Quantum Field Theory in a Non-Commutative Space: Theoretical Predictions and Numerical Results on the Fuzzy

Marco Panero

Symmetry, integrability and geometry: methods and applications, Tome 2 (2006)

Résumé

We review some recent progress in quantum field theory in non-commutative space, focusing onto the fuzzy sphere as a non-perturbative regularisation scheme. We first introduce the basic formalism, and discuss the limits corresponding to different commutative or non-commutative spaces. We present some of the theories which have been investigated in this framework, with a particular attention to the scalar model. Then we comment on the results recently obtained from Monte Carlo simulations, and show a preview of new numerical data, which are consistent with the expected transition between two phases characterised by the topology of the support of a matrix eigenvalue distribution.

Détail

Keywords: non-commutative geometry; matrix models; non-perturbative effects; phase transitions.

Bibliographie

[1] 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 | DOI | MR | Zbl

[2] Landi G., An introduction to noncommutative spaces and their geometry, hep-th/9701078

[3] Douglas M. R., Nekrasov N. A., “Noncommutative field theory”, Rev. Modern Phys., 73 (2001), 977–1029 ; hep-th/0106048 | DOI | MR

[4] Szabo R. J., Symmetry, gravity and noncommutativity, hep-th/0606233 | MR

[5] Karabali D., Nair V. P., Randjbar-Daemi S., Fuzzy spaces, the M(atrix) model and the quantum Hall effect, hep-th/0407007 | MR

[6] Connes A., Douglas M. R., Schwarz A. S., “Noncommutative geometry and matrix theory: Compactification on tori”, JHEP, 02 (1998), 003, 35 pp., ages ; hep-th/9711162 | DOI | MR

[7] Douglas M. R., Hull C. M., “D-branes and the noncommutative torus”, JHEP, 2 (1998), Paper 008, 5 pp., ages ; hep-th/9711165 | MR | Zbl

[8] Alekseev A. Y., Recknagel A., Schomerus V., “Non-commutative world-volume geometries: Branes on SU(2) and fuzzy spheres”, JHEP, 9 (1999), Paper 023, 20 pp., ages ; hep-th/9908040 | MR | Zbl

[9] Seiberg N., Witten E., “String theory and noncommutative geometry”, JHEP, 9 (1999), Paper 032, 93 pp., ages ; hep-th/9908142 | MR | Zbl

[10] Kar S., “Non-commutativity, zero modes and D-brane geometry”, Nuclear Phys. B, 577 (2000), 171–182 ; hep-th/9911251 | DOI | MR | Zbl

[11] Kar S., “D-branes, cyclic symmetry and noncommutative geometry”, Modern Phys. Lett. A, 18 (2003), 1053–1065 ; hep-th/0006073 | DOI | MR | Zbl

[12] Chamseddine A. H., Felder G., Fröhlich J., “Gravity in noncommutative geometry”, Comm. Math. Phys., 155 (1993), 205–217 ; hep-th/9209044 | DOI | MR | Zbl

[13] Groenewold H. J., “On the principles of elementary quantum mechanics”, Physica, 12 (1946), 405–460 | DOI | MR | Zbl

[14] Moyal J. E., “Quantum mechanics as a statistical theory”, Proc. Cambridge Phil. Soc., 45 (1949), 99–124 | DOI | MR | Zbl

[15] Szabo R. J., “Quantum field theory on noncommutative spaces”, Phys. Rept., 378 (2003), 207–299 ; hep-th/0109162 | DOI | MR | Zbl

[16] Doplicher S., Fredenhagen K., Roberts J. E., “Space-time quantization induced by classical gravity”, Phys. Lett. B, 331 (1994), 39–44 | DOI | MR

[17] Doplicher S., Spacetime and fields, a quantum texture, hep-th/0105251

[18] Filk T., “Divergencies in a field theory on quantum space”, Phys. Lett. B, 376 (1996), 53–58 | DOI | MR

[19] Chen G. H., Wu Y. S., On critical phenomena in a noncommutative space, hep-th/0103020

[20] Jain A., Joglekar S. D., “Causality violation in non-local quantum field theory”, Internat. J. Modern Phys. A, 19 (2004), 3409–3426 ; hep-th/0307208 | DOI

[21] Chaichian M., Kulish P. P., Nishijima K., Tureanu A., “On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT”, Phys. Lett. B, 604 (2004), 98–102 ; hep-th/0408069 | DOI | MR

[22] Minwalla S., Van Raamsdonk M., Seiberg N., “Noncommutative perturbative dynamics”, JHEP, 2 (2000), Paper 020, 31 pp., ages ; hep-th/9912072 | Zbl

