Communications in Mathematical Physics

, Volume 256, Issue 2, pp 305–374 | Cite as

Renormalisation of ϕ4-Theory on Noncommutative ℝ4 in the Matrix Base

Article

Abstract

We prove that the real four-dimensional Euclidean noncommutative ϕ4-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative ℝ4. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Minwalla, S., Van Raamsdonk, M., Seiberg, N.: Noncommutative perturbative dynamics. JHEP 0002, 020 (2000)Google Scholar
  2. 2.
    Chepelev, I., Roiban, R.: Renormalization of quantum field theories on noncommutative ℝd. I: Scalars. JHEP 0005, 037 (2000)Google Scholar
  3. 3.
    Chepelev, I., Roiban, R.: Convergence theorem for non-commutative Feynman graphs and renormalization. JHEP 0103, 001 (2001)Google Scholar
  4. 4.
    Langmann, E., Szabo, R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B 533, 168 (2002)Google Scholar
  5. 5.
    Gayral, V., Gracia-Bondía, J.M., Iochum, B., Schücker, T., Várilly, J.C.: Moyal planes are spectral triples. Commun. Math. Phys. 246, 569 (2004)Google Scholar
  6. 6.
    Langmann, E.: Interacting fermions on noncommutative spaces: Exactly solvable quantum field theories in 2n+1 dimensions. Nucl. Phys. B 654, 404 (2003)Google Scholar
  7. 7.
    Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of noncommutative field theory in background magnetic fields. Phys. Lett. B 569, 95 (2003)Google Scholar
  8. 8.
    Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of quantum field theory on noncommutative phase spaces. JHEP 0401, 017 (2004)Google Scholar
  9. 9.
    Wilson, K.G., Kogut, J.B.: The Renormalization Group And The Epsilon Expansion. Phys. Rept. 12, 75 (1974)Google Scholar
  10. 10.
    Polchinski, J.: Renormalization And Effective Lagrangians. Nucl. Phys. B 231, 269 (1984)Google Scholar
  11. 11.
    Keller, G., Kopper, C., Salmhofer, M.: Perturbative renormalization and effective Lagrangians in φ44. Helv. Phys. Acta 65, 32 (1992)Google Scholar
  12. 12.
    Grosse, H., Wulkenhaar, R.: Power-counting theorem for non-local matrix models and renormalisation. Commun. Math. Phys. 254, 91–127 (2005)Google Scholar
  13. 13.
    Meixner, J.: Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. London Math. Soc. 9, 6 (1934)Google Scholar
  14. 14.
    Grosse, H., Wulkenhaar, R.: Renormalisation of ϕ4 theory on noncommutative ℝ2 in the matrix base. JHEP 0312, 019 (2003)Google Scholar
  15. 15.
    Masson, D.R., Repka, J.: Spectral theory of Jacobi matrices in 2(ℤ) and the Open image in new window Lie algebra. SIAM J. Math. Anal. 22, 1131 (1991)Google Scholar
  16. 16.
    Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. http://arXiv.org/abs/math.CA/9602214, 1996
  17. 17.
    Rivasseau, V., Vignes-Tourneret, F., Wulkenhaar, R.: Renormalization of noncommutative ϕ4-theory by multi-scale analysis. http://arxiv.org/abs/hep-th/0501036, 2055.
  18. 18.
    Rivasseau, V., Vignes-Tourneret, F.: Non-Commutative Renormalization. In: Proceedings of Conference, “Rigorous Quantum Field Theory” in honor of J. Bros, http://arxiv.org/abs/hep-th/0409312, 2004
  19. 19.
    Grosse, H., Wulkenhaar, R.: Renormalisation of ϕ4 theory on noncommutative ℝ4 to all orders. To appear in Lett. Math. Phys., http://arxiv.org/abs/hep-th/0403232, 2004
  20. 20.
    Gracia-Bondía, J.M., Várilly, J.C.: Algebras Of Distributions Suitable For Phase Space Quantum Mechanics. 1. J. Math. Phys. 29, 869 (1988)Google Scholar
  21. 21.
    Luminet, J.P.M., Weeks, J., Riazuelo, A., Lehoucq, R., Uzan, J.P.: Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background. Nature 425, 593 (2003)Google Scholar
  22. 22.
    Grosse, H., Wulkenhaar, R.: The β-function in duality-covariant noncommutative ϕ4-theory. Eur. Phys. J. C 35, 277–282 (2004)Google Scholar
  23. 23.
    Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 032 (1999)Google Scholar
  24. 24.
    Blau, M., Figueroa-O’Farrill, J., Hull, C., Papadopoulos, G.: A new maximally supersymmetric background of IIB superstring theory. JHEP 0201, 047 (2002)Google Scholar
  25. 25.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of Series, Produces, and Integrals. Sixth Edition. San Diego: Academic Press, 2000Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität WienWienAustria
  2. 2.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany

Personalised recommendations