Geometries with the Second Poincaré Symmetry

Abstract

The second Poincaré kinematical group serves as one of new ones in addition to the known possible kinematics. The geometries with the second Poincaré symmetry is presented and their properties are analyzed. On the geometries, the new mechanics based on the principle of relativity with two universal constants (c, l) can be established.

Possible supersymmetric kinematics

“…It has been shown in Ref. [9] that if the realizations of possible kinematical algebras are taken into account, more possible kinematics possessing the same so(3) isotropy generated by J will be obtained than in the Bacry-Lévy-Leblond's classical work [2], having dramatically different physical and geometrical meanings [4][5][6]9]. For example, the second realization of Poincaré group will not preserve the Minkowksi metric.…”

“…However, if the realization is taken into account, the ordinary and second Poincaré algebras are dramatically different from each other in physics and geometry. For example, the underlying geometries for the second Poincaré algebra must be degenerate ones rather than a 4d Minkowski space-time [4][5][6].…”

“…6) and σ(x) := 1 + l −2 η µν x µ x ν > 0 defines the domain of Beltrami-AdS spacetime. The fermionic generator Q and its conjugateQ, which obey the Majorana condition Q = CQ T and are of dimension L −1/2 , may be chosen as[19] …”

“…These problems have been studied in Ref. [4][5][6]. It has been shown that when d ± are realized in terms of the Beltrami coordinates [7,8], in the l → 0 limit d ± also contract to an iso(1, 3) algebra but the generators and thus the contracted algebra has very different physical significance.…”

Rigidity and Parallelism in the spacetime

“…A complete of list of references could be found in [2]. Properties of the Fock-Lorentz transformations and their relations with other group theoretical extensions of the special theory of relativity with two fundamental constants have been studied in [3][4][5][6].…”