Group theoretical foundations of the Quantum Theory of an interacting particle

TL;DR

This paper explores the group theoretical foundations of quantum mechanics for interacting particles, addressing symmetry obstacles and proposing an approach based on invariance properties under Galilean transformations.

Contribution

It develops a method to extend group theoretical quantum approaches to interacting particles by identifying invariance properties related to specific sub-groups of Galilean transformations.

Findings

Different wave equations correspond to specific invariance properties.

The approach overcomes symmetry obstacles in non-relativistic cases.

Potential for discovering new wave equations remains open.

Abstract

{We point out some obstacles raised by the lost of symmetry against the extension to the case of an interacting particle of the approach that {\sl deductively} establishes the Quantum Theory of a free particle according to the group theoretical methods worked out by Bargmann, Mackey and Wigner. Then we develop an approach which overcomes these difficulties in the non relativistic case. According to our approach the different specific forms of the wave equation of an interacting particle are implied by particular first order invariance properties that characterize the interaction with respect to specific sub-groups of galileian transformations. Moreover, the possibility of yet unknown forms of the wave equation is left open.