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Title: Full twisted Poincare symmetry and quantum field theory on Moyal-Weyl spaces

Abstract

We explore some general consequences of a proper, full enforcement of the 'twisted Poincare' covariance of Chaichian et al., Wess, Koch et al., and Oeckl upon many-particle quantum mechanics and field quantization on a Moyal-Weyl noncommutative space(time). This entails the associated braided tensor product with an involutive braiding (or *-tensor product in the parlance of Aschieri et al.) prescription for any coordinate pair of x, y generating two different copies of the space(time); the associated nontrivial commutation relations between them imply that x-y is central and its Poincare transformation properties remain undeformed. As a consequence, in quantum field theory (QFT) (even with space-time noncommutativity) one can reproduce notions (like spacelike separation, time- and normal-ordering, Wightman or Green's functions, etc.), impose constraints (Wightman axioms), and construct free or interacting theories which essentially coincide with the undeformed ones, since the only observable quantities involve coordinate differences. In other words, one may thus well realize quantum mechanics (QM) and QFT's where the effect of space(time) noncommutativity amounts to a practically unobservable common noncommutative translation of all reference frames.

Authors:
 [1];  [2];  [3];  [4];  [3];  [3]
  1. Dip. di Matematica e Applicazioni, Universita 'Federico II', V. Claudio 21, 80125 Naples (Italy)
  2. (Italy)
  3. (Germany)
  4. Arnold Sommerfeld Center for Theoretical Physics, Universitaet Muenchen, Theresienstr. 37, 80333 Munich (Germany)
Publication Date:
OSTI Identifier:
20933307
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 10; Other Information: DOI: 10.1103/PhysRevD.75.105022; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; COMMUTATION RELATIONS; COORDINATES; GREEN FUNCTION; POINCARE GROUPS; QUANTIZATION; QUANTUM FIELD THEORY; QUANTUM MECHANICS; SPACE-TIME; SYMMETRY; TRANSFORMATIONS

