On a Lorentz-invariant interpretation of noncommutative space–time and its implications on noncommutative QFT

Abstract

By invoking the concept of twisted Poincaré symmetry of the algebra of functions on a Minkowski space-time, we demonstrate that the noncommutative space-time with the commutation relations [x µ , x ν ] = iθ µν , where θ µν is a constant real antisymmetric matrix, can be interpreted in a Lorentz-invariant way. The implications of the twisted Poincaré symmetry on QFT on such a space-time is briefly discussed. The presence of the twisted symmetry gives justification to all the previous treatments within NC QFT using Lorentz invariant quantities and the representations of the usual Poincaré symmetry. PACS: 11.10.Nx Consequently, the commutators of operators are replaced by Moyal brackets and the equiv-

Coherent states and N dimensional coordinate noncommutativity

“…Implementation of such translation symmetries will necessitate the introduction of a gauge potential. The present treatment differs from ones in which Poincaré symmetry is achieved through the introduction of a "twist" [4,5] into the action of that algebra while maintaining noncommutativity on a two dimensional plane. The noncommutativity discussed here is fully N dimensional and involves an algebra larger than the one generated by the coordinates.…”

Twist symmetry and gauge invariance

“…The study of noncommutative quantum field theories (NC QFT) with Heisenberg-like commutation relations [1] (for a review, see [2]) has got a new impetus after it was realized that, although they violate Lorentz invariance, they are however subject to a Lorentz-invariant interpretation due to their twisted Poincaré symmetry [3].…”

Twists of quantum groups and noncommutative field theory

“…[1,2]). In this paper we would like to draw attention to interrelations between noncommutative quantum field theories and quantum groups [3]. Recently, an active research takes place in noncommutative field theory related to noncommutative geometry (see the reviews [4,5] and references therein).…”

“…It has been observed by several authors (see e.g. [3,9,10,11]) that the twist theory of quantum groups provides a very useful tool for constructing non-commutative geometry of space-time, including vector bundles, measure, and equations of motion and their solutions.…”

“…One can check that with the deformed coproduct (15) these CR are invariant under the action of M µν [3]. One has to mention that the interpretation of the Weyl -Moyal product using abelian twist has been known for some time in the deformation quantization (see [17,18] and references therein).…”