Position-dependent noncommutative products: classical construction and field theory

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Elsevier Science BV
Google Scholar
Research Projects
Organizational Units
Journal Issue
We look in Euclidean R-4 for associative star products realizing the commutation relation [x(mu), x(upsilon)] i Theta(mu upsilon)(x), where the noncommutativity parameters Theta(mu upsilon) depend on the position coordinates x. We do this by adopting Rieffel's deformation theory (originally formulated for constant Theta and which includes the Moyal product as a particular case) and find that, for a topology R-2 x R-2, there is only one class of such products which are associative. It corresponds to a noncommutativity matrix whose canonical form has components Theta(12) = -Theta(21) = 0 and Theta(34) = -Theta(43) = theta(x(1), x(2)), with theta(x (1), x(2)) an arbitrary positive smooth bounded function. In Minkowski space-time, this describes a position-dependent space-like or magnetic noncommutativity. We show how to generalize our construction to n >= 3 arbitrary dimensions and use it to find traveling noncommutative lumps generalizing noncommutative solitons discussed in the literature. Next we consider Euclidean lambda phi(4) field theory on such a noncommutative background. Using a zeta-like regulator, the covariant perturbation method and working in configuration space, we explicitly compute the UV singularities. We find that, while the two-point UV divergences are nonlocal, the four-point UV divergences are local, in accordance with recent results for constant Ѳ.
© 2005 Elsevier B.V. All rights reserved. The authors thank J.C. Várilly for helpful comments. V.G. wishes to acknowledge the hospitality of the Department of Theoretical Physics of Universidad Complutense de Madrid, where this work was started. J.M.G.B. is very grateful to B. Booss-Bavnbek for making available to him unpublished notes on the Duhamel expansion. He also thanks MEC, Spain for support through a ‘Ramón y Cajal’ contract. F.R.R. is grateful to MEC, Spain for financial support through grant No. BFM2002-00950.
Unesco subjects
[1] N. Seiberg, E. Witten, String theory and noncommutative geometry, JHEP 9909 (1999) 032. [2] J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949) 99. [3] V. Gayral, J.M. Gracia-Bondía, B. Iochum, T. Schücker, J.C. Várilly, Moyal planes are spectral triples, Commun. Math. Phys. 264 (2004) 569. [4] A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry, Commun. Math. Phys. 182 (1996) 155. [5] J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, Boston, 2001. [6] J. Gomis, T. Mehen, Space–time noncommutative field theories and unitarity, Nucl. Phys. B 591 (2000) 265. [7] V. Gayral, J.M. Gracia-Bondía, F. Ruiz Ruiz, Trouble with space-like noncommutative field theory, Phys. Lett. B 610 (2005) 141. [8] D. Bahns, S. Doplicher, K. Fredenhagen, G. Piacitelli, Field theory on noncommutative spacetimes: Quasiplanar Wick products, Phys. Rev. D 71 (2005) 025022. [9] D. Bahns, S. Doplicher, K. Fredenhagen, G. Piacitelli, On the unitary problem in space–time noncommutative theories, Phys. Lett. B 533 (2002) 178. [10] K. Fujikawa, Path integral for space–time noncommutative field theory, Phys. Rev. D 70 (2004) 085006. [11] K. Landsteiner, E. López, M.H.G. Tytgat, Excitations in hot noncommutative theories, JHEP 0009 (2000) 027; C.P. Martín, F. Ruiz Ruiz, Paramagnetic dominance, the sign of the beta function and UV/IR mixing in noncommutative U(1), Nucl. Phys. B 597 (2001) 197. [12] F. Ruiz Ruiz, Gauge-fixing independence of IR divergences in noncommutative U(1), perturbative tachyonic instabilities and supersymmetry, Phys. Lett. B 502 (2001) 274. [13] S. Minwalla, M.V. Raamsdonk, N. Seiberg, Noncommutative