Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 086, 36 pages      arXiv:1403.0808      https://doi.org/10.3842/SIGMA.2014.086
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Matrix Bases for Star Products: a Review

Fedele Lizzi a, b, c and Patrizia Vitale a, b
a) Dipartimento di Fisica, Università di Napoli Federico II, Napoli, Italy
b) INFN, Sezione di Napoli, Italy
c) Institut de Ciéncies del Cosmos, Universitat de Barcelona, Catalonia, Spain

Received March 04, 2014, in final form August 11, 2014; Published online August 15, 2014

Abstract
We review the matrix bases for a family of noncommutative $\star$ products based on a Weyl map. These products include the Moyal product, as well as the Wick-Voros products and other translation invariant ones. We also review the derivation of Lie algebra type star products, with adapted matrix bases. We discuss the uses of these matrix bases for field theory, fuzzy spaces and emergent gravity.

Key words: noncommutative geometry; star products; matrix models.

pdf (661 kb)   tex (119 kb)

References

  1. Abe Y., Construction of fuzzy spaces and their applications to matrix models, arXiv:1002.4937.
  2. Alekseev A.Y., Recknagel A., Schomerus V., Brane dynamics in background fluxes and non-commutative geometry, J. High Energy Phys. 2000 (2000), no. 5, 010, 25 pages, hep-th/0003187.
  3. Ambjørn J., Makeenko Y.M., Nishimura J., Szabo R.J., Nonperturbative dynamics of noncommutative gauge theory, Phys. Lett. B 480 (2000), 399-408, hep-th/0002158.
  4. Aschieri P., Dimitrijević M., Kulish P., Lizzi F., Wess J., Noncommutative spacetimes: symmetries in noncommutative geometry and field theory, Lecture Notes in Physics, Vol. 774, Springer-Verlag, Berlin, 2009.
  5. Balachandran A.P., Kürkçüoğlu S., Gupta K.S., Edge currents in non-commutative Chern-Simons theory from a new matrix model, J. High Energy Phys. 2003 (2003), no. 9, 007, 17 pages, hep-th/0306255.
  6. Balachandran A.P., Kürkçüoğlu S., Vaidya S., Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114.
  7. Banks T., Fischler W., Shenker S.H., Susskind L., M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997), 5112-5128, hep-th/9610043.
  8. Basu P., Chakraborty B., Scholtz F.G., A unifying perspective on the Moyal and Voros products and their physical meanings, J. Phys. A: Math. Theor. 44 (2011), 285204, 11 pages, arXiv:1101.2495.
  9. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), 61-110.
  10. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), 111-151.
  11. Beiser S., Römer H., Waldmann S., Convergence of the Wick star product, Comm. Math. Phys. 272 (2007), 25-52, math.QA/0506605.
  12. Berezin F.A., General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174.
  13. Blaschke D.N., Grosse H., Schweda M., Non-commutative ${\rm U}(1)$ gauge theory on ${\mathbb R}^4_{\Theta}$ with oscillator term and BRST symmetry, Europhys. Lett. 79 (2007), 61002, 3 pages, arXiv:0705.4205.
  14. Blaschke D.N., Hohenegger S., Schweda M., Divergences in non-commutative gauge theories with the Slavnov term, J. High Energy Phys. 2005 (2005), no. 11, 041, 29 pages, arXiv:1302.2903.
  15. Bratteli O., Inductive limits of finite dimensional $C^{\ast}$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195-234.
  16. Cahill K.E., Glauber R.J., Ordered expansions in boson amplitude operators, Phys. Rev. 177 (1969), 1857-1881.
