Abstract
Kappa-Minkowski space-time is an example of noncommutative space-time with potentially interesting phenomenological consequences. However, the construction of field theories on this space, although operationally well-defined, is plagued with ambiguities. A part of ambiguities can be resolved by clarifying the geometrical picture of gauge transformations on the κ-Minkowski space-time. To this end we use the twist approach to construct the noncommutative U(1) gauge theory coupled to fermions. However, in this approach we cannot maintain the kappa-Poincaré symmetry; the corresponding symmetry of the twisted kappa-Minkowski space is the twisted igl(1,3) symmetry. We construct an action for the gauge and matter fields in a geometric way, as an integral of a maximal form. We use the Seiberg-Witten map to relate noncommutative and commutative degrees of freedom and expand the action to obtain the first order corrections in the deformation parameter.
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References
H.S. Snyder, Quantized space-time, Phys. Rev. 71 (1947) 38 [INSPIRE].
V. Chari, A.N. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge U.K. (1995).
W.J. Fairbairn and C. Meusburger, Quantum deformation of two four-dimensional spin foam models, arXiv:1012.4784 [INSPIRE].
A. Connes, M.R. Douglas and A.S. Schwarz, Noncommutative geometry and matrix theory: Compactification on tori, JHEP 02 (1998) 003 [hep-th/9711162] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
J. Lukierski, H. Ruegg, A. Nowicki and V.N. Tolstoi, Q deformation of Poincaré algebra, Phys. Lett. B 264 (1991) 331 [INSPIRE].
J. Lukierski, A. Nowicki and H. Ruegg, New quantum Poincaré algebra and κ-deformed field theory, Phys. Lett. B 293 (1992) 344 [INSPIRE].
P. Kosinski and P. Maslanka, The Duality between κ-Poincaré algebra and kappa Poincaré group, hep-th/9411033 [INSPIRE].
G. Amelino-Camelia, Relativity in space-times with short distance structure governed by an observer independent (Planckian) length scale, Int. J. Mod. Phys. D 11 (2002) 35 [gr-qc/0012051] [INSPIRE].
G. Amelino-Camelia, Testable scenario for relativity with minimum length, Phys. Lett. B 510 (2001) 255 [hep-th/0012238] [INSPIRE].
J. Magueijo and L. Smolin, Lorentz invariance with an invariant energy scale, Phys. Rev. Lett. 88 (2002) 190403 [hep-th/0112090] [INSPIRE].
J. Kowalski-Glikman and S. Nowak, Doubly special relativity theories as different bases of κ-Poincaré algebra, Phys. Lett. B 539 (2002) 126 [hep-th/0203040] [INSPIRE].
G. Gubitosi and F. Mercati, Relative Locality in κ-Poincaré, arXiv:1106.5710 [INSPIRE].
S. Hossenfelder, Bounds on an energy-dependent and observer-independent speed of light from violations of locality, Phys. Rev. Lett. 104 (2010) 140402 [arXiv:1004.0418] [INSPIRE].
G. Amelino-Camelia, M. Matassa, F. Mercati and G. Rosati, Taming Nonlocality in Theories with Planck-Scale Deformed Lorentz Symmetry, Phys. Rev. Lett. 106 (2011) 071301 [arXiv:1006.2126] [INSPIRE].
S. Hossenfelder, Reply to arXiv:1006.2126 by Giovanni Amelino-Camelia et al., arXiv:1006.4587 [INSPIRE].
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, The principle of relative locality, Phys. Rev. D 84 (2011) 084010 [arXiv:1101.0931] [INSPIRE].
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Relative locality: A deepening of the relativity principle, Gen. Rel. Grav. 43 (2011) 2547 [arXiv:1106.0313] [INSPIRE].
L. Freidel and L. Smolin, Gamma ray burst delay times probe the geometry of momentum space, arXiv:1103.5626 [INSPIRE].
G. Amelino-Camelia and L. Smolin, Prospects for constraining quantum gravity dispersion with near term observations, Phys. Rev. D 80 (2009) 084017 [arXiv:0906.3731] [INSPIRE].
