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


SIGMA 6 (2010), 047, 13 pages      arXiv:1003.4877      http://dx.doi.org/10.3842/SIGMA.2010.047
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Translation-Invariant Noncommutative Renormalization

Adrian Tanasa a, b
a) Centre de Physique Théorique, CNRS, UMR 7644, École Polytechnique, 91128 Palaiseau, France
b) Institutul de Fizică şi Inginerie Nucleară Horia Hulubei, P.O. Box MG-6, 077125 Măgurele, România

Received March 25, 2010, in final form May 24, 2010; Published online June 08, 2010

Abstract
We review here the construction of a translation-invariant scalar model which was proved to be perturbatively renormalizable on Moyal space. Some general considerations on nonlocal renormalizability are given. Finally, we present perspectives for generalizing these quantum field theoretical techniques to group field theory, a new setting for quantum gravity.

Key words: noncommutative quantum field theory; Moyal space; locality; translation-invariance.

pdf (298 kb)   ps (190 kb)   tex (75 kb)

References

  1. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  2. Grosse H., Wulkenhaar R., Renormalization of φ4-theory on noncommutative R4 in the matrix base, Comm. Math. Phys. 256 (2005), 305-374, hep-th/0401128.
  3. Grosse H., Wulkenhaar R., The β-function in duality-covariant noncommutative φ4-theory, Eur. Phys. J. C 35 (2004), 277-282, hep-th/0402093.
    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.
  4. Gurau R., Magnen J., Rivasseau V., Tanasa A., A translation-invariant renormalizable non-commutative scalar model, Comm. Math. Phys. 287 (2008), 275-290, arXiv:0802.0791.
  5. Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), 207-299, hep-th/0109162.
  6. Witten E., Noncommutative geometry and string field theory, Nuclear Phys. B 268 (1986), 253-294.
  7. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  8. Connes A, Douglas M.R., Schwarz A., Noncommutative geometry and matrix theory: compactification on tori, J. High Energy Phys. 1998 (1998), no. 2, 003, 35 pages, hep-th/9711162.
  9. Douglas M.R., Hull C., D-branes and the noncommutative torus, J. High Energy Phys. 1998 (1998), no. 2, 008, 5 pages, hep-th/9711165.
  10. Freidel L., Livine E.R., 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett. 96 (2006), 221301, 4 pages, hep-th/0512113.
  11. Joung E., Mourad J., Noui K., Three dimensional quantum geometry and deformed Poincaré symmetry, J. Math. Phys. 50 (2009), 052503, 29 pages, arXiv:0806.4121.
  12. Susskind L., The quantum hall fluid and non-commutative Chern Simons theory, hep-th/0101029.
    Polychronakos A.P., Quantum Hall states on the cylinder as unitary matrix Chern-Simons theory, J. High Energy Phys. 2001 (2001), no. 6, 070, 28 pages, hep-th/0106011.
    Hellerman S., Van Raamsdonk M., Quantum Hall physics equals noncommutative field theory?, J. High Energy Phys. 2001 (2001), no. 10, 039, 18 pages, hep-th/0103179.
  13. Heslop P., Sibold K., Quantized equations of motion in non-commutative theories, Eur. Phys. J. C 41 (2005), 545-556, hep-th/0411161.
    Liao Y., Sibold K., Time-ordered perturbation theory on noncommutative space-time. II. Unitarity, Eur. Phys. J. C 25 (2002), 479-486, hep-th/0206011.
    Liao Y., Sibold K., Time-ordered perturbation theory on noncommutative space-time. Basic rules, Eur. Phys. J. C 25 (2002), 469-477, hep-th/0205269.
    Denk S., Schweda M., Time ordered perturbation theory for nonlocal interactions: applications to NCQFT, J. High Energy Phys. 2003 (2003), no. 3, 032, 22 pages, hep-th/0306101.
    Bahns D., Doplicher S., Fredenhagen K., Piacitelli G., On the unitarity problem in space/time noncommutative theories, Phys. Lett. B 533 (2002), 178-181, hep-th/0201222.
  14. de Goursac A., Masson T., Wallet J.-C., Noncommutative ε-graded connections and application to Moyal space, arXiv:0811.3567.
  15. Magnen J., Rivasseau V., Tanasa A., Commutative limit of a renormalizable noncommutative model, Europhys. Lett. 86 (2009), 11001, 6 pages, arXiv:0807.4093.
