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Deformed Heisenberg algebra and its Hilbert space representations

Latévi M. Lawson, Ibrahim Nonkané, Kinvi Kangni

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.

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.
Paper Structure (10 sections, 4 theorems, 59 equations)

This paper contains 10 sections, 4 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Pseudo-Hermiticity in quantum mechanics
  3. Position-deformed Heisenberg algebra
  4. Hilbert space representations
  5. Position space representations
  6. Fourier transform and its inverse representations
  7. Spectrum of a free particle
  8. Conclusion
  9. Proof of the proposition 4.2
  10. Proof of the proposition 4.3

Key Result

Proposition 2.3

A non-Hermitian operator $\hat{H}$ is Hermitian with respect to the pseudo-inner product $\langle .|. \rangle_{{S}_+}$ if we have

Theorems & Definitions (12)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • proof
  • Remark 4.4
  • ...and 2 more