On \(\kappa \)-deformation and triangular quasibialgebra structure. (English) Zbl 1192.81225
Summary: We show that, up to terms of order \(\kappa ^{ - 5}\), the \(\kappa \)-deformed Poincaré algebra can be endowed with a triangular quasibialgebra structure. The universal R matrix and coassociator are given explicitly to the first few orders. In the context of \(\kappa \)-deformed quantum field theory, we argue that this structure, assuming it exists to all orders, ensures that states of any number of identical particles, in any representation, can be defined in a \(\kappa \)-covariant fashion.
MSC:
| 81T10 | Model quantum field theories |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 16T05 | Hopf algebras and their applications |
Keywords:
R matrix and coassociatorSoftware:
FORMReferences:
| [1] | Lukierski, J.; Nowicki, A.; Ruegg, H., New quantum Poincaré algebra and \(k\)-deformed field theory, Phys. Lett. B, 293, 344 (1992) · Zbl 0834.17022 |
| [2] | Lukierski, J.; Ruegg, H., Quantum kappa Poincaré in any dimension, Phys. Lett. B, 329, 189 (1994) |
| [3] | Majid, S.; Ruegg, H., Bicrossproduct structure of kappa Poincaré group and noncommutative geometry, Phys. Lett. B, 334, 348 (1994) · Zbl 1112.81328 |
| [4] | Lukierski, J., Quantum deformations of Einstein’s relativistic symmetries, AIP Conf. Proc., 861, 398 (2006) |
| [5] | Lukierski, J.; Nowicki, A., Doubly special relativity versus \(κ\)-deformation of relativistic kinematics, Int. J. Mod. Phys. A, 18, 7 (2003) · Zbl 1021.81024 |
| [6] | Young, C. A.S.; Zegers, R., Covariant particle statistics and intertwiners of the kappa-deformed Poincaré algebra, Nucl. Phys. B, 797, 537 (2008) · Zbl 1234.81091 |
| [7] | Young, C. A.S.; Zegers, R., Covariant particle exchange for \(κ\)-deformed theories in \(1 + 1\) dimensions, Nucl. Phys. B, 804, 342-360 (2008) · Zbl 1190.81071 |
| [8] | Daszkiewicz, M.; Lukierski, J.; Woronowicz, M., \(κ\)-Deformed statistics and classical fourmomentum addition law · Zbl 1169.81349 |
| [9] | Dimitrijevic, M.; Jonke, L.; Moller, L.; Tsouchnika, E.; Wess, J.; Wohlgenannt, M., Deformed field theory on kappa-spacetime, Eur. Phys. J. C, 31, 129 (2003) · Zbl 1032.81529 |
| [10] | Daszkiewicz, M.; Imilkowska, K.; Kowalski-Glikman, J.; Nowak, S., Scalar field theory on kappa-Minkowski space-time and doubly special relativity, Int. J. Mod. Phys. A, 20, 4925 (2005) · Zbl 1153.83425 |
| [11] | Agostini, A.; Amelino-Camelia, G.; D’Andrea, F., Hopf-algebra description of noncommutative-spacetime symmetries, Int. J. Mod. Phys. A, 19, 5187 (2004) · Zbl 1078.81036 |
| [12] | Freidel, L.; Kowalski-Glikman, J.; Nowak, S., Field theory on \(κ\)-Minkowski space revisited: Noether charges and breaking of Lorentz symmetry · Zbl 1178.81257 |
| [13] | Lukierski, J.; Ruegg, H.; Zakrzewski, W. J., Classical quantum mechanics of free kappa relativistic systems, Ann. Phys., 243, 90 (1995) · Zbl 0856.70012 |
| [14] | Kosinski, P.; Lukierski, J.; Maslanka, P., Local \(D = 4\) field theory on kappa-deformed Minkowski space, Phys. Rev. D, 62, 025004 (2000) |
| [15] | Grosse, H.; Wohlgenannt, M., On kappa-deformation and UV/IR mixing, Nucl. Phys. B, 748, 473 (2006) |
| [16] | Kim, H. C.; Rim, C.; Yee, J. H., Casimir energy of a spherical shell in \(κ\)-Minkowski spacetime |
| [17] | Amelino-Camelia, G.; Arzano, M., Coproduct and star product in field theories on Lie-algebra non-commutative space-times, Phys. Rev. D, 65, 084044 (2002) |
| [18] | Frappat, L.; Sciarrino, A., Lattice space-time from Poincaré and kappa Poincaré algebras, Phys. Lett. B, 347, 28 (1995) |
| [19] | Moller, L., A symmetry invariant integral on kappa-deformed spacetime, JHEP, 0512, 029 (2005) |
| [20] | Kosinski, P.; Maslanka, P.; Lukierski, J.; Sitarz, A., Towards kappa-deformed \(D = 4\) relativistic field theory, Czech. J. Phys., 48, 1407 (1998) · Zbl 0948.81016 |
| [21] | Kresic-Juric, S.; Meljanac, S.; Stojic, M., Covariant realizations of kappa-deformed space, Eur. Phys. J. C, 51, 229 (2007) · Zbl 1189.81114 |
