Twist of quantum groups and noncommutative field theory

P. P. Kulish

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 19, Tome 335 (2006), pp. 188-204

Résumé

The role of quantum universal enveloping algebras of symmetries in constructing a noncommutative geometry of space-time and corresponding field theory is discussed. It is shown that in the framework of the twist theory of quantum groups, the noncommutative (super) space-time defined by coordinates with Heisenberg commutation relations, is (super) Poincaré invariant as well as the corresponding field theory. Noncommutative parameters of global transformations are introduced.

Bibliographie

[1] P. Aschieri, M. Dimitrijevic, F. Meyer, J. Wess, Noncommutative geometry and gravity, arXiv: /hep-th/0510059 | MR

[2] L. Alvarez-Gaumé, F. Meyer, M. A. Vázquez-Mozo, Comments on noncommutative gravity, arXiv: /hep-th/0605113 | MR

[3] M. Chaichian, P. P. Kulish, K. Nishijima, A. Tureanu, “On 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

[4] M. R. Douglas, N. A. Nekrasov, “Noncommutative field theory”, Rev. Mod. Phys., 73 (2001), 977–1029 | DOI | MR | Zbl

[5] R. J. Szabo, “Quantum field theories on noncommutative spaces”, Phys. Rep., 378 (2003), 207–299 | DOI | MR | Zbl

[6] V. G. Drinfeld, “Quantum groups”, Proceedings of the International Congress of Mathematicians 1986, ed. A. M. Gleason, American Mathematical Society, Providence, 1987, 798–820 | MR

[7] N. Yu. Reshetikhin, L. A. Takhtajan, L. D. Faddeev, “Quantization of Lie groups and Lie algebras”, Algebra and Analysis, 1 (1989), 178–206 | MR

[8] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[9] P. P. Kulish, A. I. Mudrov, “Twist related geometries on $q$-Minkowski space”, Proc. Steklov Math. Inst., 226, 1999, 97–111 ; arXiv: /math.QA/9901019 | MR | Zbl

[10] R. Oeckl, “Untwisting noncommutative $R^d$ and the equivalence of quantum field theories”, Nucl. Phys. B, 581 (2000), 559–574 ; arXiv: /hep-th/0003018 | DOI | MR | Zbl

[11] J. Madore, An Introduction to Noncommutative Differential Geometry and Its Applications, Cambridge University Press, Cambridge, 2000

[12] J. Wess, Deformed coordinate space derivatives, arXiv: /hep-th/0408080 | MR

[13] J. Lukierski, M. Woronowicz, “New Lie-algebraic and quadratic deformations of Minkovski space from twisted Poincare symmetries”, Phys. Lett. B, 633 (2006), 116–124 ; arXiv: /hep-th/0508083 | DOI | MR

[14] N. Seiberg, E. Witten, “String theory and noncommutative geometry”, J. High Energy Phys., 9909 (1999), 032 ; arXiv: /hep-th/9908142 | DOI | MR | DOI

[15] D. Sternheimer, “Deformation quantization: Twenty years after”, Particles, Fields, and Gravitation, eds. J. Rembelinski, AIP Press, New York, 1998, 107–145 | MR | Zbl

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

[17] A. Giaqunto, J. J. Zhang, “Bialgebra actions, twists, and universal deformation formulas”, J. Pure. Appl. Algebra, 128 (1998), 133–151 ; arXiv: /hep-th/9411140 | DOI | MR

[18] P. Bonneau, M. Gerstenhaber, A. Giaqunto, D. Sternheimer, “Quantum groups and deformation quantization: Explicit approaches and implicit aspects”, J. Math. Phys., 45 (2004), 3703–3741 | DOI | MR | Zbl

[19] R. Banerjee, C. Lee, S. Siwach, Derformed conformal and super-Poincaré symmetries in the non-(anti)commutative spaces, arXiv: /hep-th/0511205

[20] M. Chaichian, P. Presnajder, A. Tureanu, “New concept of relativistic invariance in NC space-time: Twisted Poincaré symmetry and its implication”, Phys. Rev. Lett., 94 (2005), 151602 ; arXiv: /hep-th/0409096 | DOI

[21] A. P. Balachandran, G. Mangano, A. Pinzul, S. Vaidya, Spin and statistics on the Groenwald-Moyal plane: Pauli-forbidden levels and transitions, arXiv: /hep-th/0508002 | MR

[22] A. Tureanu, “Twist and spin-statistic relation in noncommutative quantum field theory”, Phys. Lett., in press

[23] C. Gonera, P. Kosinski, P. Maslanka, S. Giller, “Space-time symmetry of noncommutative field theory”, Phys. Lett. B, 622 (2005), 192–197 | DOI | MR

[24] P. P. Kulish, “Noncommutative geometry and quantum field theory”, Noncommutative geometry and representation theory in mathematical physics, Contem. Math., 391, American Mathematical Society, Providence, RI, 2005, 213–221 | MR | Zbl

[25] M. Chaichian, A. Tureanu, “Twist symmetry and gauge invariance”, Phys. Lett. B, 637 (2006), 199–202 ; arXiv: /hep-th/0604025 | DOI | MR

[26] N. Seiberg, “Noncommutative superspace, $\mathcal{N} = 1/2$ supersymmetry, field theory and string theory”, J. High Energy Phys., 0306 (2003), 010 ; arXiv: /hep-th/0305248 | DOI | MR

[27] E. Ivanov, O. Lechtenfeld, B. Zupnik, “Nilpotent deformations of $N=2$ superspace”, J. High Energy Phys., 0402 (2004), 012 ; arXiv: /hep-th/0308012 | DOI | MR

[28] Y. Kobayashi, S. Sasaki, “Lorentz invariant and supersymmetric interpretation of noncommutative quantum field theory”, Int. J. Mod. Phys. A, 20 (2005), 7175–7188 ; arXiv: /hep-th/0410164 | DOI | MR | Zbl

[29] B. Zupnik, “Twist-deformed supersymmetries in non-anticommutative superspace”, Phys. Lett. B, 627 (2005), 208–216 ; arXiv: /hep-th/0506043 | DOI | MR

[30] M. Arai, M. Chaichian, K. Nishijima, A. Tureanu, Non-anticommutative supersymmetric field theory and quantum shift, arXiv: /hep-th/0604029 | MR