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arXiv:hep-th/0505246 · Physics Letters B

Coherent states of the deformed Heisenberg–Weyl algebra in non-commutative space

Qi-Jun Yin, Jian-zu Zhang A B

Published Apr 1, 2005• 24 citations
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Abstract

In two-dimensional space a subtle point that for the case of both space–space and momentum–momentum non-commuting, different from the case of only space–space non-commuting, the deformed Heisenberg–Weyl algebra in non-commutative space is not completely equivalent to the undeformed Heisenberg–Weyl algebra in commutative space is clarified. It follows that there is no well-defined procedure to construct the deformed position–position coherent state or the deformed momentum–momentum coherent state from the undeformed position–momentum coherent state. Identifications of the deformed position–position and deformed momentum–momentum coherent states with the lowest energy states of a cold Rydberg atom in special conditions and a free particle, respectively, are demonstrated.

DOI
10.1016/j.physletb.2005.03.040

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In two-dimensional space a subtle point that for the case of both space–space and momentum–momentum non-commuting, different from the case of only space–space non-commuting, the deformed Heisenberg–Weyl algebra in non-commutative space is not completely equivalent to the undeformed Heisenberg–Weyl algebra in commutative space is clarified. It follows that there is no well-defined procedure to construct the deformed position–position coherent state or the deformed momentum–momentum coherent state from the undeformed position–momentum coherent state. Identifications of the deformed position–position and deformed momentum–momentum coherent states with the lowest energy states of a cold Rydberg atom in special conditions and a free particle, respectively, are demonstrated.
The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose–Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.
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.
We consider a non-canonical phase-space deformation of the Heisenberg– Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg–Pauli–Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of time-frequency analysis
At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.
Abstract We consider a non-canonical phase-space deformation of the Heisenberg–Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg–Pauli–Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of time-frequency analysis.