The Geometry of Noncommutative Spacetimes
Abstract
:1. Introduction
2. Noncommutative Spacetime—An Operational Approach
3. Noncommutative Geometry à la Connes
- —a dense -subalgebra of a -algebra ;
- —a separable Hilbert space with a faithful representation via bounded operators;
- —an unbounded self-adjoint operator on with a compact resolvent.
- —the algebra of smooth functions on ,
- —the space of square summable sections of the spinor bundle S over ,
- —the (curved) Dirac operator associated with S,
4. Causality in Noncommutative Spacetimes
5. The Foundations of Quantum Field Theory Revisited
Acknowledgments
Conflicts of Interest
References
- Akofor, E.; Balachandran, A.P.; Joseph, A. Quantum fields on the Groenewold–Moyal plane. Int. J. Mod. Phys. A 2008, 23, 1637–1677. [Google Scholar] [CrossRef]
- Snyder, H.S. Quantized Space-Time. Phys. Rev. 1947, 71, 38–41. [Google Scholar] [CrossRef]
- Yang, C.N. On Quantized Space-Time. Phys. Rev. 1947, 72, 874. [Google Scholar] [CrossRef]
- Connes, A. Noncommutative Geometry; Academic Press: Cambridge, MA, USA, 1994. [Google Scholar]
- Madore, J. An Introduction to Noncommutative Differential Geometry and its Physical Applications; London Mathematical Society Lecture Note Series; Cambridge University Press: Cambridge, UK, 1999; Volume 257. [Google Scholar]
- Majid, S. Foundations of Quantum Group Theory; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Connes, A. C*-algèbres et géométrie différentielle. C. R. Acad. Sci. Paris Sér. 1980, 290, A599–A604. [Google Scholar]
- Woronowicz, S.L. Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys. 1989, 122, 125–170. [Google Scholar] [CrossRef]
- Dubois-Violette, M.; Kerner, R.; Madore, J. Noncommutative differential geometry of matrix algebras. J. Math. Phys. 1990, 31, 316–322. [Google Scholar] [CrossRef]
- Malament, D. In defense of dogma: Why there cannot be a relativistic quantum mechanical theory of (localizable) particles. In Perspectives on Quantum Reality; Clifton, R., Ed.; Springer: Berlin/Heidelberg, Germany, 1996; pp. 1–10. [Google Scholar]
- Reeh, H.; Schlieder, S. Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Feldern. Il Nuovo Cimento 1961, 22, 1051–1068. [Google Scholar] [CrossRef]
- Yngvason, J. Localization and entanglement in relativistic quantum physics. In The Message of Quantum Science: Attempts Towards a Synthesis; Blanchard, P., Fröhlich, J., Eds.; Springer: Berlin/Heidelberg, Germany, 2015; pp. 325–348. [Google Scholar]
- Haag, R. Local Quantum Physics: Fields, Particles, Algebras; Theoretical and Mathematical Physics; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- Hossenfelder, S. Minimal Length Scale Scenarios for Quantum Gravity. Living Rev. Relat. 2013, 16, 2. [Google Scholar] [CrossRef] [PubMed]
- Garay, L.J. Quantum graivty and minimum length. Int. J. Mod. Phys. A 1995, 10, 145–165. [Google Scholar] [CrossRef]
- Doplicher, S.; Fredenhagen, K.; Roberts, J.E. Spacetime quantization induced by classical gravity. Phys. Lett. B 1994, 331, 39–44. [Google Scholar] [CrossRef]
- Doplicher, S.; Fredenhagen, K.; Roberts, J.E. The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 1995, 172, 187–220. [Google Scholar] [CrossRef]
- Franco, N.; Eckstein, M. An algebraic formulation of causality for noncommutative geometry. Class. Quantum Gravity 2013, 30, 135007. [Google Scholar] [CrossRef]
- Franco, N.; Eckstein, M. Exploring the causal structures of almost commutative geometries. Symmetry Integr. Geom. Methods Appl. 2014, 10, 010. [Google Scholar] [CrossRef]
- Franco, N.; Eckstein, M. Noncommutative geometry, Lorentzian structures and causality. In Mathematical Structures of the Universe; Eckstein, M., Heller, M., Szybka, S., Eds.; Copernicus Center Press: Kraków, Poland, 2014; pp. 315–340. [Google Scholar]
- Eckstein, M.; Franco, N. Causal structure for noncommutative geometry. In Proceedings of the Frontiers of Fundamental Physics 14, Marseille, France, 15–18 July 2014.
