Deformed symmetry in Snyder space and relativistic particle dynamics

Abstract

We describe the deformed Poincare-conformal symmetries implying the covariance of the noncommutative space obeying Snyder's algebra. Relativistic particle models invariant under these deformed symmetries are presented. A gauge (reparametrisation) independent derivation of Snyder's algebra from such models is given. The algebraic transformations relating the deformed symmetries with the usual (undeformed) ones are provided. Finally, an alternative form of an action yielding Snyder's algebra is discussed where the mass of a relativistic particle gets identified with the inverse of the noncommutativity parameter. *

Covariant realizations of kappa-deformed space

“…For example, Snyder space is of this type and covariant realizations in terms of undeformed space exist [46].…”

Emergence of special and doubly special relativity

“…It is worth of noting that the SR commutator (99) appears frequently when the first quantization of relativistic systems is discussed, see e.g., Refs. [55][56][57]81]. The roughness of a typical relativistic path can be evaluated by rewriting (94) in a time-sliced version as ∆t m x ′′ , t ′′ | 1 − (x i (t + ∆t) −x i (t)) 2 (∆t) 2 c 2 |x ′ , t ′ = x ′′ , t ′′ | (x i (τ + ∆t) −x i (τ )) 2 |x ′ , t ′ , (100) which gives for ∆x i (τ ) ≡x i (τ + ∆t) −x i (τ ) c∆t = x ′′ , t ′′ | |∆x i (τ )| 1 + m 2 c 2 (∆x i (τ )) 2 |x ′ , t ′ .…”

Plane waves in noncommutative fluids

“…The equation (26) shows that the current density is not conserved in the noncommutative case which is a consequence of a lack of translation symmetry. The violation of the translation invariance has been previously analysed in [48].…”

“…The noncommutative fluid was constructed as a noncommutative field theory [13,14] in the realization method approach [15,16,17,18,19,20,21,22,23,24,25,26,27] by generalizing the first order action functional of the commutative perfect relativistic fluid [28,29,30,31]. In this approach one obtains the fluid dynamics of the long wavelength degrees of freedom of the system bypassing the statistical analysis of the microscopic degrees of freedom which in the noncommutative spaces is not well understood yet (see for tentative approaches [32,33,34]).…”