Quantum Fields on Noncommutative Spacetimes: Theory and Phenomenology

Aiyalam P. Balachandran ; Alberto Ibort ; Giuseppe Marmo ; Mario Martone

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

In the present work we review the twisted field construction of quantum field theory on noncommutative spacetimes based on twisted Poincaré invariance. We present the latest development in the field, in particular the notion of equivalence of such quantum field theories on a noncommutative spacetime, in this regard we work out explicitly the inequivalence between twisted quantum field theories on Moyal and Wick–Voros planes; the duality between deformations of the multiplication map on the algebra of functions on spacetime $\mathscr F(\mathbb R^4)$ and coproduct deformations of the Poincaré–Hopf algebra $H\mathscr P$ acting on $\mathscr F(\mathbb R^4)$; the appearance of a nonassociative product on $\mathscr F(\mathbb{R}^4)$ when gauge fields are also included in the picture. The last part of the manuscript is dedicated to the phenomenology of noncommutative quantum field theories in the particular approach adopted in this review. CPT violating processes, modification of two-point temperature correlation function in CMB spectrum analysis and Pauli-forbidden transition in $\mathrm Be^4$ are all effects which show up in such a noncommutative setting. We review how they appear and in particular the constraint we can infer from comparison between theoretical computations and experimental bounds on such effects. The best bound we can get, coming from Borexino experiment, is $\gtrsim10^{24}$ TeV for the energy scale of noncommutativity, which corresponds to a length scale $\lesssim 10^{-43}$ m. This bound comes from a different model of spacetime deformation more adapted to applications in atomic physics. It is thus model dependent even though similar bounds are expected for the Moyal spacetime as well as argued elsewhere.

Détail

Keywords: noncommutative spacetime; quantum field theory; twisted field construction; Poincaré–Hopf algebra.

Bibliographie

[1] Doplicher S., Fredenhagen K., Roberts J. E., “Spacetime quantization induced by classical gravity”, Phys. Lett. B, 331 (1994), 33–44 | DOI | MR

[2] Seiberg N., Witten E., “String theory and noncommutative geometry”, J. High Energy Phys., 1999:9 (1999), 032, 93 pp., arXiv: hep-th/9908142 | DOI | MR

[3] Doplicher S., Fredenhagen K., Roberts J. E., “The quantum structure of spacetime at the Planck scale and quantum fields”, Comm. Math. Phys., 172 (1995), 187–220, arXiv: hep-th/0303037 | DOI | MR | Zbl

[4] Drinfel'd V. G., “Quasi-Hopf algebras”, Leningrad Math. J., 1 (1990), 1419–1457 | MR

[5] 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, arXiv: hep-th/0408069 | DOI | MR

[6] Wess J., “Deformed coordinates spaces: derivatives”, Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model (Vrnjacka Banja, Serbia, 2003), eds. G. Djordjevic, L. Nesic and J. Wess, World Scientific, 2005, 122–128, arXiv: hep-th/0408080

[7] Balachandran A. P., Pinzul A., Quereshi B. A., “Twisted Poincaré invariant quantum field theories”, Phys. Rev. D, 77 (2008), 025021, 9 pp., arXiv: 0708.1779 | DOI | MR | Zbl

[8] Dito G., Sternheimer D., “Deformation quantization: genesis, developments and metamorphoses”, Deformation Quantization (Strasbourg, 2001), v. 1, IRMA Lect. Math. Theor. Phys., ed. G. Halbout, de Gruyter, Berlin, 2002, 9–54, arXiv: math.QA/02201168 | MR | Zbl

[9] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[10] Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[11] Aschieri P., Lectures on Hopf algebras, quantum groups and twists, arXiv: hep-th/0703013

[12] Galluccio S., Lizzi F., Vitale P., “Twisted noncommutative field theory with the Wick–Voros and Moyal products”, Phys. Rev. D, 78 (2008), 085007, 14 pp., arXiv: 0810.2095 | DOI | MR

[13] Aschieri P., Lizzi F., Vitale P., “Twisting all the way: from classical mechaincs to quantum fields”, Phys. Rev. D, 77 (2008), 025037, 16 pp., arXiv: 0708.3002 | DOI | MR

[14] Bahns D., Doplicher S., Fredenhagen K., Piacitelli G., “On the unitarity problem in space-time noncommutative theories”, Phys. Lett. B, 533 (2002), 178–181, arXiv: hep-th/0201222 | DOI | MR | Zbl

