Local Currents for a Deformed Algebra of Quantum Mechanics with a Fundamental Length Scale

TL;DR

This paper investigates explicit representations of a stable deformed quantum algebra with a fundamental length, relating it to the Heisenberg algebra and constructing a generalized harmonic oscillator Hamiltonian.

Contribution

It introduces a new framework for representing a deformed quantum algebra with a fundamental length and extends local current algebra to this setting.

Findings

Explicit representations of the deformed algebra are constructed.

The relation to the standard Heisenberg algebra is clarified.

A generalized harmonic oscillator Hamiltonian is formulated.

Abstract

We explore some explicit representations of a certain stable deformed algebra of quantum mechanics, considered by R. Vilela Mendes, having a fundamental length scale. The relation of the irreducible representations of the deformed algebra to those of the (limiting) Heisenberg algebra is discussed, and we construct the generalized harmonic oscillator Hamiltonian in this framework. To obtain local currents for this algebra, we extend the usual nonrelativistic local current algebra of vector fields and the corresponding group of diffeomorphisms, modeling the quantum configuration space as a commutative spatial manifold with one additional dimension.