distances on state and quasi-state spaces of c*-algebra, (arxiv 1505.06061). |
| [63] | D. Zaev, Lp-Wasserstein distances on state and quasi-state spaces of c*-algebra, (arxiv 1505.06061). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
