Quantum Groups and Von Neumann Theorem

TL;DR

This paper explores the q-deformation of the Weyl-Heisenberg algebra and its relation to the von Neumann theorem, revealing how the deformation parameter distinguishes different quantum representations and their implications in quantum mechanics and field theory.

Contribution

It establishes a connection between q-deformation parameters and unitarily inequivalent representations in quantum mechanics and field theory.

Findings

q-deformation labels Weyl systems in quantum mechanics

Deformation parameter distinguishes inequivalent representations in quantum field theory

Provides insights into the structure of quantum representations

Abstract

We discuss the $q$ deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the $q$-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory.