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


SIGMA 6 (2010), 048, 20 pages      arXiv:1003.5788      http://dx.doi.org/10.3842/SIGMA.2010.048
Contribution to the Special Issue “Noncommutative Spaces and Fields”

On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory

Axel de Goursac
Département de Mathématiques, Université Catholique de Louvain, Chemin du Cyclotron, 2, 1348 Louvain-la-Neuve, Belgium

Received March 30, 2010, in final form June 01, 2010; Published online June 09, 2010

Abstract
The harmonic term in the scalar field theory on the Moyal space removes the UV-IR mixing, so that the theory is renormalizable to all orders. In this paper, we review the three principal interpretations of this harmonic term: the Langmann-Szabo duality, the superalgebraic approach and the noncommutative scalar curvature interpretation. Then, we show some deep relationship between these interpretations.

Key words: noncommutative QFT; gauge theory; renormalization; Heisenberg algebra.

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References

  1. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  2. Landi G., An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics, New Series m: Monographs, Vol. 51, Springer-Verlag, Berlin, 1997.
  3. Moyal J.E., Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99-124.
  4. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of space-time at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, hep-th/0303037.
  5. Minwalla S., Van Raamsdonk M., Seiberg N., Noncommutative perturbative dynamics, J. High Energy Phys. 2000 (2000), no. 2, 020, 31 pages, hep-th/9912072.
  6. Gayral V., Heat-kernel approach to UV/IR mixing on isospectral deformation manifolds, Ann. Henri Poincaré 6 (2005), 991-1023, hep-th/0412233.
  7. Gayral V., Non compact isospectral deformations and quantum field theory, hep-th/0507208.
  8. Grosse H., Wulkenhaar R., Renormalisation of φ4 theory on noncommutative R4 in the matrix base, Comm. Math. Phys. 256 (2005), 305-374, hep-th/0401128.
  9. Langmann E., Szabo R.J., Duality in scalar field theory on noncommutative phase spaces, Phys. Lett. B 533 (2002), 168-177, hep-th/0202039.
  10. Bieliavsky P., Gurau R., Rivasseau V., Noncommutative field theory on rank one symmetric spaces, J. Noncommut. Geom. 3 (2009), 99-123, arXiv:0806.4255.
  11. de Goursac A., Masson T., Wallet J.-C., Noncommutative ε-graded connections, arXiv:0811.3567.
  12. Buric M., Wohlgenannt M., Geometry of the Grosse-Wulkenhaar model, J. High Energy Phys. 2010 (2010), no. 3, 053, 17 pages, arXiv:0902.3408.
  13. Buric M., Grosse H., Madore J., Gauge fields on noncommutative geometries with curvature, arXiv:1003.2284.
  14. Gracia-Bondia J.M., Varilly J.C., Algebras of distributions suitable for phase space quantum mechanics. I, J. Math. Phys. 29 (1988), 869-879.
  15. Varilly J.C., Gracia-Bondia J.M., Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra, J. Math. Phys. 29 (1988), 880-887.
  16. Gayral V., Gracia-Bondia J.M., Iochum B., Schucker T., Varilly J.C., Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), 569-623, hep-th/0307241.
  17. Cagnache E., D'Andrea F., Martinetti P., Wallet J.-C., The spectral distance on the Moyal plane, arXiv:0912.0906.
  18. Filk T., Divergencies in a field theory on quantum space, Phys. Lett. B 376 (1996), 53-58.
  19. Magnen J., Rivasseau V., Tanasa A., Commutative limit of a renormalizable noncommutative model, Europhys. Lett. 86 (2009), 11001, 6 pages, arXiv:0807.4093.
  20. Panero M., Quantum field theory in a non-commutative space: theoretical predictions and numerical results on the fuzzy sphere, SIGMA 2 (2006), 081, 14 pages, hep-th/0609205.
  21. 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.
  22. Gurau R., Rivasseau V., Vignes-Tourneret F., Propagators for noncommutative field theories, Ann. Henri Poincaré 7 (2006), 1601-1628, hep-th/0512071.
  23. Vignes-Tourneret F., Renormalisation of non commutative field theories, PhD Thesis, math-ph/0612014.
  24. Rivasseau V., Vignes-Tourneret F., Wulkenhaar R., Renormalization of noncommutative φ4-theory by multi-scale analysis, Comm. Math. Phys. 262 (2006), 565-594, hep-th/0501036.
  25. Gurau R., Magnen J., Rivasseau V., Vignes-Tourneret F., Renormalization of non-commutative Φ44 field theory in x space, Comm. Math. Phys. 267 (2006), 515-542, hep-th/0512271.
  26. Gurau R., Tanasa A., Dimensional regularization and renormalization of non-commutative QFT, Ann. Henri Poincaré 9 (2008), 655-683, arXiv:0706.1147.
  27. Grosse H., Wulkenhaar R., Renormalisation of φ4-theory on noncommutative R2 in the matrix base, J. High Energy Phys. 2003 (2003), no. 12, 019, 26 pages, hep-th/0307017.
  28. Gurau R., Rivasseau V., Parametric representation of noncommutative field theory, Comm. Math. Phys. 272 (2007), 811-835, math-ph/0606030.