- Franco, N.; Eckstein, M. Causality in noncommutative two-sheeted space-times. J. Geom. Phys. 2015, 96, 42–58. [Google Scholar] [CrossRef]
- Eckstein, M.; Miller, T. Causality for nonlocal phenomena. Ann. Henri Poincaré 2017, 1–48. [Google Scholar] [CrossRef]
- Eckstein, M.; Miller, T. Causal evolution of wave packets. Phys. Rev. A 2017, 95, 032106. [Google Scholar] [CrossRef]
- Miller, T. Polish spaces of causal curves. J. Geom. Phys. 2017, 116, 295–315. [Google Scholar] [CrossRef]
- Eckstein, M.; Franco, N.; Miller, T. Noncommutative geometry of Zitterbewegung. Phys. Rev. D 2017, in press. [Google Scholar]
- Strocchi, F. An Introduction to the Mathematical Structure of Quantum Mechanics; World Scientific: Singapore, 2008. [Google Scholar]
- Keyl, M. Fundamentals of quantum information theory. Phys. Rep. 2002, 369, 431–548. [Google Scholar] [CrossRef]
- Dixmier, J. C*-Algebras; North-Holland Publishing Company: Amsterdam, The Netherlands, 1977. [Google Scholar]
- Kadison, R.V.; Ringrose, J.R. Fundamentals of the Theory of Operator Algebras; Academic Press: Cambridge, MA, USA, 1986. [Google Scholar]
- Petz, D. An Invitation to the Algebra of Canonical Commutation Relations; Leuven University Press: Leuven, Belgium, 1990. [Google Scholar]
- Heisenberg, W. The Physical Principles of the Quantum Theory; Courier Corporation: North Chelmsford, MA, USA, 1930. [Google Scholar]
- Shull, C.G. Single-slit diffraction of neutrons. Phys. Rev. 1969, 179, 752–754. [Google Scholar] [CrossRef]
- Friedrich, T.; Nestke, A. Dirac Operators in Riemannian Geometry. In Graduate Studies in Mathematics; American Mathematical Society: Providence, RI, USA, 2000; Volume 25. [Google Scholar]
- Várilly, J. Dirac Operators and Spectral Geometry. Lecture Notes. 2006. Available online: www.impan.pl/swiat-matematyki/notatki-z-wyklado~/varilly_dosg.pdf (accessed on 14 March 2017).
- Connes, A. On the spectral characterization of manifolds. J. Noncommut. Geom. 2013, 7, 1–82. [Google Scholar] [CrossRef]
- Krajewski, T. Classification of finite spectral triples. J. Geom. Phys. 1998, 28, 1–30. [Google Scholar] [CrossRef]
- van Suijlekom, W.D. Noncommutative Geometry and Particle Physics; Springer: Berlin, Germany, 2015. [Google Scholar]
- Chamseddine, A.H.; Connes, A.; Marcolli, M. Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 2007, 11, 991–1089. [Google Scholar] [CrossRef]
- Marcolli, M. Building cosmological models via noncommutative geometry. Int. J. Geom. Methods Mod. Phys. 2011, 8, 1131–1168. [Google Scholar] [CrossRef]
- Sakellariadou, M. Cosmological consequences of the noncommutative spectral geometry as an approach to unification. J. Phys. Conf. Ser. 2011, 283, 012031. [Google Scholar] [CrossRef]
- Christensen, E.; Ivan, C.; Schrohe, E. Spectral triples and the geometry of fractals. J. Noncommut. Geom. 2012, 6, 249–274. [Google Scholar] [CrossRef]
- Connes, A.; Landi, G. Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 2001, 221, 141–159. [Google Scholar] [CrossRef]
- Rieffel, M.A. C*-algebras associated with irrational rotations. Pac. J. Math. 1981, 93, 415–429. [Google Scholar] [CrossRef]
- Chakraborty, P.S.; Pal, A. Equivariant spectral triples on the quantum SUq(2) group. K-Theory 2003, 28, 107–126. [Google Scholar] [CrossRef]
