Quantum Spacetime: a Disambiguation

Gherardo Piacitelli

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

We review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the “canonical quantum spacetime”. The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance.

Détail

Keywords: quantum spacetime; covariance; noncommutative geometry; doubly special relativity.

Bibliographie

[1] Doplicher S., “Spacetime and fields, a quantum texture”, AIP Conf. Proc., 589, 2001, 204–213, arXiv: hep-th/0105251 | DOI

[2] Doplicher S., “Quantum field theory on quantum spacetime”, J. Phys. Conf. Ser., 53 (2006), 793–798, arXiv: hep-th/0608124 | DOI

[3] Mead C. A., “Possible connection between gravitation and fundamental length”, Phys. Rev., 135 (1964), B849–B862 | DOI | MR

[4] Mead C. A., Wilczek F., “Walking the Planck length through history”, Phys. Today, 54 (2001), 15–15 | DOI

[5] Ciafaloni M., Veneziano G., “Superstring collisions at planckian energies”, Phys. Lett. B, 197 (1987), 81–88 | DOI

[6] Maggiore M., “A generalized uncertainty principle in quantum gravity”, Phys. Lett. B, 304 (1993), 65–69, arXiv: hep-th/9301067 | DOI | MR

[7] 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, arXiv: hep-th/0303037 | DOI | MR | Zbl

[8] Maggiore M., “The algebraic structure of the generalized uncertainty principle”, Phys. Lett. B, 319 (1993), 83–86, arXiv: hep-th/9309034 | DOI | MR

[9] Snyder H. S., “Quantized space-time”, Phys. Rev., 71 (1947), 38–41 | DOI | MR | Zbl

[10] Bahns D., Doplicher S., Fredenhagen K., Piacitelli G., “Ultraviolet finite quantum field theory on quantum spacetime”, Comm. Math. Phys., 237 (2003), 221–241, arXiv: hep-th/0301100 | DOI | MR | Zbl

[11] Amelino-Camelia G., “Doubly-special relativity: facts, myths and some key open issues”, Symmetry, 2 (2010), 230–271, arXiv: 1003.3942 | DOI

[12] Dabrowski L., Godlinski M., Piacitelli G., “Lorentz covariant $k$-Minkowski spacetime”, Phys. Rev. D, 81 (2010), 125024, 8 pp., arXiv: 0912.5451 | DOI

[13] Cuntz J., “Bivariant $k$-theory and the Weyl algebra”, $K$-theory, 35 (2005), 93–137, arXiv: math/0401295 | DOI | MR | Zbl

[14] Rieffel M. A., “On the operator algebra for the space-time uncertainty relations”, Operator Algebras and Quantum Field Theory (Rome, 1996), Int. Press, Cambridge, MA, 1997, 374–382, arXiv: funct-an/9701011 | MR | Zbl

[15] Estrada R., Gracia-Bondia J. M., Varilly J. C., “On asymptotic expansions of twisted products”, J. Math. Phys., 30 (1989), 2789–2796 | DOI | MR | Zbl

[16] Aschieri P., Dimitrijević M., Kulish P., Lizzi F., Wess J., Noncommutative spacetimes: symmetries in noncommutative geometry and field theory, Springer, Dordrecht, 2009

[17] Weyl H., Gruppentheorie und Quantenmechanik, Hirzel, Leipzig, 1928 | Zbl

[18] von Neumann J., “Die Eindeutigkeit der Schrödingerschen Operatoren”, Math. Ann., 104 (1931), 570–578 | DOI | MR | Zbl

[19] Piacitelli G., Normal ordering of operator products on quantum spacetime and quantum field theory, PhD Thesis, Universitá di Padova, 2002

[20] Bahns D., Doplicher S., Fredenhagen K., Piacitelli G., Quantum geometry on quantum spacetime: distance, area and volume operators, arXiv: 1005.2130

[21] Djemai A. E. F., “Introduction to Dubois–Violette's noncommutative differential geometry”, Internat. J. Theoret. Phys., 34 (1995), 801–887 | DOI | MR | Zbl