[23] Hinchliffe I., Kersting N., Ma Y. L., “Review of the phenomenology of noncommutative geometry”, Internat. J. Modern Phys. A, 19 (2004), 179–204 ; hep-ph/0205040 | DOI | MR | Zbl

[24] Gubser S. S., Sondhi S. L., “Phase structure of non-commutative scalar field theories”, Nuclear Phys. B, 605 (2001), 395–424 ; hep-th/0006119 | DOI | MR | Zbl

[25] Castorina P., Zappalà D., “Nonuniform symmetry breaking in noncommutative $\lambda\Phi^4$ theory”, Phys. Rev. D, 68 (2003), 065008, 7 pp., ages ; hep-th/0303030 | DOI | MR

[26] Langmann E., Szabo R. J., “Duality in scalar field theory on noncommutative phase spaces”, Phys. Lett. B, 533 (2002), 168–177 ; hep-th/0202039 | DOI | MR | Zbl

[27] Govindarajan T. R., Kürkçüoǧlu S., Panero M., “Nonlocal regularisation of noncommutative field theories”, Modern Phys. Lett. A, 21 (2006), 1851–1863 ; hep-th/0604061 | DOI | MR | Zbl

[28] 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 | DOI | MR | Zbl

[29] Rivasseau V., Vignes-Tourneret F., Wulkenhaar R., “Renormalization of noncommutative $\phi^4$-theory by multi-scale analysis”, Comm. Math. Phys., 262 (2006), 565–594 ; hep-th/0501036 | DOI | MR | Zbl

[30] Grosse H., Steinacker H., “Renormalization of the noncommutative $\phi^3$ model through the Kontsevich model”, Nuclear Phys. B, 746 (2006), 202–226 ; hep-th/0512203 | DOI | MR | Zbl

[31] Grosse H., Steinacker H., A nontrivial solvable noncommutative $\phi^3$ model in 4 dimensions, hep-th/0603052

[32] Grosse H., Steinacker H., Exact renormalization of a noncommutative $\phi^3$ model in 6 dimensions, hep-th/0607235

[33] Ambjørn J., Makeenko Y. M., Nishimura J., Szabo R. J., “Finite $N$ matrix models of noncommutative gauge theory”, JHEP, 11 (1999), Paper 029, 17 pp., ages ; hep-th/9911041 | MR | Zbl

[34] 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 | DOI | MR | Zbl

[35] Ambjørn J., Makeenko Y. M., Nishimura J., Szabo R. J., “Lattice gauge fields and discrete noncommutative Yang–Mills theory”, JHEP, 5 (2000), Paper 023, 49 pp., ages ; hep-th/0004147

[36] Azuma T., Bal S., Nagao K., Nishimura J., “Nonperturbative studies of fuzzy spheres in a matrix model with the Chern–Simons term”, JHEP, 5 (2004), Paper 005, 36 pp., ages ; hep-th/0401038 | MR

[37] Bietenholz W., Hofheinz F., Nishimura J., “The renormalizability of 2D Yang–Mills theory on a non-commutative geometry”, JHEP, 9 (2002), Paper 009, 18 pp., ages ; hep-th/0203151 | MR

[38] Bietenholz W., Hofheinz F., Nishimura J., “Simulating non-commutative field theory”, Nuclear Phys. Proc. Suppl., 119 (2003), 941–946 ; hep-lat/0209021 | DOI | Zbl

[39] Ambjørn J., Catterall S., “Stripes from (noncommutative) stars”, Phys. Lett. B, 549 (2002), 253–259 ; hep-lat/0209106 | DOI | MR | Zbl

[40] Bietenholz W., Hofheinz F., Nishimura J., “Phase diagram and dispersion relation of the non-commutative lambda $\phi^4$ model in $d=3$”, JHEP, 6 (2004), Paper 042, 36 pp., ages ; hep-th/0404020 | MR

[41] Bietenholz W., Bigarini A., Hofheinz F., Nishimura J., Susaki Y., Volkholz J., “Numerical results for U(1) gauge theory on 2d and 4d non-commutative spaces”, Fortsch. Phys., 53 (2005), 418–425 ; hep-th/0501147 | DOI | MR

[42] Bietenholz W., Nishimura J., Susaki Y., Volkholz J., A non-perturbative study of 4d U(1) non-commutative gauge theory – the fate of one-loop instability, hep-th/0608072 | MR

[43] Madore J., “The fuzzy sphere”, Classical Quantum Gravity, 9 (1992), 69–87 | DOI | MR | Zbl

[44] Alexanian G., Balachandran A. P., Immirzi G., Ydri B., “Fuzzy $\mathbb C\rm P^2$”, J. Geom. Phys., 42 (2002), 28–53 ; hep-th/0103023 | DOI | MR | Zbl

