Quantum kinematics

Abstract

The FRT quantum group and space theory is reformulated from the standard mathematical basis to an arbitrary one. The $N$-dimensional quantum vector Cayley-Klein spaces are described in Cartesian basis and the quantum analogs of $(N-1)$-dimensional constant curvature spaces are introduced. Part of the 4-dimensional constant curvature spaces are interpreted as the non-commutative analogs of $(1+3)$ kinematics. A different unifications of Cayley-Klein and Hopf structures in a kinematics are described with the help of permutations. All permutations which lead to the physically nonequivalent kinematics are found and the corresponding non-commutative $(1+3)$ kinematics are investigated. As a result the quantum (anti) de Sitter, Minkowski, Newton, Galilei kinematics with the fundamental length, the fundamental mass and the fundamental velocity are obtained.

Talk given at the Int. conference "Non-Commutative Geometry and Representation Theory in Mathematical Physics", 5-10 July, 2004, Karlstad, Sweden and at XI Int. conference "Symmetry Metods in Physics", 21-24 June, 2004, Prague, Czech Republic, 24 pages, 1 figure