On the uniqueness of the unitary representations of the non commutative Heisenberg-Weyl algebra

TL;DR

This paper investigates the conditions under which the Stone-von Neumann theorem applies to the non-commutative Heisenberg-Weyl algebra, establishing the uniqueness of its unitary representations except along a critical parameter line.

Contribution

It extends the Stone-von Neumann theorem to a non-commutative setting, identifying conditions for the uniqueness of unitary representations.

Findings

Uniqueness of unitary representations holds outside a critical parameter line.

The classical Stone-von Neumann theorem is generalized to non-commutative cases.

The critical line delineates where the theorem's applicability breaks down.

Abstract

In this paper we discuss the uniqueness of the unitary representations of the non commutative Heisenberg-Weyl algebra. We show that, apart from a critical line for the non commutative position and momentum parameters, the Stone-von Neumann theorem still holds, which implies uniqueness of the unitary representation of the Heisenberg-Weyl algebra.