Time dependent quantum generators for the Galilei group

TL;DR

This paper extends the theory of time-dependent quantum generators for the Galilei group from (1+1) dimensions to higher dimensions, providing a broader framework for understanding quantum symmetries with explicit time dependence.

Contribution

It generalizes the approach of Doebner and Mann to (2+1) and (3+1) dimensions, establishing the formal structure of quantum generators in higher-dimensional Galilei groups.

Findings

Derived generators of ray representations in (2+1) and (3+1) dimensions.

Identified differences in formal structure compared to lower dimensions.

Extended the applicability of time-dependent quantum generators to higher-dimensional settings.

Abstract

In 1995 Doebner and Mann introduced an approach to the ray representations of the Galilei group in ($1+1$)-dimensions, giving rise to quantum generators with an explicit dependence on time. Recently (2004) Wawrzycki proposed a generalization of Bargmann's theory: in his paper he introduce phase exponents that are explicitely dependent by 4-space point. In order to find applications of such generalization, we extend the approach of Doebner and Mann to higher dimensions: as a result, we determine the generators of the ray representation in ($2+1$) and ($3+1$) dimensions. The differences of the outcoming formal apparatus with respect to the smaller dimension case are established.