  16. Minwalla S., Van Raamsdonk M., Seiberg N., Noncommutative perturbative dynamics, J. High Energy Phys. 2000 (2000), no. 2, 020, 31 pages, hep-th/9912072.
  17. 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.
    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.
  18. Steinacker H., Emergent geometry and gravity from matrix models: an introduction, Classical Quantum Gravity 27 (2010), 133001, 46 pages, arXiv:1003.4134.
    Grosse H., Steinacker H., Wohlgenannt M., Emergent gravity, matrix models and UV/IR mixing, J. High Energy Phys. 2008 (2008), no. 4, 023, 30 pages, arXiv:0802.0973.
    Steinacker H., Emergent gravity from noncommutative gauge theory, J. High Energy Phys. 2007 (2007), no. 12, 049, 36 pages, arXiv:0708.2426.
  19. Langmann E., Szabo R.J., Duality in scalar field theory on noncommutative phase spaces, Phys. Lett. B 533, (2002), 168-177, hep-th/0202039.
  20. Fischer A., Szabo R.J., UV/IR duality in noncommutative quantum field theory, arXiv:1001.3776.
    Fischer A., Szabo R.J., Duality covariant quantum field theory on noncommutative Minkowski space, J. High Energy Phys. 2009 (2009), no. 2, 031, 36 pages, arXiv:0810.1195.
  21. de Goursac A., Wallet J.-C., Symmetries of noncommutative scalar field theory, arXiv:0911.2645.
  22. Langmann E., Szabo R.J., Zarembo K., Exact solution of noncommutative field theory in background magnetic fields, Phys. Lett. B 569 (2003), 95-101, hep-th/0303082.
    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.
  23. Gurau R., Tanasa A., Dimensional regularization and renormalization of non-commutative quantum field theory, Ann. Henri Poincaré 9 (2008), 655-683, arXiv:0706.1147.
  24. Tanasa A., Vignes-Tourneret F., Hopf algebra of non-commutative field theory, J. Noncommut. Geom. 2 (2008), 125-139, arXiv:0707.4143.
  25. Rivasseau V., Tanasa A., Parametric representation of "covariant" noncommutative QFT models, Comm. Math. Phys. 279 (2007), 355-379, math-ph/0701034.
  26. Tanasa A., Overview of the parametric representation of renormalizable non-commutative field theory, J. Phys. Conf. Ser. 103 (2008), 012012, 10 pages, arXiv:0709.2270.
  27. Tanasa A., Feynman amplitudes in renormalizable non-commutative quantum field theory, in Modern Encyclopedia of Mathematical Physics, Editors I. Aref'eva and D. Sternheimer, Springer, Berlin, arXiv:0711.3355.
  28. Gurau R., Malbouisson A., Rivasseau V., Tanasa A., Non-commutative complete Mellin representation for Feynman amplitudes, Lett. Math. Phys. 81 (2007), 161-175, arXiv:0705.3437.
  29. Aluffi P., Marcolli M., Feynman motives of banana graphs, arXiv:0807.1690.
  30. de Goursac A., Tanasa A., Wallet J.-C., Vacuum configurations for renormalizable non-commutative scalar models, Eur. Phys. J. C 53 (2007), 459-466, arXiv:0709.3950.
  31. Ben Geloun J., Tanasa A., One-loop β functions of a translation-invariant renormalizable noncommutative scalar model, Lett. Math. Phys. 86 (2008), 19-32, arXiv:0806.3886.
  32. Tanasa A., Kreimer D., Combinatorial Dyson-Schwinger equation in noncommutative field theory, arXiv:0907.2182.
  33. Gurau R., Rosten O.J., Wilsonian renormalization of noncommutative scalar field theory, J. High Energy Phys. 2009 (2009), no. 7, 064, 45 pages, arXiv:0902.4888.
  34. Tanasa A., Parametric representation of a translation-invariant renormalizable noncommutative model, J. Phys. A: Math. Theor. 42 (2009), 365208, 18 pages, arXiv:0807.2779.
  35. Dudal D., Sorella S., Vandersickel N., Verschelde H., New features of the gluon and ghost propagator in the infrared region from the Gribov-Zwanziger approach, Phys. Rev. D 77 (2008), 071501, 5 pages, arXiv:0711.4496.
    Dudal D., Gracey J., Sorella S., Vandersickel N., Verschelde H., A refinement of the Gribov-Zwanziger approach in the Landau gauge: infrared propagators in harmony with the lattice results, Phys. Rev. D 78 (2008), 065047, 30 pages, arXiv:0806.4348.
  36. Palma G., Patil S., UV/IR mode mixing and the CMB, Phys. Rev. D 80 (2009), 083010, 11 pages, arXiv:0906.4727.
    Patil S., On Semi-classical degravitation and the cosmological constant problems, arXiv:1003.3010.