[45] Balachandran A. P., Dolan B. P., Lee J. H., Martin X., O'Connor D., “Fuzzy complex projective spaces and their star-products”, J. Geom. Phys., 43 (2002), 184–204 ; hep-th/0107099 | DOI | MR | Zbl

[46] Hammou A. B., Lagraa M., Sheikh-Jabbari M. M., “Coherent state induced star-product on $\mathbb R_\lambda^3$ and the fuzzy sphere”, Phys. Rev. D, 66 (2002), 025025, 11 pp., ages ; hep-th/0110291 | DOI | MR

[47] Medina J., O'Connor D., “Scalar field theory on fuzzy $S^4$”, JHEP, 11 (2003), Paper 051, 13 pp., ages ; hep-th/0212170 | MR

[48] Vaidya S., Ydri B., “On the origin of the UV-IR mixing in noncommutative matrix geometry”, Nuclear Phys. B, 671 (2003), 401–431 ; hep-th/0305201 | DOI | MR | Zbl

[49] Dolan B. P., O'Connor D., “A fuzzy three sphere and fuzzy tori”, JHEP, 10 (2003), Paper 060, 16 pp., ages ; hep-th/0306231 | MR

[50] Balachandran A. P., Kürkçüoǧlu S., Vaidya S., Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114

[51] Sheikh-Jabbari M. M., Inherent holography in fuzzy spaces and an $N$-tropic approach to the cosmological constant problem, hep-th/0605110 | MR

[52] Martin X., “A matrix phase for the $\phi^4$ scalar field on the fuzzy sphere”, JHEP, 4 (2004), Paper 077, 20 pp., ages ; hep-th/0402230 | MR

[53] García Flores F., O'Connor D., Martin X., “Simulating the scalar field on the fuzzy sphere”, PoS(LAT2005)262, Proceedings for the XXIIIrd International Symposium on Lattice Field Theory, 2006, 6 pages; hep-lat/0601012

[54] Medina J., Bietenholz W., Hofheinz F., O'Connor D., “Field theory simulations on a fuzzy sphere: an alternative to the lattice”, PoS(LAT2005)263, Proceedings for the XXIIIrd International Symposium on Lattice Field Theory, 2006, 6 pages; hep-lat/0509162

[55] Azuma T., Bal S., Nagao K., Nishimura J., “Absence of a fuzzy $S^4$ phase in the dimensionally reduced 5d Yang–Mills–Chern–Simons model”, JHEP, 7 (2004), Paper 066, 11 pp., ages ; hep-th/0405096 | MR

[56] Anagnostopoulos K. N., Azuma T., Nagao K., Nishimura J., “Impact of supersymmetry on the nonperturbative dynamics of fuzzy spheres”, JHEP, 9 (2005), Paper 046, 27 pp., ages ; hep-th/0506062 | MR

[57] Azuma T., Bal S., Nagao K., Nishimura J., “Perturbative versus nonperturbative dynamics of the fuzzy $S^2\times S^2$”, JHEP, 9 (2005), Paper 047, 20 pp., ages ; hep-th/0506205 | MR

[58] O'Connor D., Ydri B., Monte Carlo simulation of a NC gauge theory on the fuzzy sphere, hep-lat/0606013

[59] Panero M., Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere, hep-th/0608202 | MR

[60] Grosse H., Klimčík C., Prešnajder P., “Towards finite quantum field theory in noncommutative geometry”, Internat. J. Theor. Phys., 35 (1996), 231–244 ; hep-th/9505175 | DOI | MR | Zbl

[61] Chu C. S., Madore J., Steinacker H., “Scaling limits of the fuzzy sphere at one loop”, JHEP, 8 (2001), Paper 038, 17 pp., ages ; hep-th/0106205 | MR

[62] Dolan B. P., O'Connor D., Prešnajder P., “Matrix $\phi^4$ models on the fuzzy sphere and their continuum limits”, JHEP, 3 (2002), Paper 013, 15 pp., ages ; hep-th/0109084

[63] Dolan B. P., O'Connor D., Private communication

[64] Steinacker H., “A non-perturbative approach to non-commutative scalar field theory”, JHEP, 3 (2005), Paper 075, 39 pp., ages ; hep-th/0501174 | MR

[65] Klimčík C., “Gauge theories on the noncommutative sphere”, Comm. Math. Phys., 199 (1998), 257–279 ; hep-th/9710153 | DOI | MR | Zbl

[66] Carow-Watamura U., Watamura S., “Noncommutative geometry and gauge theory on fuzzy sphere”, Comm. Math. Phys., 212 (2000), 395–413 ; hep-th/9801195 | DOI | MR | Zbl

