Quantum kinematics

TL;DR

This paper reformulates quantum group theory in arbitrary bases, introduces quantum analogs of curvature spaces, and derives various non-commutative kinematics including quantum de Sitter, Minkowski, Newton, and Galilei models with fundamental physical constants.

Contribution

It presents a novel reformulation of quantum group and space theory in arbitrary bases and constructs non-commutative kinematics with fundamental constants.

Findings

Derived quantum (anti) de Sitter, Minkowski, Newton, Galilei kinematics.

Identified permutations leading to physically distinct non-commutative kinematics.

Established quantum analogs of constant curvature spaces in Cartesian basis.

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.