Quantum mechanics and field theory with momentum defined on an anti-de-Sitter space

TL;DR

This paper explores relativistic quantum dynamics on anti-de-Sitter space, introducing conjugate coordinate operators, revealing a non-commutative space-time algebra, and analyzing the implications for time discreteness and wave function overlaps.

Contribution

It develops a novel framework for quantum mechanics on anti-de-Sitter space with non-commuting coordinates and examines the resulting algebra and wave function behavior.

Findings

Space-time coordinates do not commute, leading to a non-standard algebra.

Time becomes discrete, complicating the continuous limit.

High energy behavior challenges the definition of action functionals.

Abstract

Relativistic dynamics with energy and momentum resricted to an anti-de-Sitter space is presented, specifically in the introduction of coordiate operators conjugate to such momenta. Definition of functions of these operators, their differentiation and integration, all necessary for the development of dynamics is presented. The resulting algebra differs from the standard Heisenberg one, notably in that the space-time coordinates do not commute among each other. The resulting time variable is discrete and the limit to continuous time presents difficulties. A parallel approach, in which an overlap function, between position and momentum states, is obtained from solutions of wave equations on this curved space are also investigated. This approach, likewise, has problems in the that high energy behavior of these overlap functions precludes a space-time definition of action functionals.