perturbative dynamics, JHEP 0002 (2000) 020. [14] A. Matusis, L. Susskind, N. Toumbas, The IR/UV connection in the noncommutative gauge theories, JHEP 0012 (2000) 002. [15] J.M. Gracia-Bondía, F. Lizzi, G. Marmo, P. Vitale, Infinitely many star-products to play with, JHEP 0204 (2002) 026. [16] F. Lizzi, G. Mangano, G. Miele, M. Peloso, Cosmological perturbations and short distance physics from noncommutative geometry, JHEP 0206 (2002) 049. [17] L. Dolan, C.R. Nappi, Noncommutativity in a time dependent background, Phys. Lett. B 551 (2003) 369. [18] X. Calmet, M. Wohlgennant, Effective field theories on noncommutative space–time, Phys. Rev. D 68 (2003) 025016. [19] W. Behr, A. Sykora, Construction of gauge theories on curved noncommutative space–time, Nucl. Phys. B 698 (2004) 473. [20] A. Hashimoto, K. Thomas, Dualities, twists and gauge theories with nonconstant noncommutativity, JHEP 0501 (2005) 033. [21] L. Cornalba, R. Schiappa, Nonassociative star product deformations for D-brane worldvolumes in curved backgrounds, Commun. Math. Phys. 225 (2002) 33. [22] M.A. Rieffel, Deformation Quantization for Actions of Rd , Memoirs of the American Mathematical Society, vol. 506, American Mathematical Society, Providence, RI, 1993. [23] R. Gopakumar, S. Minwalla, A. Strominger, Noncommutative solitons, JHEP 0005 (2000) 020. [24] A.O. Barvinsky, G.A. Vilkovisky, Covariant perturbation theory (II). Second order in the curvature. General algorithms, Nucl. Phys. B 333 (1990) 471. [25] P.B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah–Singer Theorem, CRC Press, Boca Raton, 1995. [26] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157. [27] C.D. Fosco, G. Torroba, Noncommutative theories and general coordinate transformations, Phys. Rev. D 71 (2005) 065012. [28] H.J. Groenewold, On the principles of elementary quantum mechanics, Physica 12 (1946) 405–460. [29] M.S. Bartlett, J.E. Moyal, The exact transition probabilities of quantum-mechanical oscillators calculated by the phase-space method, Proc. Cambridge Philos. Soc. 45 (1949) 545–553. [30] J.M. Gracia-Bondía, J.C. Várilly, Algebras of distributions suitable for phase-space quantum mechanics. I, J. Math. Phys. 29 (1988) 869; J.C. Várilly, J.M. Gracia-Bondía, Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra, J. Math. Phys. 29 (1988) 880. [31] V. Gayral, Heat-kernel approach to UV/IR mixing on isospectral deformation manifolds, hep-th/0412233, Ann. Inst. H. Poincaré, in press. [32] V. Gayral, B. Iochum, J.C. Várilly, Dixmier traces on noncompact isospectral deformations, in preparation. [33] A. Connes, G. Landi, Noncommutative manifolds, the instanton algebra and isospectral deformations, Commun. Math. Phys. 221 (2001) 141. [34] A. Connes, M. Dubois-Violette, Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, Commun. Math. Phys. 230 (2002) 539. [35] A. Connes, H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys. 198 (1998) 198. [36] C. Itzykson, J.B. Zuber, Quantum Field Theory, McGraw–Hill, New York, 1980. [37] V. Gayral, B. Iochum, The spectral action for Moyal planes, hep-th/0402147, J. Math. Phys., in press. [38] H. Araki, Expansional in Banach algebras, Ann. Sci. École Norm. Sup. 6 (1973) 67. [39] A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994. [40] F. Ruiz Ruiz, UV/IR mixing and the Goldstone theorem in noncommutative field theory, Nucl. Phys. B 637 (2002) 143. [41] R. Oeckl, Untwisting noncommutative Rd and the equivalence of quantum field theories, Nucl. Phys. B 581 (2000) 559; M. Chaichian, P.P. Kulish, K. Nishijima, A. Tureanu, On a Lorentz-invariant interpretation of noncommutative space–time and its implications on noncommutative quantum field theory, Phys. Lett. B 604 (2004) 98.