  17. Chatzistavrakidis A., Steinacker H., Zoupanos G., On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes, Fortschr. Phys. 58 (2010), 537-552, arXiv:0909.5559.
  18. Chatzistavrakidis A., Steinacker H., Zoupanos G., Intersecting branes and a standard model realization in matrix models, J. High Energy Phys. 2011 (2011), no. 9, 115, 36 pages, arXiv:1107.0265.
  19. Chu C.-S., Madore J., Steinacker H., Scaling limits of the fuzzy sphere at one loop, J. High Energy Phys. 2001 (2001), no. 8, 038, 17 pages, hep-th/0106205.
  20. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  21. D'Andrea F., Lizzi F., Várilly J.C., Metric properties of the fuzzy sphere, Lett. Math. Phys. 103 (2013), 183-205, arXiv:1209.0108.
  22. de Goursac A., Wallet J.-C., Wulkenhaar R., Noncommutative induced gauge theory, Eur. Phys. J. C 51 (2007), 977-987, hep-th/0703075.
  23. de Goursac A., Wallet J.-C., Wulkenhaar R., On the vacuum states for noncommutative gauge theory, Eur. Phys. J. C 56 (2008), 293-304, arXiv:0803.3035.
  24. Dijkgraaf R., Verlinde E., Verlinde H., Matrix string theory, Nuclear Phys. B 500 (1997), 43-61, \mboxhep-th/9703030.
  25. Estrada R., Gracia-Bondía J.M., Várilly J.C., On asymptotic expansions of twisted products, J. Math. Phys. 30 (1989), 2789-2796.
  26. Falomir H., Franchino Vi\ nas S.A., Pisani P.A.G., Vega F., Boundaries in the Moyal plane, J. High Energy Phys. 2013 (2013), no. 12, 024, 20 pages, arXiv:1307.4464.
  27. Galluccio S., Lizzi F., Vitale P., Twisted noncommutative field theory with the Wick-Voros and Moyal products, Phys. Rev. D 78 (2008), 085007, 14 pages, arXiv:0810.2095.
  28. Galluccio S., Lizzi F., Vitale P., Translation invariance, commutation relations and ultraviolet/infrared mixing, J. High Energy Phys. 2009 (2009), no. 9, 054, 18 pages, arXiv:0907.3640.
  29. Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Várilly J.C., Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), 569-623, hep-th/0307241.
  30. Géré A., Vitale P., Wallet J.-C., Quantum gauge theories on noncommutative 3-d space, arXiv:1312.6145.
  31. Géré A., Wallet J.-C., Spectral theorem in noncommutative field theories I: Jacobi dynamics, arXiv:1402.6976.
  32. Gracia-Bondía J.M., Lizzi F., Marmo G., Vitale P., Infinitely many star products to play with, J. High Energy Phys. 2002 (2002), no. 4, 026, 35 pages, hep-th/0112092.
  33. Gracia-Bondía J.M., Várilly J.C., Algebras of distributions suitable for phase-space quantum mechanics. I, J. Math. Phys. 29 (1988), 869-879.
  34. Groenewold H.J., On the principles of elementary quantum mechanics, Physica 12 (1946), 405-460.
  35. Grosse H., Lizzi F., Steinacker H., Noncommutative gauge theory and symmetry breaking in matrix models, Phys. Rev. D 81 (2010), 085034, 12 pages, arXiv:1001.2703.
  36. Grosse H., Wulkenhaar R., Renormalisation of $\phi^4$-theory on noncommutative ${\mathbb R}^2$ in the matrix base, J. High Energy Phys. 2003 (2003), no. 12, 019, 26 pages, hep-th/0307017.
  37. Grosse H., Wulkenhaar R., Renormalisation of $\phi^4$-theory on noncommutative ${\mathbb R}^4$ in the matrix base, Comm. Math. Phys. 256 (2005), 305-374, hep-th/0401128.
  38. Gurau R., Magnen J., Rivasseau V., Tanasa A., A translation-invariant renormalizable non-commutative scalar model, Comm. Math. Phys. 287 (2009), 275-290, arXiv:0802.0791.
