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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access May 16, 2017

Quantum Riemannian geometry of phase space and nonassociativity

Edwin J. Beggs and Shahn Majid EMAIL logo
From the journal Demonstratio Mathematica

Abstract

Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of ℂℙn where the commutation relations have the canonical form [wi, w̄j] = iλδij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.

MSC 2010: 81R50; 58B32; 83C57

References

[1] Majid S., Almost commutative Riemannian geometry: wave operators, Commun. Math. Phys., 2012, 310, 569-60910.1007/s00220-012-1416-0Search in Google Scholar

[2] Brody D. C., Hughston L. P., Geometric quantum mechanics, J. Geom. Phys., 2001, 38, 19-5310.1016/S0393-0440(00)00052-8Search in Google Scholar

[3] Beggs E. J., Majid S., Semiclassical differential structures, Pac. J. Math., 2006, 224, 1-4410.2140/pjm.2006.224.1Search in Google Scholar

[4] Beggs E. J., Majid S., Bar categories and star operations, Alg. and Representation Theory, 2009, 12, 103-15210.1007/s10468-009-9141-xSearch in Google Scholar

[5] Beggs E. J., Majid S., .-Compatible connections in noncommutative Riemannian geometry, J. Geom. Phys., 2011, 61, 95-12410.1016/j.geomphys.2010.09.002Search in Google Scholar

[6] Beggs E. J., Majid S., Gravity induced by quantum spacetime, Class. Quantum. Grav., 2014, 31, 03502010.1088/0264-9381/31/3/035020Search in Google Scholar

[7] Beggs E. J., Majid S., Poisson Riemannian geometry, J. Geom. Phys., 2017, 114, 450-49110.1016/j.geomphys.2016.12.012Search in Google Scholar

[8] Connes A., Noncommutative geometry, Academic Press, 1994Search in Google Scholar

[9] Fedosov B. V., Deformation quantisation and index theory, Akademie Verlag, 1996Search in Google Scholar

[10] Hawkins E., Noncommutative rigidity, Commun. Math. Phys., 2004, 246, 218-23210.1007/s00220-004-1036-4Search in Google Scholar

[11] Majid S., Reconstruction and quantisation of Riemannian structures [arXiv:1307.2778 (math.QA)]Search in Google Scholar

[12] Beggs E. J., Majid S., Poisson complex geometry and nonassociativity (in preparation)Search in Google Scholar

[13] Beggs E. J., Smith S. P., Noncommutative complex differential geometry, J. Geom. Phys., 2013, 72, 7-3310.1016/j.geomphys.2013.03.018Search in Google Scholar

[14] Dubois-Violette M., Masson T., On the first-order operators in bimodules, Lett. Math. Phys., 1996, 37, 467-47410.1007/BF00312677Search in Google Scholar

[15] Dubois-Violette M., Michor P. W., Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys., 1996, 20, 218-23210.1016/0393-0440(95)00057-7Search in Google Scholar

[16] Majid S., Noncommutative Riemannian geometry of graphs, J. Geom. Phys., 2013, 69, 74-9310.1016/j.geomphys.2013.02.004Search in Google Scholar

[17] J. Mourad, Linear connections in noncommutative geometry, Class. Quantum Grav., 1995, 12, 965-97410.1088/0264-9381/12/4/007Search in Google Scholar

[18] Penrose R., talk at Workshop on ‘Noncommutative Geometry and Physics: fundamental structure of space and time’, Newton Institute, 2006Search in Google Scholar

[19] Aldrovandi R., Pereira J. G., Teleparallel Gravity: An Introduction, Springer, 201310.1007/978-94-007-5143-9Search in Google Scholar

[20] Majid S., Meaning of noncommutative geometry and the Planck-scale quantum group, Springer Lect. Notes Phys., 2000, 541, 227-27610.1007/3-540-46634-7_10Search in Google Scholar

[21] Majid S., Hopf algebras for physics at the Planck scale, Class. Quantum Grav., 1988, 5, 1587-160710.1088/0264-9381/5/12/010Search in Google Scholar

Received: 2016-08-03
Accepted: 2016-09-05
Published Online: 2017-05-16
Published in Print: 2017-04-25

© 2017 Shahn Majid and Edwin J. Beggs

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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