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International Journal of Geometric Methods in Modern Physics

On the Heisenberg algebra in Bargmann–Fock space with natural cutoffs

Kourosh Nozari, M. Roushan

Published Feb 16, 2016• 7 citations
Publisher

Abstract

We construct a generalized Hilbert space representation of quantum mechanics in [Formula: see text]-dimensions based on Bargmann–Fock space with natural cutoffs as a minimal length, minimal momentum and maximal momentum in order to have a general framework for Heisenberg algebra in Bargmann–Fock space.
DOI
10.1142/s0219887816500547

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The commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of nonzero minimal uncertainties in the positions and momenta. The usual quantum mechanical behavior is recovered as a limiting case for not too small and not too large distances and momenta.
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.
Position deformation of a Heisenberg algebra and Hilbert space representation of both maximal length and minimal momentum uncertainties may lead to loss of Hermiticity of some operators that generate this algebra. Consequently, the Hamiltonian operator constructed from these operators is also not Hermitian. In this paper, with an appropriate positive-definite Dyson map, we establish the Hermiticity of these operators by means of a similarity transformation. We then construct Hilbert space representations associated with these Hermitian operators that generate a Hermitian Heisenberg algebra. With the help of these representations, we establish the path integral formulation of any systems in this Hermitian algebra. Finally, using the path integral of a free particle as an example, we demonstrate that the Euclidean propagator, action, and kinetic energy of this system are constrained by the standard classical mechanics limits.