[15] Balachandran A. P., Govindarajan T. R., Mangano G., Pinzul A., Quereshi B. A., Vaidya S., “Statistics and UV-IR mixing with twisted Poincaré invariance”, Phys. Rev. D, 75 (2007), 045009, 7 pp., arXiv: hep-th/0608179 | DOI | MR

[16] Balachandran A. P., Martone M., “Twisted quantum fields on Moyal and Wick–Voros planes are inequivalent”, Modern Phys. Lett. A, 24 (2009), 1721–1730, arXiv: 0902.1247 | DOI | MR | Zbl

[17] Balachandran A. P., Ibort A., Marmo G., Martone M., “Inequivalence of QFT's on noncommutative spacetimes: Moyal versus Wick–Voros”, Phys. Rev. D, 81 (2010), 085017, 8 pp., arXiv: 0910.4779 | DOI | Zbl

[18] Grosse H., “On the construction of Möller operators for the nonlinear Schrödigner equation”, Phys. Lett. B, 86 (1979), 267–271 | DOI | MR

[19] Zamolodchikov A. B., Zamolodchikov Al. B., “Factorized $S$-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models”, Ann. Physics, 120 (1979), 253–291 | DOI | MR

[20] Faddeev L. D., “Quantum completely integrable models in field theory”, Mathematical Physics Reviews, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., 1, Harwood Academic, Chur, 1980, 107–155 | MR

[21] Balachandran A. P., Pinzul A., Qureshi B. A., “UV-IR mixing in non-commutative plane”, Phys. Lett. B, 634 (2006), 434–436, arXiv: hep-th/0508151 | DOI | MR

[22] Balachandran A. P., Pinzul A., Queiroz A. R., “Twisted Poincaré invariance, noncommutative gauge theories and UV-IR mixing”, Phys. Lett. B, 668 (2008), 241–245, arXiv: 0804.3588 | DOI | MR

[23] Balachandran A. P., Ibort A., Marmo G., Martone M., Covariant quantum fields on noncommutative spacetimes, in preparation

[24] Balachandran A. P., Mangano G., Pinzul A., Vaidya S., “Spin and statistics on the Groenwald–Moyal plane: Pauli-forbidden levels and transitions”, Internat. J. Modern Phys. A, 21 (2006), 3111–3126, arXiv: hep-th/0508002 | DOI | MR | Zbl

[25] Balachandran A. P., Martone M., “Spacetime from symmetry: the Moyal plane from the Poincaré–Hopf algebra”, Modern Phys. Lett. A, 24 (2009), 1811–1821, arXiv: 0902.3409 | DOI | MR | Zbl

[26] Balachandran A. P., Pinzul A., Qureshi B. A., Vaidya S., “Twisted gauge and gravity theories on the Groenewold–Moyal plane”, Phys. Rev. D, 76 (2007), 105025, 10 pp., arXiv: 0708.0069 | DOI | MR

[27] Balachandran A. P., Pinzul A., Qureshi B. A., Vaidya S., “$S$ matrix on the Moyal plane: locality versus Lorentz invariance”, Phys. Rev. D, 77 (2008), 025020, 8 pp., arXiv: 0708.1379 | DOI | MR | Zbl

[28] Balachandran A. P., Qureshi B. A., “Poincaré quasi-Hopf symmetry and nonassociative spacetime algebra from twisted gauge theories”, Phys. Rev. D, 81 (2010), 065006, 6 pp., arXiv: 0903.0478 | DOI | Zbl

[29] Chu C.-S., Greene B. R., Shiu G., “Remarks on inflation and noncommutative geometry”, Modern Phys. Lett. A, 16 (2001), 2231–2240, arXiv: hep-th/0011241 | DOI | MR | Zbl

[30] Lizzi F., Mangano G., Miele G., Peloso M., “Cosmological perturbations and short distance physics from noncommutative geometry”, J. High Energy Phys., 2002:6 (2002), 049, 16 pp., arXiv: hep-th/0203099 | DOI | MR

[31] Brandenberger R., Ho P.-M., “Noncommutative spacetime, stringy spacetime uncertainty principle and density fluctuations”, Phys. Rev. D, 66 (2002), 023517, 10 pp., arXiv: hep-th/0203119 | DOI | MR

[32] Huang Q.-G., Li M., “CMB power spectrum from noncommutative spacetime”, J. High Energy Phys., 2003:6 (2003), 014, 7 pp., arXiv: hep-th/0304203 | DOI | MR

