The Normal Ordering Procedure and Coherent State of the Q-Deformed Generalized Heisenberg Algebra

Copyright: © 2014 Chung WS. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abstract In this paper we discuss the normal ordering procedure of the q-deformed generalized Heisenberg algebra. We also obtain the coherent state for some types of characteristic functions.


Introduction
In the last few years quantum algebras and quantum groups have been the subject of intensive research in several physics and mathematics fields. Quantum groups or q-deformed Lie algebra implies some specific deformation of classical Lie algebra. From the mathematical point of view, it is a non-commutative associative Hopf algebra. The structure and representation theory of quantum groups have been developed extensively by Jimbo [1] and Drinfeld [2]. In the study of the basic hypergeometric function Jackson [3] invented the Jackson derivative and integral, which is now called q-derivative and q-integral. Jackson's pioneering research enabled theoretical physicists and mathematician to study the new physics or mathematics related to the q-calculus. Much was accomplished in this direction and work is under way to find the meaning of the deformed theory. By using the q-calculus, Arik and Coon [4] proposed the q-deformation of the Heisenberg algebra as follows; † † 1 aa qa a − = † † [ , ] ,[ , ] N a a N a a = = − (1) Where † N N = is called a number operator and † † ( ) a a = Following the approach of the authors of ref [4], several deformed Heisenberg algebra has been proposed in the literature [5][6][7][8][9][10]. In most of deformed Heisenberg algebra, authors adopted the same commutation relations between the number operator and step operators and deformed the commutation relation between a and † a In 2000, the new generalization of the Heisenberg algebra was introduced by Rego-Monteiro and Curado [11][12][13], which takes the

Ha a f H aH F H a a a f H H
where H is a hamiltonian of the physical system under consideration and f(H) is an analytic function of H, called a characteristic function of the algebra. In this algebra, the commutation relations between the number operator and step operators were changed into the more general form which is characterized in terms of the function of the number operator. The authors of the ref [11,12] called this function a characteristic function and discussed the cases when the characteristic function is linear and quadratic in the number operator [12].
In this paper we change the algebra (2) by introducing the parameter q and discuss the normal ordering procedure of the q-deformed generalized Heisenberg algebra (GHA). We use some operator identities to construct the q-deformed generalized Stirling operator of the second kind and its generating function. We also present the q-deformed generalized Heisenberg algebra whose characteristic function is a MÄobius transformation. Finally we discuss some characteristic functions giving a Klauder's coherent state.

Q Deformed Generalized Heisenberg Algebra
In this section we discuss the representation theory of the q-deformed generalized Heisenberg algebra. The q-deformed generalized Heisenberg algebra takes the following form;

Ha a f H aH f H a a a aa qa a f H qH
Where H is a hamiltonian of the physical system under consideration and f(H) is an analytic function of H, called a characteristic function of the algebra. The deformation parameter is related to the concrete form of f(H) and a large class of type Heisenberg algebra can be obtained by choosing the function f(H). From now on we restrict our concern to the case of 0 < q < 1 For example, if we take f(H) = 1 + qH, the algebra (3) reduces to the q-deformed Heisenberg algebra where the hamiltonian is related to the number operator N as follows The choice of f(H) gives a lot of deformed algebra, which is the reason why f(H) is called a characteristic function of the algebra. Now let us introduce the q-Casimir operator as follows; We demand that the q-Casimir operator obeys † When q goes to a unity, the q-Casimir operator reduces to an ordinary Casimir operator. If Now let us construct the irreducible representation of the algebra (3) by introducing the ground state 0 with the lowest eigenvalue of H obeying

Journal of Generalized Lie Theory and Applications
Applying H on † a n yields † † † H a n a f H n a n є f (10) Which means that † a n is an eigen state of H with eigen value So all eigen values of H are determined from 0 Acting a on 1 n + we get a H n f H a n є a n f є a n (13) We get Which shows that 1 a n + is also an eigen state of H with eigen value n є and 1 a n + is proportional to n The representation of the q-deformed generalized Heisenberg algebra is then given by n n a n N n a n N n Where The relation between step operators and hamiltonian is given by † 0 Here Pq is a q-projection operator satisfying † † , , q q q q q aP qP a Pqa qa P P H HP = = = (20) Where f-number is defined as The (n+1)-th eigenvalue e n+1 of the Hamiltonian depends on the previous eigenvalue e n ( ) 1 n n e f e + = (24) So this algebra is sometimes called a one step algebra.
The representation can be rewritten in terms of the f-number as follows; † 0 0 f f a n N n n a n N n n = + + = − (25)

Normal Ordering Process and F-stirling Operator
Now we discuss the normal ordering process for the q-deformed generalized Heisenberg algebra. From the second relation of the eq.(3), we have For an arbitrary function ( ) Then we have the following formulas; Where S(I, j,H) = 0 for i < j and we used the following formulas The recurrence relations are then given by If we set 0 ( | ) P H x I = , we get The eq.(42) can be written in terms of a sum of partial fractions Therefore the f-Stirling operator of the second kind takes the following form;

The Deformed Heisenberg Algebra Related to the Mäobius Transformation
In this section we are going to find the representation for the algebra defined by the relation given in the eq.(3) considering Where , , , α β γ δ are real. The ordinary Heisenberg algebra and q-deformed Heisenberg algebra are obtained from the suitable choice of , , , α β γ δ inverse of the MÄobius transformation is given by In the choice of the characteristic function given in the eq.(46), the algebra (3) Now let us introduce the following characteristic function; Where we assume that α > 0; 0 < γ < 1. Then we have the following The inverse of the characteristic function is given by The n-the iterate of f is then given by Where we assumed that 0 0, 0, 0 1, The representation takes the following form;

Coherent state
We define a coherent state as a eigenvector of the annihilation operator as follows: , a z z z = Where z is a complex number. The coherent state can be represented by using the number state as follows : Inserting the eq.(58) into the eq.(57), we obtain the following relations :

Some Characteristic Functions giving Klauder's Coherent State
In this section we discuss the some characteristic functions giving q-deformed Klauder's coherent state (KCS). The KCS should satisfy the normalizability The coherent state is then given by We can easily find the weighting function The coherent state is then given by Now we have to find the weighting function w(x) so that the coherent sate may satisfy the completeness, which implies We can easily find the weighting function Where we used

Conclusion
In this paper we discussed the normal ordering procedure of the q-deformed generalized Heisenberg algebra, where we introduced the q-deformed generalized Stirling operator of the second kind instead of the Stirling number of the second kind and constructed its generating function. We also discussed the q-deformed generalized Heisenberg algebra whose characteristic function is a MÄobius transformation. Finally we discussed some types of characteristic functions giving a Klauder's coherent state. In fact, it is possible to construct more general algebra as † † † 0 0 ( ), ( ) , It is tempting to investigate, as we did in this paper, the above algebra for some interesting characteristic function. We hope that this topic and its related one will become clear in the near future.