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Modern Physics Letters A

DEFORMED BOSON ALGEBRA AND PROJECTION OPERATOR OF VACUUM IN NONCOMMUTATIVE PHASE SPACE

Bing-Sheng Lin, Yong Guan, Sicong Jing

Published Mar 14, 2009• 1 citation
Publisher

Abstract

In this paper we introduce a new formalism to analyze Fock space structure of noncommutative phase space (NCPS). Based on this new formalism, we derive deformed boson commutation relations and study corresponding deformed Fock space, especially its vacuum structure, which leads to get a form of the vacuum projection operator. As an example of applications of such an operator, we define two-mode coherent state in the NCPS and show its completeness relation.

DOI
10.1142/s0217732309028564

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In this paper we introduce a new formalism to analyze Fock space structure of noncommutative phase space (NCPS). Based on this new formalism, we derive deformed boson commutation relations and study corresponding deformed Fock space, especially its vacuum structure, which leads to get a form of the vacuum projection operator. As an example of applications of such an operator, we define two-mode coherent state in the NCPS and show its completeness relation.
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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.
We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space. Based on the deformed boson algebra, we construct coherent state representations. We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations. It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.
In this paper, by using the Wess–Zumino formalism of noncommutative differential calculus, we show that the concept of nonlinear coherent states originates from noncommutative geometry. For this purpose, we first formulate the differential calculus on a GL p, q(2) quantum plane. By using the commutation relations between coordinates and their interior derivatives, we then construct the two-parameter (p, q)-deformed quantum phase space together with the associated deformed Heisenberg commutation relations. Finally, by applying the obtained results for the quantum harmonic oscillator we construct the associated coherent states, which can be identified as nonlinear coherent states. Furthermore, we show that some of the well-known deformed (nonlinear) coherent states, such as two-parameter (p, q)-deformed coherent states, Maths-type q-deformed coherent states, Phys-type q-deformed coherent states and Quesne deformed coherent states, can be easily obtained from our treatment.
Recent studies on nonperturbation aspects of noncommutative quantum mechanics explored a new type of boson commutation relations at the deformed level, described by deformed annihilation–creation operators in noncommutative space. This correlated boson commutator correlates different degrees of freedom, and shows an essential influence on dynamics. This Letter devotes to the development of formalism of deformed two-photon squeezed states in noncommutative space. General representations of deformed annihilation–creation operators and the consistency condition for the electromagnetic wave with a single mode of frequency in noncommunicative space are obtained. Two-photon squeezed states are studied. One finds that variances of the dimensionless Hermitian quadratures of the annihilation operator in one degree of freedom include variances in the other degree of freedom. Such correlations show the new feature of spatial noncommutativity and allow a deeper understanding of the correlated boson commutator.