[33] Huang Q.-G., Li M., “Noncommutative inflation and the CMB multipoles”, J. Cosmol. Astropart. Phys., 2003:11 (2003), 001, 10 pp., arXiv: astro-ph/0308458 | DOI | MR

[34] Tsujikawa S., Maartens R., Branderberger R., “Noncommutative inflation and the CMB”, Phys. Lett. B, 574 (2003), 141–148, arXiv: astro-ph/0308169 | DOI | MR | Zbl

[35] Balachandran A. P., Queiroz A. R., Marques A. M., Teotonio-Sobrinho P., “Quantum fields with noncommutative target spaces”, Phys. Rev. D, 77 (2008), 105032, 11 pp., arXiv: 0706.0021 | DOI | MR | Zbl

[36] Barosi L., Brito F. A., Queiroz A. R., “Noncommutative field gas driven inflation”, J. Cosmol. Astropart. Phys., 2008:4 (2008), 005, 16 pp., arXiv: 0801.0810 | DOI | MR

[37] Fatollahi A. H., Hajirahimi M., “Noncommutative black-body radiation: implications on cosmic microwave background”, Europhys. Lett., 75 (2006), 542–547, arXiv: astro-ph/0607257 | DOI | MR

[38] Fatollahi A. H., Hajirahimi M., “Black-body radiation of noncommutative gauge fields”, Phys. Lett. B, 641 (2006), 381–385, arXiv: hep-th/0611225 | DOI | MR

[39] Akofor E., Balachandran A. P., Jo S. G., Joseph A., Qureshi B. A., “Direction-dependent CMB power spectrum and statistical anisotropy from noncommutative geometry”, J. High Energy Phys., 2008:5 (2008), 092, 21 pp., arXiv: 0710.5897 | DOI | Zbl

[40] Akofor E., Balachandran A. P., Joseph A., Pekowsky L., Qureshi B. A., “Constraints from cosmic microwave background on spacetime noncommutativity and causality violation”, Phys. Rev. D, 79 (2009), 063004, 5 pp., arXiv: 0806.2458 | DOI

[41] Dodelson S., Modern cosmology, Academic Press, San Diego, 2003

[42] Mukhanov V., Physical foundations of cosmology, Cambridge University Press, New York, 2005 | MR | Zbl

[43] Mukhanov V. F., Feldman H. A., Bradenberger R. H., “Theory of cosmological perturbations”, Phys. Rep., 215 (1992), 203–333 | DOI | MR

[44] Komatsu E. et al., “Five-year Wilkinson microwave anisotropy probe (WMAP) observations: cosmological interpretation”, Astrophys. J. Suppl., 180 (2009), 330–376, arXiv: 0803.0547 | DOI

[45] Dunkey J. et al., “Five-year Wilkinson microwave anisotropy probe (WMAP) observations: likelihoods and parameters from the WMAP data”, Astrophys. J. Suppl., 180 (2009), 306–329, arXiv: 0803.0586 | DOI

[46] Nolta M. R. et al., “Five-year Wilkinson microwave anisotropy probe (WMAP) observations: angular power spectra”, Astrophys. J. Suppl., 180 (2009), 296–305, arXiv: 0803.0593 | DOI

[47] Reichardte C. L. et al., “High resolution CMB power spectrum from the complete ACBAR data set”, Astrophys. J., 694 (2009), 1200–1219, arXiv: 0801.1491 | DOI

[48] Kuo C. L. et al., “Improved measurements of the CMB power spectrum with ACBAR”, Astrophys. J., 664 (2007), 687–701, arXiv: astro-ph/0611198 | DOI

[49] Kuo C. L. et al., “High resolution observations of the CMB power spectrum with ACBAR”, Astrophys. J., 600 (2004), 32–51, arXiv: astro-ph/0212289 | DOI

[50] Mason B. S. et al., “The anisotrpy of the microwave backgound to $I=3500$: deep field observations with the cosmic background imager”, Astrophys. J., 591 (2007), 540–555, arXiv: astro-ph/0205384 | DOI

[51] Sivers J. L. et al., “Implications of the cosmic background imager polarization data”, Astrophys. J., 660 (2007), 976–987, arXiv: astro-ph/0509203 | DOI

[52] Sivers J. L. et al., “Cosmological parameters from cosmic background imager observations and comparisons with BOOMERANG, DASI, and MAXIMA”, Astrophys. J., 591 (2003), 599–622, arXiv: astro-ph/0205387 | DOI