[22] Piacitelli G., “Twisted covariance as a non-invariant restriction of the fully covariant DFR model”, Comm. Math. Phys., 295 (2010), 701–729, arXiv: 0902.0575 | DOI | MR | Zbl

[23] Piacitelli G., “Twisted covariance and Weyl quantisation”, AIP Conf. Proc., 1196, 2009, 219–224, arXiv: 0901.3109 | DOI

[24] Chaichian M., Kulish P. P., Nishijima K., Tureanu A., “On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT”, Phys. Lett. B, 604 (2004), 98–102, arXiv: hep-th/0408069 | DOI | MR

[25] Wess J., “Deformed coordinate spaces; derivatives”, Proceedings of the BW2003 Workshop on Mathematical Theoretical and Phenomenological Challenges Beyond the Standard Model: Perspectives of Balkans Collaboration (2003, Vrnjacka Banja, Serbia), Vrnjacka Banja, 2003, 122–128, arXiv: hep-th/0408080

[26] Dabrowski L., Piacitelli G., Poincaré covariant $k$-Minkowski spacetime, arXiv: 1006.5658

[27] Borowiec A., Pachoł A., $\kappa$-Minkowski spacetimes and DSR algebras: fresh look and old problems, arXiv: 1005.4429

[28] Corinaldesi E., “Some aspects of the problem of measurability in quantum electrodynamics”, Nuovo Cimento, 10, suppl. (1953), 83–100 | DOI | MR | Zbl

[29] Bahns D., Ultraviolet finiteness of the averaged Hamiltonian on the noncommutative Minkowski space, arXiv: hep-th/0405224

[30] Bahns D., Doplicher S., Fredenhagen K., Piacitelli G., “Field theory on noncommutative spacetimes: quasiplanar Wick products”, Phys. Rev. D, 71 (2005), 025022, 12 pp., arXiv: hep-th/0408204 | DOI | MR

[31] Bahns D., Doplicher S., Fredenhagen K., Piacitelli G., “On the unitarity problem in space-time noncommutative theories”, Phys. Lett. B, 533 (2002), 178–181, arXiv: hep-th/0201222 | DOI | MR | Zbl

[32] Gomis J., Mehen T., “Space-time noncommutative field theories and unitarity”, Nuclear Phys. B, 591 (2000), 265–276, arXiv: hep-th/0005129 | DOI | MR | Zbl

[33] Piacitelli G., “Non local theories: new rules for old diagrams”, J. High Energy Phys., 2004:8 (2004), 031, 13 pp., arXiv: hep-th/0403055 | DOI | MR

[34] Filk T., “Divergencies in a field theory on quantum space”, Phys. Lett. B, 376 (1996), 53–58 | DOI | MR | Zbl

[35] 't Hooft G., Veltman M. J. G., “DIAGRAMMAR”, NATO Adv. Study Inst. Ser. B Phys., 4, Plenum Publishing Corp., New York, 1974, 177–322

[36] Liao Y., Sibold K., “Time-ordered perturbation theory on non-commutative spacetime: basic rules”, Eur. Phys. J. C Part. Fields, 25 (2002), 469–477, arXiv: hep-th/0205269 | DOI | MR | Zbl

[37] Liao Y., Sibold K., “Time-ordered perturbation theory on noncommutative spacetime. II. Unitarity”, Eur. Phys. J. C Part. Fields, 25 (2002), 479–486, arXiv: hep-th/0206011 | DOI | MR | Zbl

[38] Bahns D., Perturbative methods on the noncommutative Minkowski space, PhD Thesis, Universität Hamburg, 2003

[39] Denk S., Schweda M., “Time ordered perturbation theory for non-local interactions: applications to NCQFT”, J. High Energy Phys., 2003:9 (2003), 032, 22 pp., arXiv: hep-th/0306101 | DOI | MR

[40] Roepstorff G., Path integral approach to quantum physics. An introduction, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1994 | MR | Zbl

[41] Bahns D., Schwinger functions in noncommutative quantum field theory, arXiv: 0908.4537

[42] Minwalla S., Van Raamsdonk M., Seiberg N., “Noncommutative perturbative dynamics”, J. High Energy Phys., 2000:2 (2000), 020, 31 pp., arXiv: hep-th/9912072 | DOI | MR