Quantum Riemannian geometry of phase space and nonassociativity

• Autorzy: Beggs E. J. , Majid S.

Identyfikatory

DOI

10.1515/dema-2017-0009

• Języki publikacji: EN

Abstrakty

EN

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, wj] = 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.

Słowa kluczowe

EN

noncommutative geometry; quantum gravity; Poisson geometry; Riemannian geometry; quantum mechanics

PL

geometria nieprzemienna; grawitacja kwantowa; geometria Poissona; geometria Riemanna; mechanika kwantowa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2017
• Tom: Vol. 50, nr 1
• Strony: 83--93
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Beggs E. J.

• Department of Mathematics, Swansea University, Singleton Parc, Swansea SA2 8PP, United Kingdom

autor

Majid S.

• Queen Mary, University of London, School of Mathematics, Mile End Rd, London E1 4NS, United Kingdom

Bibliografia

• [1] Majid S., Almost commutative Riemannian geometry: wave operators, Commun. Math. Phys., 2012, 310, 569-609
• [2] Brody D. C., Hughston L. P., Geometric quantum mechanics, J. Geom. Phys., 2001, 38, 19-53
• [3] Beggs E. J., Majid S., Semiclassical differential structures, Pac. J. Math., 2006, 224, 1-44
• [4] Beggs E. J., Majid S., Bar categories and star operations, Alg. and Representation Theory, 2009, 12, 103-152
• [5] Beggs E. J., Majid S., ∗-Compatible connections in noncommutative Riemannian geometry, J. Geom. Phys., 2011, 61, 95-124
• [6] Beggs E. J., Majid S., Gravity induced by quantum spacetime, Class. Quantum. Grav., 2014, 31, 035020
• [7] Beggs E. J., Majid S., Poisson Riemannian geometry, J. Geom. Phys., 2017, 114, 450-491
• [8] Connes A., Noncommutative geometry, Academic Press, 1994
• [9] Fedosov B. V., Deformation quantisation and index theory, Akademie Verlag, 1996
• [10] Hawkins E., Noncommutative rigidity, Commun. Math. Phys., 2004, 246, 218-232
• [11] Majid S., Reconstruction and quantisation of Riemannian structures [arXiv:1307.2778 (math.QA)]
• [12] Beggs E. J., Majid S., Poisson complex geometry and nonassociativity (in preparation)
• [13] Beggs E. J., Smith S. P., Noncommutative complex differential geometry, J. Geom. Phys., 2013, 72, 7-33
• [14] Dubois-Violette M., Masson T., On the first-order operators in bimodules, Lett. Math. Phys., 1996, 37, 467-474
• [15] Dubois-Violette M., Michor P. W., Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys., 1996, 20, 218-232
• [16] Majid S., Noncommutative Riemannian geometry of graphs, J. Geom. Phys., 2013, 69, 74-93
• [17] J. Mourad, Linear connections in noncommutative geometry, Class. Quantum Grav., 1995, 12, 965-974
• [18] Penrose R., talk at Workshop on ‘Noncommutative Geometry and Physics: fundamental structure of space and time’, Newton Institute, 2006
• [19] Aldrovandi R., Pereira J. G., Teleparallel Gravity: An Introduction, Springer, 2013
• [20] Majid S., Meaning of noncommutative geometry and the Planck-scale quantum group, Springer Lect. Notes Phys., 2000, 541, 227-276
• [21] Majid S., Hopf algebras for physics at the Planck scale, Class. Quantum Grav., 1988, 5, 1587-1607

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).