Deformed Heisenberg algebra and its Hilbert space representations

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.

arXiv:2602.15801 [math-ph]

(or arXiv:2602.15801v2 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2602.15801 Focus to learn more arXiv-issued DOI via DataCite

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