Noncommutative spaces and Poincaré symmetry

Abstract

We present a framework which unifies a large class of non-commutative spacetimes that can be described in terms of a deformed Heisenberg algebra. The commutation relations between spacetime coordinates are up to linear order in the coordinates, with structure constants depending on the momenta plus terms depending only on the momenta. The possible implementations of the action of Lorentz transformations on these deformed phase spaces are considered, together with the consistency requirements they introduce. It is found that Lorentz transformations in general act nontrivially on tensor products of momenta. In particular the Lorentz group element which acts on the left and on the right of a composition of two momenta is different, and depends on the momenta involved in the process. We conclude with two representative examples, which illustrate the mentioned effect.

Nonassociative Snyder ϕ4 quantum field theory

“…[31,32]. A further generalization of Snyder spacetime deformations was recently introduced in [36][37][38]. Also several nonassociative star/cross product geometries and related quantum field theories have been discussed recently in [39].…”

Constrains of Charge-to-Mass Ratios on Noncommutative Phase Space

“…In consideration of that the momentum operators are defined as the derivatives of the action with respect to the noncommutative coordinates, algebra (2) can appear naturally as a result of algebra (1). A more general investigation on the whole Poincare group was conducted recently in [27] and relativistic corrections to the algebra of position variables and spinorbital interaction were studied in [28]. Therefore, in case that the gauge problem can be cured by using the SeibergWitten (SW) map [3,9,10], it is interesting to study the physical effects when both the nontrivial algebras (1) and (2) exist.…”

Probing noncommutativities of phase space by using persistent charged current and its asymmetry

“…And recently, a more general extension on the generators of the whole Poincaré group was conducted in Ref. [52]. Therefore, there are strong phenomenological motivations to study the extensions (1) and (2) simultaneously.…”