  39. Hammou A.B., Lagraa M., Sheikh-Jabbari M.M., Coherent state induced star product on ${\mathbb R}^3_\lambda$ and the fuzzy sphere, Phys. Rev. D 66 (2002), 025025, 11 pages, hep-th/0110291.
  40. Hayakawa M., Perturbative analysis on infrared aspects of noncommutative QED on ${\mathbb R}^4$, Phys. Lett. B 478 (2000), 394-400, hep-th/9912094.
  41. Hoppe J., Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. Thesis, Massachusetts Institute of Technology, 1982, reprinted in Soryushiron Kenkyu 80 (1989), 145-202.
  42. Ishibashi N., Kawai H., Kitazawa Y., Tsuchiya A., A large-$N$ reduced model as superstring, Nuclear Phys. B 498 (1997), 467-491, hep-th/9612115.
  43. Iso S., Kimura Y., Tanaka K., Wakatsuki K., Noncommutative gauge theory on fuzzy sphere from matrix model, Nuclear Phys. B 604 (2001), 121-147, hep-th/0101102.
  44. Kobayashi S., Asakawa T., Angles in fuzzy disc and angular noncommutative solitons, J. High Energy Phys. 2013 (2013), no. 4, 145, 22 pages, arXiv:1206.6602.
  45. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  46. Landi G., Lizzi F., Szabo R.J., From large $N$ matrices to the noncommutative torus, Comm. Math. Phys. 217 (2001), 181-201, hep-th/9912130.
  47. Lizzi F., Spisso B., Noncommutative field theory: numerical analysis with the fuzzy disk, Internat. J. Modern Phys. A 27 (2012), 1250137, 26 pages, arXiv:1207.4998.
  48. Lizzi F., Vitale P., Zampini A., From the fuzzy disc to edge currents in Chern-Simons theory, Modern Phys. Lett. A 18 (2003), 2381-2387, hep-th/0309128.
  49. Lizzi F., Vitale P., Zampini A., The fuzzy disc, J. High Energy Phys. 2003 (2003), no. 8, 057, 16 pages, hep-th/0306247.
  50. Lizzi F., Vitale P., Zampini A., The beat of a fuzzy drum: fuzzy Bessel functions for the disc, J. High Energy Phys. 2005 (2005), no. 9, 080, 32 pages, hep-th/0506008.
  51. Lizzi F., Vitale P., Zampini A., The fuzzy disc: a review, J. Phys. Conf. Ser. 53 (2006), 830-842.
  52. Lizzi F., Zampini A., Szabo R.J., Geometry of the gauge algebra in non-commutative Yang-Mills theory, J. High Energy Phys. 2001 (2001), no. 8, 032, 54 pages, hep-th/0107115.
  53. Madore J., The fuzzy sphere, Classical Quantum Gravity 9 (1992), 69-87.
  54. Madore J., An introduction to noncommutative differential geometry and its physical applications, London Mathematical Society Lecture Note Series, Vol. 257, 2nd ed., Cambridge University Press, Cambridge, 1999.
  55. Madore J., Schraml S., Schupp P., Wess J., Gauge theory on noncommutative spaces, Eur. Phys. J. C 16 (2000), 161-167, hep-th/0001203.
  56. Man'ko O.V., Man'ko V.I., Marmo G., Alternative commutation relations, star products and tomography, J. Phys. A: Math. Gen. 35 (2002), 699-719, quant-ph/0112110.
  57. Man'ko V.I., Marmo G., Vitale P., Phase space distributions and a duality symmetry for star products, Phys. Lett. A 334 (2005), 1-11, hep-th/0407131.
  58. Man'ko V.I., Marmo G., Vitale P., Zaccaria F., A generalization of the Jordan-Schwinger map: the classical version and its $q$ deformation, Internat. J. Modern Phys. A 9 (1994), 5541-5561, hep-th/9310053.
  59. Martinetti P., Vitale P., Wallet J.-C., Noncommutative gauge theories on ${\mathbb R}_{\theta}^2$ as matrix models, J. High Energy Phys. 2013 (2013), no. 9, 051, 26 pages, arXiv:1303.7185.
  60. Matusis A., Susskind L., Toumbas N., The IR/UV connection in non-commutative gauge theories, J. High Energy Phys. 2000 (2000), no. 12, 002, 18 pages, hep-th/0002075.