  29. Rivasseau V., Tanasa A., Parametric representation of 'critical' noncommutative QFT models, Comm. Math. Phys. 279 (2008), 355-379, math-ph/0701034.
  30. Tanasa A., Vignes-Tourneret F., Hopf algebra of non-commutative field theory, J. Noncommut. Geom. 2 (2008), 125-139, arXiv:0707.4143.
  31. Tanasa A., Kreimer D., Combinatorial Dyson-Schwinger equations in noncommutative field theory, arXiv:0907.2182.
  32. Disertori M., Rivasseau V., Two- and three-loops beta function of noncommutative Φ44 theory, Eur. Phys. J. C 50 (2007), 661-671, hep-th/0610224.
  33. Disertori M., Gurau R., Magnen J., Rivasseau V., Vanishing of beta function of non-commutative φ44 theory to all orders, Phys. Lett. B 649 (2007), 95-102, hep-th/0612251.
  34. Grosse H., Wulkenhaar R., The β-function in duality-covariant noncommutative φ4-theory, Eur. Phys. J. C 35 (2004), 277-282, hep-th/0402093.
  35. de Goursac A., Tanasa A., Wallet J.-C., Vacuum configurations for renormalizable non-commutative scalar models, Eur. Phys. J. C 53 (2008), 459-466, arXiv:0709.3950.
  36. de Goursac A., Noncommutative geometry, gauge theory and renormalization, PhD Thesis, arXiv:0910.5158.
  37. de Goursac A., Wallet J.-C., Symmetries of noncommutative scalar field theory, arXiv:0911.2645.
  38. Langmann E., Szabo R.J., Zarembo K., Exact solution of quantum field theory on noncommutative phase spaces, J. High Energy Phys. 2004 (2004), no. 1, 017, 69 pages, hep-th/0308043.
  39. Vignes-Tourneret F., Renormalization of the orientable non-commutative Gross-Neveu model, Ann. Henri Poincaré 8 (2007), 427-474, math-ph/0606069.
  40. 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.
  41. Geloun J.B., Tanasa A., One-loop β functions of a translation-invariant renormalizable noncommutative scalar model, Lett. Math. Phys. 86 (2008), 19-32, arXiv:0806.3886.
  42. Tanasa A., Translation-invariant noncommutative renormalization, SIGMA 6 (2010), 047, 13 pages, arXiv:1003.4877.
  43. Blaschke D.N., Gieres F., Kronberger E., Schweda M., Wohlgenannt M., Translation-invariant models for non-commutative gauge fields, J. Phys. A: Math. Theor. 41 (2008), 252002, 7 pages, arXiv:0804.1914.
  44. Blaschke D.N., Rofner A., Schweda M., Sedmik R.I.P., One-loop calculations for a translation invariant non-commutative gauge model, Eur. Phys. J. C 62 (2009), 433-443, arXiv:0901.1681.
  45. de Goursac A., Wallet J.-C., Wulkenhaar R., Noncommutative induced gauge theory, Eur. Phys. J. C 51 (2007), 977-987, hep-th/0703075.
  46. Wallet J.-C., Noncommutative induced gauge theories on Moyal spaces, J. Phys. Conf. Ser. 103 (2008), 012007, 20 pages, arXiv:0708.2471.
  47. Matusis A., Susskind L., Toumbas N., The IR/UV connection in the non-commutative gauge theories, J. High Energy Phys. 2000 (2000), no. 12, 002, 18 pages, hep-th/0002075.
  48. Grosse H., Wohlgenannt M., Induced gauge theory on a noncommutative space, Eur. Phys. J. C 52 (2007), 435-450, hep-th/0703169.
  49. Blaschke D.N., Grosse H., Schweda M., Non-commutative U(1) gauge theory on RΘ4 with oscillator term and BRST symmetry, Europhys. Lett. 79 (2007), 61002, 3 pages, arXiv:0705.4205.
  50. Blaschke D.N., Grosse H., Kronberger E., Schweda M., Wohlgenannt M., Loop calculations for the non-commutative U(1) gauge field model with oscillator term, arXiv:0912.3642.
  51. de Goursac A., On the effective action of noncommutative Yang-Mills theory, J. Phys. Conf. Ser. 103 (2008), 012010, 16 pages, arXiv:0710.1162.
  52. 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.
  53. Ilderton A., Lundin J., Marklund M., Strong field, noncommutative QED, SIGMA 6 (2010), 041, 27 pages, arXiv:1003.4184.
  54. Fischer A., Szabo R.J., UV/IR duality in noncommutative quantum field theory, arXiv:1001.3776.
  55. Folland G.B., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.
  56. Dubois-Violette M., Dérivations et calcul différentiel non commutatif, C.R. Acad. Sci. Paris Sér. I Math. 307 (1988), 403-408.
  57. Masson T., Examples of derivation-based differential calculi related to noncommutative gauge theories, Int. J. Geom. Methods Mod. Phys. 5 (2008), 1315-1336, arXiv:0810.4815.
  58. Wulkenhaar R., Non-compact spectral triples with finite volume, arXiv:0907.1351.
  59. Cagnache E., Masson T., Wallet J.-C., Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus, J. Noncommut. Geom., to appear, arXiv:0804.3061.
  60. Madore J., Mourad J., Quantum space-time and classical gravity, J. Math. Phys. 39 (1998), 423-442, gr-qc/9607060.
  61. Dubois-Violette M., Madore J., Masson T., Mourad J., On curvature in noncommutative geometry, J. Math. Phys. 37 (1996), 4089-4102, q-alg/9512004.
  62. Hollands S., Wald R.M., On the renormalization group in curved spacetime, Comm. Math. Phys. 237 (2003), 123-160, gr-qc/0209029.

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