- Dąbrowski, L.; Sitarz, A. Dirac operator on the standard Podleś quantum sphere. Banach Center Publ. 2003, 61, 49–58. [Google Scholar]
- Gayral, V.; Iochum, B. The spectral action for Moyal planes. J. Math. Phys. 2005, 46, 043503. [Google Scholar] [CrossRef]
- Strohmaier, A. On noncommutative and pseudo-Riemannian geometry. J. Math. Phys. 2006, 56, 175–195. [Google Scholar] [CrossRef]
- van Suijlekom, W.D. The noncommutative Lorentzian cylinder as an isospectral deformation. J. Math. Phys. 2004, 45, 537–556. [Google Scholar] [CrossRef]
- Paschke, M.; Sitarz, A. Equivariant Lorentzian spectral triples. arXiv, 2006; arXiv:math-ph/0611029. [Google Scholar]
- Barrett, J. Lorentzian version of the noncommutative geometry of the standard model of particle physics. J. Math. Phys. 2007, 48, 012303. [Google Scholar] [CrossRef]
- Franco, N. Lorentzian Approach to Noncommutative Geometry. Ph.D. Thesis, University of Namur, Namur, Belgium, 2011. [Google Scholar]
- Verch, R. Quantum Dirac field on Moyal–Minkowski spacetime—Illustrating quantum field theory over Lorentzian spectral geometry. Acta Phys. Pol. B Suppl. 2011, 4, 507–530. [Google Scholar] [CrossRef]
- van den Dungen, K.; Paschke, M.; Rennie, A. Pseudo-Riemannian spectral triples and the harmonic oscillator. J. Math. Phys. 2013, 73, 37–55. [Google Scholar] [CrossRef]
- Franco, N. Temporal Lorentzian spectral triples. Rev. Math. Phys. 2014, 26, 1430007. [Google Scholar] [CrossRef]
- van den Dungen, K.; Rennie, A. Indefinite Kasparov modules and pseudo-Riemannian manifolds. Ann. Henri Poincaré 2016, 17, 3255–3286. [Google Scholar] [CrossRef]
- van den Dungen, K. Krein spectral triples and the fermionic action. Math. Phys. Anal. Geom. 2016, 19, 4. [Google Scholar] [CrossRef]
- Besnard, F.; Bizi, N. On the definition of spacetimes in noncommutative geometry, Part I. arXiv, 2016; arXiv:1611.07830. [Google Scholar]
- Besnard, F. On the definition of spacetimes in noncommutative geometry, Part II. arXiv, 2016; arXiv:1611.07842. [Google Scholar]
- Bizi, N.; Brouder, C.; Besnard, F. Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and the standard model. arXiv, 2016; arXiv:1611.07062. [Google Scholar]
- Bognár, J. Indefinite Inner Product Spaces; Springer: Berlin, Germany, 1974. [Google Scholar]
- Baum, H. Spin-Strukturen und Dirac-Operatoren über Pseudoriemannschen Mannigfaltigkeiten; Teubner-Texte zur Mathematik; Leipzig: Teubner, Germany, 1981; Volume 41. [Google Scholar]
- Franco, N.; Wallet, J.C. Metrics and causality on Moyal planes. In Noncommutative Geometry and Optimal Transport; Contemporary Mathematics; American Mathematical Society: Providence, RI, USA, 2016; Volume 676, pp. 147–173. [Google Scholar]
- Rieffel, M.A. Metrics on state spaces. Doc. Math. 1999, 4, 559–600. [Google Scholar]
- Beem, J.; Ehrlich, P.; Easley, K. Global Lorentzian Geometry; Monographs and Textbooks in Pure and Applied Mathematics; CRC Press: Boca Raton, FL, USA, 1996; Volume 202. [Google Scholar]
- Minguzzi, E. Time functions as utilities. Commun. Math. Phys. 2010, 298, 855–868. [Google Scholar] [CrossRef]
- Miller, T. On the causality and K-causality between measures. Universe 2017, in press. [Google Scholar]
