Deforming the Maxwell-Sim algebra

Abstract

The Maxwell algebra is a non-central extension of the Poincaré algebra, in which the momentum generators no longer commute, but satisfy [P µ , P ν ] = Z µν . The charges Z µν commute with the momenta, and transform tensorially under the action of the angular momentum generators. If one constructs an action for a massive particle, invariant under these symmetries, one finds that it satisfies the equations of motion of a charged particle interacting with a constant electromagnetic field via the Lorentz force. In this paper, we explore the analogous constructions where one starts instead with the ISim subalgebra of Poincaré, this being the symmetry algebra of Very Special Relativity. It admits an analogous non-central extension, and we find that a particle action invariant under this Maxwell-Sim algebra again describes a particle subject to the ordinary Lorentz force. One can also deform the ISim algebra to DISim b , where b is a non-trivial dimensionless parameter. We find that the motion described by an action invariant under the corresponding MaxwellDISim algebra is that of a particle interacting via a Finslerian modification of the Lorentz force. In an appendix is it shown that the DISim b algebra is isomorphic to the extended Schrödinger algebra with b = 1 1−z .

The symmetries of the Carroll superparticle

“…We have constructed the super-Killing equations and showed that the symmetries form an infinite dimensional superalgebra. After taking the flat limit we found that among the symmetries of the N = 1 Carroll superparticle we have a supersymmetric extension of the Lifshitz Carroll algebra [30] with dynamical exponent z = 0. The bosonic part of this algebra has appeared as a symmetry of warped conformal field theories [22].…”

“…These dilatations, together with the super-Carroll transformations, form a supersymmetric extension of the Lifshitz Carroll algebra [30] with dynamical exponent z=0. The Lifshitz Carroll algebra with z=0 has appeared in a recent study of warped conformal field theories [22].…”

Modified Newton's gravity in Finsler space as a possible alternative to dark matter hypothesis

“…And in the framework of Finsler geometry, the four-velocity vector is treated as independent variable [12]. The explicit DISIM b (2) invariant Finslerian line element and the respective Lie algebra was first proposed thirty years ago [13,14,15]. Finsler geometry is a natural and fundamental generalization of Riemann geometry.…”

Geometrical Models of the Locally Anisotropic Space-Time

“…In particular, the requirement of SIM(2) symmetry was sufficient to show [39] that neutrinos may have mass along with the lepton number conservation, and it is important that this result can not be obtained within the framework of Lorentz-invariant approach without introducing sterile neutrinos. However, a significant drawback of the VSR is that is regarded only as a phenomenological parameter and VSR can not say anything of its [40] using alternative methods (see also [41]). In particular, the inhomogeneous 8-parametric group of relativistic symmetry of metric (1) with its Lie algebra (4) were obtained using the method of continuous deformations of algebra ISIM (2).…”