×

zbMATH — the first resource for mathematics

Renormalisation of \(\phi^4\)-theory on noncommutative \(\mathbb R^{4}\) in the matrix base. (English) Zbl 1075.82005
The papers shows that the real four-dimensional Euclidean noncommutative \(\phi^4\)-model is renormalisable to all orders in perturbation theory. The proof is hard, intricate and lengthy. A long introduction displayed its heuristics and its strategy. Loosely speaking one re-write the problem in the harmonic oscillator base of the Moyal plane and then one uses Meixner polynomials to solve the free theory. The direct calculation of the Feynman graphs is intractable, and to circumvent this hard task, one refers to a renormalization method based on flow equations which they have previously written for non-local (dynamical) matrix models. The matrix Polchinski equation is solved by using ribbon graphs drawn on Riemann surfaces.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81T75 Noncommutative geometry methods in quantum field theory
81T08 Constructive quantum field theory
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Minwalla, S., Van Raamsdonk, M., Seiberg, N.: Noncommutative perturbative dynamics. JHEP 0002, 020 (2000) · Zbl 0959.81108 · doi:10.1088/1126-6708/2000/02/020
[2] Chepelev, I., Roiban, R.: Renormalization of quantum field theories on noncommutative ?d. I: Scalars. JHEP 0005, 037 (2000) · Zbl 0990.81756 · doi:10.1088/1126-6708/2000/05/037
[3] Chepelev, I., Roiban, R.: Convergence theorem for non-commutative Feynman graphs and renormalization. JHEP 0103, 001 (2001) · Zbl 0990.81756 · doi:10.1088/1126-6708/2001/03/001
[4] Langmann, E., Szabo, R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B 533, 168 (2002) · Zbl 0994.81116 · doi:10.1016/S0370-2693(02)01650-7
[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) · Zbl 1084.58008 · doi:10.1007/s00220-004-1057-z
[6] Langmann, E.: Interacting fermions on noncommutative spaces: Exactly solvable quantum field theories in 2n+1 dimensions. Nucl. Phys. B 654, 404 (2003) · Zbl 1010.81082 · doi:10.1016/S0550-3213(03)00006-3
[7] Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of noncommutative field theory in background magnetic fields. Phys. Lett. B 569, 95 (2003) · Zbl 1059.81608 · doi:10.1016/j.physletb.2003.07.020
[8] Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of quantum field theory on noncommutative phase spaces. JHEP 0401, 017 (2004) · Zbl 1243.81205 · doi:10.1088/1126-6708/2004/01/017
[9] Wilson, K.G., Kogut, J.B.: The Renormalization Group And The Epsilon Expansion. Phys. Rept. 12, 75 (1974) · doi:10.1016/0370-1573(74)90023-4
[10] Polchinski, J.: Renormalization And Effective Lagrangians. Nucl. Phys. B 231, 269 (1984) · doi:10.1016/0550-3213(84)90287-6
[11] Keller, G., Kopper, C., Salmhofer, M.: Perturbative renormalization and effective Lagrangians in ?44. Helv. Phys. Acta 65, 32 (1992)
[12] Grosse, H., Wulkenhaar, R.: Power-counting theorem for non-local matrix models and renormalisation. Commun. Math. Phys. 254, 91-127 (2005) · Zbl 1079.81049 · doi:10.1007/s00220-004-1238-9
[13] Meixner, J.: Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. London Math. Soc. 9, 6 (1934) · Zbl 0008.16205 · doi:10.1112/jlms/s1-9.1.6
[14] Grosse, H., Wulkenhaar, R.: Renormalisation of ?4 theory on noncommutative ?2 in the matrix base. JHEP 0312, 019 (2003) · Zbl 1142.81365 · doi:10.1088/1126-6708/2003/12/019
[15] Masson, D.R., Repka, J.: Spectral theory of Jacobi matrices in ?2(?) and the Lie algebra. SIAM J. Math. Anal. 22, 1131 (1991) · Zbl 0729.33011 · doi:10.1137/0522073
[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] Rivasseau, V., Vignes-Tourneret, F., Wulkenhaar, R.: Renormalization of noncommutative ?4-theory by multi-scale analysis. http://arxiv.org/abs/hep-th/0501036, 2055. · Zbl 1109.81056
[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] Grosse, H., Wulkenhaar, R.: Renormalisation of ?4 theory on noncommutative ?4 to all orders. To appear in Lett. Math. Phys., http://arxiv.org/abs/hep-th/0403232, 2004 · Zbl 1115.81055
[20] Gracia-Bondía, J.M., Várilly, J.C.: Algebras Of Distributions Suitable For Phase Space Quantum Mechanics. 1. J. Math. Phys. 29, 869 (1988) · Zbl 0652.46026
[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) · doi:10.1038/nature01944
[22] Grosse, H., Wulkenhaar, R.: The ?-function in duality-covariant noncommutative ?4-theory. Eur. Phys. J. C 35, 277-282 (2004) · Zbl 1191.81207 · doi:10.1140/epjc/s2004-01853-x
[23] Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 032 (1999) · Zbl 0957.81085 · doi:10.1088/1126-6708/1999/09/032
[24] Blau, M., Figueroa-O?Farrill, J., Hull, C., Papadopoulos, G.: A new maximally supersymmetric background of IIB superstring theory. JHEP 0201, 047 (2002) · doi:10.1088/1126-6708/2002/01/047
[25] Gradshteyn, I.S., Ryzhik, I.M.: Tables of Series, Produces, and Integrals. Sixth Edition. San Diego: Academic Press, 2000 · Zbl 0981.65001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.