  61. Minwalla S., Van Raamsdonk M., Seiberg N., Noncommutative perturbative dynamics, J. High Energy Phys. 2000 (2000), no. 2, 020, 31 pages, hep-th/9912072.
  62. Moyal J.E., Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99-124.
  63. Panero M., Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere, J. High Energy Phys. 2007 (2007), no. 5, 082, 20 pages, hep-th/0608202.
  64. Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
  65. Pimsner M., Voiculescu D., Imbedding the irrational rotation $C^{\ast}$-algebra into an AF-algebra, J. Operator Theory 4 (1980), 201-210.
  66. Pinzul A., Stern A., Edge states from defects on the noncommutative plane, Modern Phys. Lett. A 18 (2003), 2509-2516, hep-th/0307234.
  67. Rieffel M.A., $C^{\ast}$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429.
  68. Rieffel M.A., Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance, Mem. Amer. Math. Soc. 168 (2004), 67-91, math.OA/0108005.
  69. Rosa L., Vitale P., On the $\star$-product quantization and the Duflo map in three dimensions, Modern Phys. Lett. A 27 (2012), 1250207, 15 pages, arXiv:1209.2941.
  70. Sakharov A.D., Vacuum quantum fluctuations in curved space and the theory of gravitation, Sov. Phys. Dokl. 12 (1968), 1040-1041, reprinted in Gen. Relativity Gravitation 32 (2000), 365-367.
  71. Scholtz F.G., Chakraborty B., Govaerts J., Vaidya S., Spectrum of the non-commutative spherical well, J. Phys. A: Math. Theor. 40 (2007), 14581-14592, arXiv:0709.3357.
  72. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  73. Soloviev M.A., Algebras with convergent star products and their representations in Hilbert spaces, J. Math. Phys. 54 (2013), 073517, 16 pages, arXiv:1312.6571.
  74. Spisso B., Wulkenhaar R., A numerical approach to harmonic noncommutative spectral field theory, Internat. J. Modern Phys. A 27 (2012), 1250075, 25 pages, arXiv:1111.3050.
  75. Steinacker H., Emergent gravity from noncommutative gauge theory, J. High Energy Phys. 2007 (2007), no. 12, 049, 36 pages, arXiv:0708.2426.
  76. Steinacker H., Zahn J., An extended standard model and its Higgs geometry from the matrix model, arXiv:1401.2020.
  77. Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), 207-299, \mboxhep-th/0109162.
  78. Tanasa A., Vitale P., Curing the UV/IR mixing for field theories with translation-invariant star products, Phys. Rev. D 81 (2010), 065008, 12 pages, arXiv:0912.0200.
  79. Várilly J.C., Gracia-Bondía J.M., Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra, J. Math. Phys. 29 (1988), 880-887.
  80. Varshalovich D.A., Moskalev A.N., Khersonskiǐ V.K., Quantum theory of angular momentum, World Scientific Publishing Co., Inc., Teaneck, NJ, 1988.
  81. Varshovi A.A., Groenewold-Moyal product, $\alpha^\star$-cohomology, and classification of translation-invariant non-commutative structures, J. Math. Phys. 54 (2013), 072301, 9 pages, arXiv:1210.1004.
  82. Vitale P., Wallet J.-C., Noncommutative field theories on ${\mathbb R}^3_\lambda$: towards UV/IR mixing freedom, J. High Energy Phys. 2013 (2013), no. 4, 115, 36 pages, arXiv:1212.5131.
  83. Yang H.S., Emergent gravity from noncommutative space-time, Internat. J. Modern Phys. A 24 (2009), 4473-4517, hep-th/0611174.

Previous article  Next article   Contents of Volume 10 (2014)