M. Dimitrijević, L. Jonke and L. Möller, U(1) gauge field theory on kappa-Minkowski space, JHEP 09 (2005) 068 [hep-th/0504129] [INSPIRE].
M. Dimitrijević, L. Jonke, L. Möller, E. Tsouchnika, J. Wess and M. Wohlgenannt, Deformed field theory on kappa space-time, Eur. Phys. J. C 31 (2003) 129 [hep-th/0307149] [INSPIRE].
M. Dimitrijević, F. Meyer, L. Möller and J. Wess, Gauge theories on the kappa Minkowski space-time, Eur. Phys. J. C 36 (2004) 117 [hep-th/0310116] [INSPIRE].
A. Connes, Non-commutative Geometry, Academic Press, New York U.S.A. (1994).
G. Landi, An Introduction to noncommutative spaces and their geometry, hep-th/9701078 [INSPIRE].
J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, second Edition, Cambridge University Press, Cambridge U.K. (1999).
P. Aschieri, M. Dimitrijević, P. Kulish, F. Lizzi and J. Wess, Lecture notes in physics. Vol. 774: Noncommutative spacetimes: Symmetries in noncommutative geometry and field theory, Springer, Heidelberg Germany (2009).
J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C 16 (2000) 161 [hep-th/0001203] [INSPIRE].
X. Calmet, B. Jurčo, P. Schupp, J. Wess and M. Wohlgenannt, The Standard model on noncommutative space-time, Eur. Phys. J. C 23 (2002) 363 [hep-ph/0111115] [INSPIRE].
P. Aschieri, B. Jurčo, P. Schupp and J. Wess, Noncommutative GUTs, standard model and C,P,T, Nucl. Phys. B 651 (2003) 45 [hep-th/0205214] [INSPIRE].
W. Behr, N. Deshpande, G. Duplančić, P. Schupp, J. Trampetić and J. Wess, The Z → gamma gamma, g g decays in the noncommutative standard model, Eur. Phys. J. C 29 (2003) 441 [hep-ph/0202121] [INSPIRE].
B. Melić, K. Passek-Kumerički, P. Schupp, J. Trampetić and M. Wohlgennant, The Standard model on non-commutative space-time: Electroweak currents and Higgs sector, Eur. Phys. J. C 42 (2005) 483 [hep-ph/0502249] [INSPIRE].
B. Melić, K. Passek-Kumerički, J. Trampetić, P. Schupp and M. Wohlgenannt, The Standard model on non-commutative space-time: Strong interactions included, Eur. Phys. J. C 42 (2005) 499 [hep-ph/0503064] [INSPIRE].
H. Grosse and R. Wulkenhaar, Renormalization of phi 4 theory on noncommutative R 4 in the matrix base, Commun. Math. Phys. 256 (2005) 305 [hep-th/0401128] [INSPIRE].
M. Burić, D. Latas and V. Radovanović, Renormalizability of noncommutative SU(N) gauge theory, JHEP 02 (2006) 046 [hep-th/0510133] [INSPIRE].
C. Martin and C. Tamarit, Renormalisability of noncommutative GUT inspired field theories with anomaly safe groups, JHEP 12 (2009) 042 [arXiv:0910.2677] [INSPIRE].
V. Drinfel’d, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [INSPIRE].
P. Aschieri and L. Castellani, Noncommutative D = 4 gravity coupled to fermions, JHEP 06 (2009) 086 [arXiv:0902.3817] [INSPIRE].
A. Agostini, F. Lizzi and A. Zampini, Generalized Weyl systems and kappa Minkowski space, Mod. Phys. Lett. A 17 (2002) 2105 [hep-th/0209174] [INSPIRE].
M. Dimitrijević, L. Möller and E. Tsouchnika, Derivatives, forms and vector fields on the kappa-deformed Euclidean space, J. Phys. A 37 (2004) 9749 [hep-th/0404224] [INSPIRE].
