Possible quantum kinematics

Abstract

A possible mathematics for the unification of quantum mechanics and general relativity A CP-violating kinematic structure AIP Conf. Proc. 566, 317 (2001); 10.1063/1.1378641 SU(2) quantum kinematics: Rotation-observable versus angular-momentum generalized commutation relationsThe quantum group and space theory is reformulated from the standard skewsymmetric basis to an arbitrary one. The N-dimensional quantum Cayley-Klein spaces are described in Cartesian basis and the quantum analogs of ͑N −1͒-dimensional constant curvature spaces are introduced. Part of the fourdimensional constant curvature spaces are interpreted as the noncommutative analogs of ͑1+3͒ space-time models. As a result the quantum ͑anti͒ de Sitter, Minkowski, Newton, Galilei, Carroll kinematics with the fundamental length and the fundamental time are suggested.

Possible quantum kinematics. II. Nonminimal case

“…The transformation of the quantum deformation parameter Z = Jv under contraction is the important ingredient of the noncommutative quantum groups and noncommutative quantum spaces. Unlike previous papers on this subject [1], [2] second power of contraction parameters are included in the multiplier J, what make all kinematics contractions admissible. The different combinations of quantum structure and Cayley-Klein scheme of contractions and analytical continuations are described with the help of permutations σ.…”

“…The analysis in the previous paper [1] was confined to this minimal case. The "union" of two multipliers is understood as the multiplication of all parameters j k , which occur at least in one multiplier and the power of j k in the "union" is equal to its maximal power in both multipliers, for example, (j 1 j 2 2 ) (j 2 j 3 ) = j 1 j 2 2 j 3 .…”

“…It was shown in the previous paper [1] that the quantum N-dimensional Cayley-Klein space O N v (j; σ; C) is characterized by the following commutation relations of the Cartesian generators…”

“…The analysis in [1], [2] was confined to the minimal multiplier J, which has the first power multiplication of contraction parameters. This restriction imply that for certain combinations of quantum structure and Cayley-Klein scheme kinematics contractions do not exist.…”

“…Therefore the analysis of a possible space-time models has the fundamental meaning for physics. In the previous papers [1], [2] the noncommutative analogs of the possible commutative kinematics [3] was constructed starting from the mathematical theory of quantum groups and quantum vector spaces [4]. Cayley-Klein scheme of contractions and analytical continuations was applied to the five dimensional q-Euclidean space.…”