- Watcharangkool, A.; Sakellariadou, M. Noncommutative geometrical origin of the energy-momentum dispersion relation. Phys. Rev. D 2017, 95, 025027. [Google Scholar] [CrossRef]
- Schrödinger, E. Über die kräftefreie Bewegung in der relativistischen Quantenmechanik. Sitzungber. Preuss. Akad. Wiss. Phys. Math. Kl. 1930, 24, 418–428. [Google Scholar]
- Hestenes, D. The zitterbewegung interpretation of quantum mechanics. Found. Phys. 1990, 20, 1213–1232. [Google Scholar] [CrossRef]
- Penrose, R. The Road to Reality: A Complete Guide to the Laws of the Universe; Jonathan Cape: London, UK, 2004. [Google Scholar]
- Mattingly, D. Modern tests of Lorentz invariance. Living Rev. Relat. 2005, 8, 2003. [Google Scholar] [CrossRef] [PubMed]
- Brukner, Č. Quantum causality. Nat. Phys. 2014, 10, 259–263. [Google Scholar] [CrossRef]
- Besnard, F.; Bizi, N. The disappearance of causality at small scale in almost-commutative manifolds. arXiv, 2014; arXiv:1411.0878. [Google Scholar]
- Besnard, F. Two roads to noncommutative causality. J. Phys. Conf. Ser. 2015, 634, 012009. [Google Scholar] [CrossRef]
- Sitarz, A. Pointless Geometry. In Mathematical Structures of the Universe; Eckstein, M., Heller, M., Szybka, S., Eds.; Copernicus Center Press: Kraków, Poland, 2014; pp. 301–314. [Google Scholar]
- Majid, S. Quantum groups and noncommutative geometry. J. Math. Phys. 2000, 41, 3892–3942. [Google Scholar] [CrossRef]
- Heller, M.; Sasin, W.; Lambert, D. Groupoid approach to noncommutative quantization of gravity. J. Math. Phys. 1997, 38, 5840–5853. [Google Scholar] [CrossRef]
- Heller, M.; Pysiak, L.; Sasin, W. Conceptual Unification of Gravity and Quanta. Int. J. Theor. Phys. 2007, 46, 2494–2512. [Google Scholar] [CrossRef]
- Kobakhidze, A.; Manning, A.; Tureanu, A. Observable Zitterbewegung in curved spacetimes. Phys. Lett. B 2016, 757, 84–91. [Google Scholar] [CrossRef]
- Greenberg, O.W. Failure of microcausality in quantum field theory on noncommutative spacetime. Phys. Rev. D 2006, 73, 045014. [Google Scholar] [CrossRef]
- Soloviev, M.A. Failure of microcausality in noncommutative field theories. Phys. Rev. D 2008, 77, 125013. [Google Scholar] [CrossRef]
- Balachandran, A.P.; Joseph, A.; Padmanabhan, P. Causality and statistics on the Groenewold–Moyal plane. Found. Phys. 2010, 40, 692–702. [Google Scholar] [CrossRef]
- 1.In general, does not need to contain a unit. However, it always contains an approximate unit, which can be utilised to rigorously express the normalisation requirement [29].
- 3.A concrete model of a noncommutative spacetime with nonlocal events, but a rigid causal structure was developed in [63].
- 4.Such a viewpoint leads to the so-called ‘zigzag picture of the electron’ [71, Section 25.2].
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Eckstein, M. The Geometry of Noncommutative Spacetimes. Universe 2017, 3, 25. https://doi.org/10.3390/universe3010025
Eckstein M. The Geometry of Noncommutative Spacetimes. Universe. 2017; 3(1):25. https://doi.org/10.3390/universe3010025
Chicago/Turabian StyleEckstein, Michał. 2017. "The Geometry of Noncommutative Spacetimes" Universe 3, no. 1: 25. https://doi.org/10.3390/universe3010025