Citation Formats

Fiore, Gaetano, INFN, Sez. di Napoli, Complesso MSA, V. Cintia, 80126 Naples, Arnold Sommerfeld Center for Theoretical Physics, Universitaet Muenchen, Theresienstr. 37, 80333 Munich, Wess, Julius, Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Munich, and Universitaet Hamburg, II Institut fuer Theoretische Physik and DESY, Luruper Chaussee 149, 22761 Hamburg. Full twisted Poincare symmetry and quantum field theory on Moyal-Weyl spaces. United States: N. p., 2007. Web. doi:10.1103/PHYSREVD.75.105022.
Fiore, Gaetano, INFN, Sez. di Napoli, Complesso MSA, V. Cintia, 80126 Naples, Arnold Sommerfeld Center for Theoretical Physics, Universitaet Muenchen, Theresienstr. 37, 80333 Munich, Wess, Julius, Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Munich, & Universitaet Hamburg, II Institut fuer Theoretische Physik and DESY, Luruper Chaussee 149, 22761 Hamburg. Full twisted Poincare symmetry and quantum field theory on Moyal-Weyl spaces. United States. doi:10.1103/PHYSREVD.75.105022.
Fiore, Gaetano, INFN, Sez. di Napoli, Complesso MSA, V. Cintia, 80126 Naples, Arnold Sommerfeld Center for Theoretical Physics, Universitaet Muenchen, Theresienstr. 37, 80333 Munich, Wess, Julius, Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Munich, and Universitaet Hamburg, II Institut fuer Theoretische Physik and DESY, Luruper Chaussee 149, 22761 Hamburg. Tue . "Full twisted Poincare symmetry and quantum field theory on Moyal-Weyl spaces". United States. doi:10.1103/PHYSREVD.75.105022.
@article{osti_20933307,
title = {Full twisted Poincare symmetry and quantum field theory on Moyal-Weyl spaces},
author = {Fiore, Gaetano and INFN, Sez. di Napoli, Complesso MSA, V. Cintia, 80126 Naples and Arnold Sommerfeld Center for Theoretical Physics, Universitaet Muenchen, Theresienstr. 37, 80333 Munich and Wess, Julius and Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Munich and Universitaet Hamburg, II Institut fuer Theoretische Physik and DESY, Luruper Chaussee 149, 22761 Hamburg},
abstractNote = {We explore some general consequences of a proper, full enforcement of the 'twisted Poincare' covariance of Chaichian et al., Wess, Koch et al., and Oeckl upon many-particle quantum mechanics and field quantization on a Moyal-Weyl noncommutative space(time). This entails the associated braided tensor product with an involutive braiding (or *-tensor product in the parlance of Aschieri et al.) prescription for any coordinate pair of x, y generating two different copies of the space(time); the associated nontrivial commutation relations between them imply that x-y is central and its Poincare transformation properties remain undeformed. As a consequence, in quantum field theory (QFT) (even with space-time noncommutativity) one can reproduce notions (like spacelike separation, time- and normal-ordering, Wightman or Green's functions, etc.), impose constraints (Wightman axioms), and construct free or interacting theories which essentially coincide with the undeformed ones, since the only observable quantities involve coordinate differences. In other words, one may thus well realize quantum mechanics (QM) and QFT's where the effect of space(time) noncommutativity amounts to a practically unobservable common noncommutative translation of all reference frames.},
doi = {10.1103/PHYSREVD.75.105022},
journal = {Physical Review. D, Particles Fields},
number = 10,
volume = 75,
place = {United States},
year = {Tue May 15 00:00:00 EDT 2007},
month = {Tue May 15 00:00:00 EDT 2007}
}
  • After briefly reviewing the gauge symmetry in Moyal spacetimes, we analyse aspects of symmetry breaking within a quantisation program preserving the twisted Poincare symmetry. We develop the LSZ approach for Moyal spacetimes and derive a mapping for scattering amplitudes on these spacetimes from the corresponding ones on the commutative spacetime. This map applies in the presence of spontaneous breakdown of symmetries as well. We also derive Goldstone's theorem on Moyal spacetime. The formalism developed here can be directly applied to the twisted standard model.
  • On Moyal space-time, one can implement twisted Poincare symmetry with the resultant modification of symmetrization and antisymmetrization postulates for bosons and fermions. We develop the thermofield approach of Umezawa and Takahashi on such a space-time preserving the twisted Poincare symmetry of the underlying quantum field theory (qft). Implications of this twisted Poincare symmetry for qft's at finite temperature are pointed out.
  • We present a comparison of the noncommutative field theories built using two different star products: Moyal and Wick-Voros (or normally ordered). For the latter we discuss both the classical and the quantum field theory in the quartic potential case and calculate the Green's functions up to one loop, for the two- and four-point cases. We compare the two theories in the context of the noncommutative geometry determined by a Drinfeld twist, and the comparison is made at the level of Green's functions and S matrix. We find that while the Green's functions are different for the two theories, the Smore » matrix is the same in both cases and is different from the commutative case.« less
  • Einstein's theory of general relativity was formulated as a gauge theory of Lorentz symmetry by Utiyama in 1956, while the Einstein-Cartan gravitational theory was formulated by Kibble in 1961 as the gauge theory of Poincare transformations. In this framework, we propose a formulation of the gravitational theory on canonical noncommutative space-time by covariantly gauging the twisted Poincare symmetry, in order to fulfil the requirement of covariance under the general coordinate transformations, an essential ingredient of the theory of general relativity. It appears that the twisted Poincare symmetry cannot be gauged by generalizing the Abelian twist to a covariant non-Abelian twist,more » nor by introducing a more general covariant twist element. The advantages of such a formulation as well as the related problems are discussed and possible ways out are outlined.« less
  • By twisting the commutation relations between creation and annihilation operators, we show that quantum conformal invariance can be implemented in the 2-d Moyal plane. This is an explicit realization of an infinite dimensional symmetry as a quantum algebra.