[53] Pearson T. J. et al., “The anisotropy of the microwave background to $I=3500$: mosaic observations with the cosmic background imager”, Astrophys. J., 591 (2003), 556–574, arXiv: astro-ph/0205388 | DOI

[54] Readhead A. C. S. et al., “Extended mosaic observations with the cosmic background imager”, Astrophys. J., 609 (2004), 498–512, arXiv: astro-ph/0402359 | DOI

[55] Chakraborty B., Gangopadhyay S., Hazra A. G., Sholtz F. G., “Twisted Galilean symmetry and the Pauli principle at low energies”, J. Phys. A: Math. Gen., 39 (2006), 9557–9572, arXiv: hep-th/0601121 | DOI | MR | Zbl

[56] Balachandran A. P., Joseph A., Padmanabhan P., Non-Pauli transitions from spacetime noncommutativity, arXiv: 1003.2250

[57] Back H. O. et al. [Borexino collaboration], “New experimental limits on violations of the Pauli exclusion principle obtained with the Borexino counting test facility”, Eur. Phys. J. C, 37 (2004), 421–431, arXiv: hep-ph/0406252 | DOI

[58] Barabash A. S., Kornoukhov V. N., Tsipenyuk Yu. M., Chapyzhnikov B. A., “Search for anomalous carbon atoms – evidence of violation of the Pauli principle during the period of nucleosynthesis”, JETP Lett., 68 (1998), 112–116 ; Arnold R. et al., “Testing the Pauli exclusion principle with the NEMO-2 detector”, Eur. Phys. J. A, 6 (1999), 361–366 | DOI | DOI

[59] Ramberg E., Snow G. A., “Experimental limit on a small violation of the Pauli principle”, Phys. Lett. B, 238 (1990), 438–411 | DOI

[60] Suzuki Y. et al. [Kamiokande collaboration], “Study of invisible nucleon decay, $N\to\nu\nu\bar\nu$, and a forbidden nuclear transition in the Kamiokande detector”, Phys. Lett. B, 311 (1993), 357–361 | DOI

[61] Bartalucci S. et al. [VIP collaboration], “New experimental limit on the Pauli exclusion principle violation by electrons”, Phys. Lett. B, 641 (2006), 18–22 | DOI

[62] Akofor E., Balachandran A. P., Jo S. G., Joseph A., “Quantum fields on the Groenwold–Moyal plane: $C$, $P$, $T$ and $CPT$”, J. High Energy Phys., 2007:8 (2007), 045, 14 pp., arXiv: 0706.1259 | DOI | MR

[63] Joseph A., “Particle phenomenology on noncommutative spacetime”, Phys. Rev. D, 79 (2009), 096004, 9 pp., arXiv: 0811.3972 | DOI

[64] Gibbons L. K. et al., “$CP$ and $CPT$ symmetry tests from the two-pion decays of the neutral kaon with the Fermilab E731 detector”, Phys. Rev. D, 55 (1997), 6625–6715 | DOI

[65] Angelopoulos A. et al. [CPLEAR collaboration], “A determination of the CPT violation parameter $\mathrm{Re}(\delta)$ from the semileptonic decay of strangeness tagged neutral kaons”, Phys. Lett. B, 444 (1998), 52–60 | DOI

[66] Lai A. et al. [NA48 collaboration], “Search for CP violation in $K^0\to3\pi^0$ decays”, Phys. Lett. B, 610 (2005), 165–176, arXiv: hep-ex/0408053 | DOI

[67] Bennett G. W. et al. [Muon (g-2) collaboration], “Search for Lorentz and $CPT$ violation effects in muon spin precession”, Phys. Rev. Lett., 100 (2008), 091602, 5 pp., arXiv: 0709.4670 | DOI

[68] Bailey J. et al. [CERN–Mainz–Daresbury collaboration], “Final report on the CERN muon storage ring including the anomalous magnetic moment and the electric dipole moment of the muon, and a direct test of relativistic time dilation”, Nuclear Phys. B, 150 (1979), 1–75 | DOI

[69] Carey R. M. et al., “New measurement of the anomalous magnetic moment of the positive muon”, Phys. Rev. Lett., 82 (1999), 1632–1635 | DOI

[70] Brown H. N. et al. [Muon (g-2) collaboration], “Precise measurement of the positive muon anomalous magnetic moment”, Phys. Rev. Lett., 86 (2001), 2227–2231, arXiv: hep-ex/0102017 | DOI