S. Meljanac, A. Samsarov, M. Stojić and K. Gupta, Kappa-Minkowski space-time and the star product realizations, Eur. Phys. J. C 53 (2008) 295 [arXiv:0705.2471] [INSPIRE].
P. Kosinski, P. Maslanka, J. Lukierski and A. Sitarz, Generalized κ-deformations and deformed relativistic scalar fields on noncommutative Minkowski space, hep-th/0307038 [INSPIRE].
A. Agostini, G. Amelino-Camelia, M. Arzano and F. D’Andrea, Action functional for kappa-Minkowski noncommutative spacetime, hep-th/0407227 [INSPIRE].
A. Sitarz, Noncommutative differential calculus on the kappa Minkowski space, Phys. Lett. B 349 (1995) 42 [hep-th/9409014] [INSPIRE].
E. Beggs and S. Majid, Nonassociative Riemannian geometry by twisting, J. Phys. Conf. Ser. 254 (2010) 012002 [arXiv:0912.1553] [INSPIRE].
S. Meljanac and S. Kresić-Jurić, Differential structure on kappa-Minkowski space and kappa-Poincaré algebra, Int. J. Mod. Phys. A 26 (2011) 3385 [arXiv:1004.4647] [INSPIRE].
S. Meljanac and A. Samsarov, Scalar field theory on kappa-Minkowski spacetime and translation and Lorentz invariance, Int. J. Mod. Phys. A 26 (2011) 1439 [arXiv:1007.3943] [INSPIRE].
S. Meljanac, A. Samsarov, J. Trampetić and M. Wohlgenannt, Noncommutative kappa-Minkowski phi4 theory: Construction, properties and propagation, arXiv:1107.2369 [INSPIRE].
A. Borowiec and A. Pachol, kappa-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79 (2009) 045012 [arXiv:0812.0576] [INSPIRE].
B. Jurčo, S. Schraml, P. Schupp and J. Wess, Enveloping algebra valued gauge transformations for nonAbelian gauge groups on noncommutative spaces, Eur. Phys. J. C 17 (2000) 521 [hep-th/0006246] [INSPIRE].
M. Burić, D. Latas, V. Radovanović and J. Trampetić, Chiral fermions in noncommutative electrodynamics: Renormalizability and dispersion, Phys. Rev. D 83 (2011) 045023 [arXiv:1009.4603] [INSPIRE].
J. Lukierski, H. Ruegg and W.J. Zakrzewski, Classical and quantum-mechanics of free κ-relativistic systems, Annals Phys. 243 (1995) 90 [hep-th/9312153].
G. Amelino-Camelia and S. Majid, Waves on noncommutative space-time and gamma-ray bursts, Int. J. Mod. Phys. A 15 (2000) 4301 [hep-th/9907110] [INSPIRE].
OPERA collaboration, T. Adam et al., Measurement of the neutrino velocity with the OPERA detector in the CNGS beam, arXiv:1109.4897 [INSPIRE].
R.C. Myers and M. Pospelov, Ultraviolet modifications of dispersion relations in effective field theory, Phys. Rev. Lett. 90 (2003) 211601 [hep-ph/0301124] [INSPIRE].
P.A. Bolokhov and M. Pospelov, Low-energy constraints on kappa-Minkowski extension of the Standard Model, Phys. Lett. B 677 (2009) 160 [arXiv:0807.1522] [INSPIRE].
F.W. Hehl and Y.N. Obukhov, How does the electromagnetic field couple to gravity, in particular to metric, nonmetricity, torsion and curvature?, Lect. Notes Phys. 562 (2001) 479 [gr-qc/0001010] [INSPIRE].
H. Steinacker, Emergent Geometry and Gravity from Matrix Models: an Introduction, Class. Quant. Grav. 27 (2010) 133001 [arXiv:1003.4134] [INSPIRE].
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Dimitrijević, M., Jonke, L. A twisted look on kappa-Minkowski: U(1) gauge theory. J. High Energ. Phys. 2011, 80 (2011). https://doi.org/10.1007/JHEP12(2011)080
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DOI: https://doi.org/10.1007/JHEP12(2011)080