dS-AdS structures in the non-commutative Minkowski spaces

TL;DR

This paper explores non-commutative Minkowski spaces with different signatures, providing their algebraic structures, classical limits, quantizations, and explicit representations, including solutions to the Klein-Gordon equation.

Contribution

It introduces new non-commutative Minkowski space models with specific algebraic and geometric properties, including their Poisson structures, quantizations, and explicit eigenfunction constructions.

Findings

Three compatible Poisson structures identified

Explicit irreducible representations constructed

Eigen-functions of Klein-Gordon equation derived

Abstract

We consider a family of non-commutative 4d Minkowski spaces with the signature (1,3) and two types of spaces with the signature (2,2). The Minkowski spaces are defined by the common reflection equation and differ in anti-involutions. There exist two Casimir elements and the fixing of one of them leads to non-commutative "homogeneous" spaces $H_3$, $dS_3$, $AdS_3$ and light-cones. We present the quasi-classical description of the Minkowski spaces. There are three compatible Poisson structures - quadratic, linear and canonical. The quantization of the former leads to the considered Minkowski spaces. We introduce the horospheric generators of the Minkowski spaces. They lead to the horospheric description of $H_3$, $dS_3$ and $AdS_3$. The irreducible representations of Minkowski spaces $H_3$ and $dS_3$ are constructed. We find the eigen-functions of the Klein-Gordon equation in the terms of the horospheric generators of the Minkowski spaces. They give rise to eigen-functions on the $H_3$, $dS_3$, $AdS_3$ and light-cones.