Deformed Heisenberg algebra and its Hilbert space representations

TL;DR

This paper explores a deformed Heisenberg algebra with both maximal length and minimal momentum, addressing non-Hermiticity issues through pseudo-Hermitian operators and analyzing its impact on quantum levels and particle transitions.

Contribution

It introduces a new position deformation of Heisenberg algebra with maximal length and minimal momentum, and constructs corresponding Hilbert space representations using pseudo-Hermiticity.

Findings

Operators become Hermitian with a pseudo-metric operator

Deformation affects quantum energy levels and allows low-energy transitions

Constructs a consistent mathematical framework for the deformed algebra

Abstract

A deformation of Heisenberg algebra induces among other consequences a loss of Hermiticity of some operators that generate this algebra. Therefore, these operators are not Hermitian, nor is the Hamiltonian operator built from them. In the present paper, we propose a position deformation of Heisenberg algebra with both maximal length and minimal momentum uncertainties. By using a pseudo-similarity transformation to the non-Hermitian operators, we prove their Hermiticity with a suitable positive-definite pseudo-metric operator. We then construct Hilbert space representations associated with these pseudo-Hermitian operators. Finally, we study the eigenvalue problem of a free particle in this deformed space and we show that this deformation curved the quantum levels allowing particles to jump from one state to another with low energy transitions.