[67] Alekseev A. Y., Recknagel A., Schomerus V., “Brane dynamics in background fluxes and non-commutative geometry”, JHEP, 5 (2000), Paper 010, 25 pp., ages ; hep-th/0003187 | MR

[68] Hashimoto K., Krasnov K., “D-brane solutions in non-commutative gauge theory on fuzzy sphere”, Phys. Rev. D, 64 (2001), 046007, 11 pp., ages ; hep-th/0101145 | DOI | MR

[69] Kimura Y., “Noncommutative gauge theories on fuzzy sphere and fuzzy torus from matrix model”, Prog. Theoret. Phys., 106 (2001), 445–469 ; hep-th/0103192 | DOI | MR | Zbl

[70] Steinacker H., “Quantized gauge theory on the fuzzy sphere as random matrix model”, Nuclear Phys. B, 679 (2004), 66–98 ; hep-th/0307075 | DOI | MR | Zbl

[71] Iso S., Umetsu H., “Note on gauge theory on fuzzy supersphere”, Phys. Rev. D, 69 (2004), 105014, 7 pp., ages ; hep-th/0312307 | DOI | MR

[72] Kimura Y., “Nonabelian gauge field and dual description of fuzzy sphere”, JHEP, 4 (2004), Paper 058, 29 pp., ages ; hep-th/0402044 | MR

[73] Azuma T., Bal S., Nagao K., Nishimura J., “Dynamical aspects of the fuzzy $\mathbf C\rm P^2$ in the large $N$ reduced model with a cubic term”, JHEP, 5 (2006), Paper 061, 27 pp., ages ; hep-th/0405277 | MR

[74] Grosse H., Steinacker H., “Finite gauge theory on fuzzy $\mathbb C\rm P^2$”, Nuclear Phys. B, 707 (2005), 145–198 ; hep-th/0407089 | DOI | MR | Zbl

[75] Castro-Villarreal P., Delgadillo-Blando R., Ydri B., “A gauge-invariant UV-IR mixing and the corresponding phase transition for U(1) fields on the fuzzy sphere”, Nuclear Phys. B, 704 (2005), 111–153 ; hep-th/0405201 | DOI | MR | Zbl

[76] Behr W., Meyer F., Steinacker H., “Gauge theory on fuzzy $S^2\times S^2$ and regularization on noncommutative $\mathbb R^4$”, JHEP, 7 (2005), Paper 040, 38 pp., ages ; hep-th/0503041 | MR

[77] Grosse H., Klimčík C., Prešnajder P., “Field theory on a supersymmetric lattice”, Comm. Math. Phys., 185 (1997), 155–175 ; hep-th/9507074 | DOI | MR | Zbl

[78] Grosse H. Reiter G., “The fuzzy supersphere”, J. Geom. Phys., 28 (1998), 349–383 ; math-ph/9804013 | DOI | MR | Zbl

[79] Balachandran A. P., Kürkçüoǧlu S., Rojas E., “The star product on the fuzzy supersphere”, JHEP, 7 (2002), Paper 056, 22 pp., ages ; hep-th/0204170 | MR

[80] Balachandran A. P., Pinzul A., Qureshi B., “SUSY anomalies break $\mathcal N=2$ to $\mathcal N=1$: the supersphere and the fuzzy supersphere”, JHEP, 12 (2005), Paper 002, 14 pp., ages ; hep-th/0506037 | MR

[81] 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 | DOI | MR | Zbl

[82] Iso S., Umetsu H., “Gauge theory on noncommutative supersphere from supermatrix model”, Phys. Rev. D, 69 (2004), 105003, 7 pp., ages ; hep-th/0311005 | DOI | MR

[83] Hasebe K., Kimura Y., “Fuzzy supersphere and supermonopole”, Nuclear Phys. B, 709 (2005), 94–114 ; hep-th/0409230 | DOI | MR | Zbl

[84] Kürkçüoǧlu S., “Non-linear sigma models on the fuzzy supersphere”, JHEP, 3 (2004), Paper 062, 12 pp., ages ; hep-th/0311031 | MR

[85] Imai T., Kitazawa Y., Takayama Y., Tomino D., “Effective actions of matrix models on homogeneous spaces”, Nuclear Phys. B, 679 (2004), 143–167 ; hep-th/0307007 | DOI | MR | Zbl

[86] Imai T., Kitazawa Y., Takayama Y., Tomino D., “Quantum corrections on fuzzy sphere”, Nuclear Phys. B, 665 (2003), 520–544 ; hep-th/0303120 | DOI | MR | Zbl

[87] Shimamune Y., “On the phase structure of large N matrix models and gauge models”, Phys. Lett. B, 108 (1982), 407–410 | DOI

[88] Bleher P., Its A., Double scaling limit in the random matrix model: the Riemann–Hilbert approach, math-ph/0201003 | MR