| [22] | Dimitrijevic, M.; Moller, L.; Tsouchnika, E., Derivatives, forms and vector fields on the kappa-deformed Euclidean space, J. Phys. A, 37, 9749 (2004) · Zbl 1073.81053 |
| [23] | Govindarajan, T. R.; Gupta, K. S.; Harikumar, E.; Meljanac, S.; Meljanac, D., Twisted statistics in kappa-Minkowski spacetime, Phys. Rev. D, 77, 105010 (2008) |
| [24] | Meljanac, S.; Samsarov, A.; Stojic, M.; Gupta, K. S., Kappa-Minkowski space-time and the star product realizations, Eur. Phys. J. C, 53, 295 (2008) · Zbl 1189.81115 |
| [25] | Weinberg, S., The Quantum Theory of Fields. Vol. 1: Foundations (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0959.81002 |
| [26] | Kosinski, P.; Lukierski, J.; Maslanka, P.; Sobczyk, J., The Classical basis for kappa-deformed Poincaré (super)algebra and the second kappa-deformed supersymmetric Casimir, Mod. Phys. Lett. A, 10, 2599 (1995) · Zbl 1022.81577 |
| [27] | Giller, S.; Gonera, C.; Majewski, M., Deformation map for generalized \(κ\)-Poincaré and \(κ\)-Weyl algebras, Acta Phys. Pol. B, 27, 2131 (1996) · Zbl 0966.81520 |
| [28] | Kosinski, P.; Lukierski, J.; Maslanka, P., Kappa-deformed Wigner construction of relativistic wave functions and free fields on kappa-Minkowski space, Nucl. Phys. B (Proc. Suppl.), 102, 161 (2001) · Zbl 1006.81085 |
| [29] | Majid, S., Foundations of Quantum Group Theory (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0857.17009 |
| [30] | Arzano, M.; Marciano, A., Fock space, quantum fields and kappa-Poincaré symmetries, Phys. Rev. D, 76, 125005 (2007) |
| [31] | Daszkiewicz, M.; Lukierski, J.; Woronowicz, M., Towards quantum noncommutative \(κ\)-deformed field theory, Phys. Rev. D, 77, 105007 (2008) |
| [32] | Daszkiewicz, M.; Lukierski, J.; Woronowicz, M., Quantization of kappa-deformed free fields and kappa-deformed oscillators |
| [33] | Daszkiewicz, M.; Lukierski, J.; Woronowicz, M., Kappa-deformed oscillators, the choice of star product and free kappa-deformed quantum fields · Zbl 1191.81120 |
| [34] | Zakrzewski, S., Quantum Poincaré group related to the kappa-Poincaré algebra, J. Phys. A: Math. Gen., 27, 2075-2082 (1994) · Zbl 0834.17024 |
| [35] | Zakrzewski, S., Poisson Poincaré groups · Zbl 1181.22020 |
| [36] | Daszkiewicz, M., Canonical, Lie-algebraic and quadratic twist deformations of Galilei group · Zbl 1169.81341 |
| [37] | Meusburger, C.; Schroers, B. J., Generalised Chern-Simons actions for 3d gravity and kappa-Poincaré symmetry · Zbl 1192.83011 |
| [38] | Celeghini, E.; Giachetti, R.; Sorace, E.; Tarlini, M., The three-dimensional Euclidean quantum group \(E(3) q\) and its \(R\)-matrix, J. Math. Phys., 32, 1159 (1991) · Zbl 0743.17017 |
| [39] | Vermaseren, J. A.M., New features of FORM · Zbl 1344.65050 |
| [40] | Drinfel’d, V. G., On the structure of quasitriangular quasi-Hopf algebras, Funktsional. Anal. i Prilozhen.. Funktsional. Anal. i Prilozhen., Funct. Anal. Appl., 26, 1, 63-65 (1992), translation in: · Zbl 0760.17005 |
| [41] | Drinfel’d, V. G., Quasi-Hopf algebras, Algebra i Analiz. Algebra i Analiz, Leningrad Math. J., 1, 6, 1419-1457 (1990), translation in: · Zbl 0718.16033 |
| [42] | Mack, G.; Schomerus, V., Quasi-Hopf quantum symmetry in quantum theory, Nucl. Phys. B, 370, 185 (1992) |
| [43] | Kowalski-Glikman, J.; Starodubtsev, A., Can we see gravitational collapse in (quantum) gravity perturbation theory? |
| [44] | Amelino-Camelia, G.; Smolin, L.; Starodubtsev, A., Quantum symmetry, the cosmological constant and Planck scale phenomenology, Class. Quantum Grav., 21, 3095 (2004) · Zbl 1061.83025 |
| [45] | Freidel, L.; Kowalski-Glikman, J.; Smolin, L., \(2 + 1\) gravity and doubly special relativity, Phys. Rev. D, 69, 044001 (2004) |
| [46] | Goldin, G. A.; Majid, S., On the Fock space for nonrelativistic anyon fields and braided tensor products, J. Math. Phys., 45, 3770-3787 (2004) · Zbl 1071.81065 |
| [47] | Beggs, E. J.; Majid, S., Quantization by cochain twists and nonassociative differentials · Zbl 1310.81101 |
| [48] | Freidel, L.; Kowalski-Glikman, J., \(κ\)-Minkowski space, scalar field, and the issue of Lorentz invariance |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