  37. Helling R., You J., Macroscopic screening of Coulomb potentials from UV/IR-mixing, J. High Energy Phys. 2008 (2008), no. 6, 067, 10 pages, arXiv:0707.1885.
  38. Krajewski T., Rivasseau V., Tanasa A., Wang Z., Topological graph polynomials and quantum field theory. I. Heat kernel theories, J. Noncommut. Geom. 4 (2010), 29-82, arXiv:0811.0186.
  39. Tanasa A., Algebraic structures in quantum gravity, Classical Quantum Gravity 27 (2010), 095008, 17 pages, arXiv:0909.5631.
  40. 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.
  41. Blaschke D.N., Rofner A., Sedmik R., One-loop calculations and detailed analysis of the localized non-commutative p–2 U(1) gauge model, SIGMA 6 (2010), 037, 20 pages, arXiv:0908.1743.
    Blaschke D.N., Kronberger E., Rofner A., Schweda M., Sedmik R., Wohlgenannt M., On the problem of renormalizability in non-commutative gauge field models - a critical review, Fortschr. Phys. 58 (2010), 364-372, arXiv:0908.0467.
    Blaschke D.N., Rofner A., Schweda M., Sedmik R., Improved localization of a renormalizable non-commutative translation invariant U(1) gauge model, Europhys. Lett. 86 (2009), 51002, 10 pages, arXiv:0903.4811.
    Blaschke D.N., Rofner A., Schweda M., Sedmik R., One-loop calculations for a translation invariant non-commutative gauge model, Eur. Phys. J. C 62 (2009), 433-443, arXiv:0901.1681.
    Blaschke D.N., Gieres F., Kronberger E., Schweda M., Sedmik R., Wohlgenannt M., Translation-invariant models for non-commutative gauge fields, J. Phys. A: Math. Theor. 41 (2008), 252002, 7 pages, arXiv:0804.1914.
    Tanasa A., Scalar and gauge translation-invariant noncommutative models, Romanian J. Phys. 53 (2008), 1207-1212, arXiv:0808.3703.
    Vilar L.C.Q., Ventura O.S., Tedesco D.G., Lemes V.E.R., Renormalizable noncommutative U(1) gauge theory without IR/UV mixing, arXiv:0902.2956.
  42. de Goursac A., Noncommutative geometry, gauge theory and renormalization, PhD Thesis, arXiv:0910.5158.
    de Goursac A., Wallet J.-C., Wulkenhaar R., On the vacuum states for non-commutative gauge theory, Eur. Phys. J. C 56 (2008), 293-304, arXiv:0803.3035.
    de Goursac A., On the effective action of noncommutative Yang-Mills theory, J. Phys. Conf. Ser. 103 (2008), 012010, 19 pages, arXiv:0710.1162.
    de Goursac A., Wallet J.-C., Wulkenhaar R., Noncommutative induced gauge theory, Eur. Phys. J. C 51 (2007), 977-987, hep-th/0703075.
    Grosse H., Wohlgenannt M., Induced gauge theory on a noncommutative space, Eur. Phys. J. C 52 (2007), 435-450, hep-th/0703169.
  43. Freidel L., Group field theory: an overview, Internat. J. Theoret. Phys. 44 (2005), 1769-1783, hep-th/0505016.
    Oriti D., Quantum gravity as a group field theory: a sketch, J. Phys. Conf. Ser. 33 (2006), 271-278, gr-qc/0512048.
    Oriti D., The group field theory approach to quantum gravity, gr-qc/0607032.
    Oriti D., The group field theory approach to quantum gravity: some recent results, arXiv:0912.2441.
  44. Magnen J., Noui K., Rivasseau V., Smerlak M., Scaling behaviour of three-dimensional group field theory, Classical Quantum Gravity 26 (2009), 185012, 20 pages, arXiv:0906.5477.
    Freidel L., Gurau R., Oriti D., Group field theory renormalization - the 3d case: power counting of divergences, Phys. Rev. D 80 (2009), 044007, 20 pages, arXiv:0905.3772.
    Ben Geloun J., Krajewski T., Magnen J., Rivasseau V., Linearized group field theory and power counting theorems, Classical Quantum Gravity, to appear, arXiv:1002.3592.
    Ben Geloun J., Magnen J., Rivasseau V., Bosonic colored group field theory, Classical Quantum Gravity, to appear, arXiv:0911.1719.
    Gurau R., Colored group field theory, arXiv:0907.2582.
  45. Engle J., Livine E., Pereira R., Rovelli C., LQG vertex with finite Immirzi parameter, Nuclear Phys. B 799 (2008), 136-149, arXiv:0711.0146.
  46. Freidel L., Krasnov K., A new spin Foam model for 4D gravity, Classical Quantum Gravity 25 (2008), 125018, 36 pages, arXiv:0708.1595.

Previous article   Next article   Contents of Volume 6 (2010)