A General q-Oscillator Algebra

It is well-known that the Macfarlane-Biedenharn q-oscillator and its generalization has no Hopf structure, whereas the Hong Yan q-oscillator can be endowed with a Hopf structure. In this letter, we demonstrate that it is possible to construct a general q-oscillator algebra which includes the Macfarlane-Biedenharn oscillator algebra and the Hong Yan oscillator algebra as special cases. E-mail address: [email protected] E-mail address: [email protected] The harmonic oscillator has often emerged as a basic theoretical tool for investigating a variety of physical systems. It appears naturally in quantum optics and the theory of coherent radiation. The q-analog of the harmonic oscillator was first proposed by Arik and Coon [1] and subsequently applied to the realization of the q-deformed su(2) algebra by Macfarlane and Biedenharn [2, 3]. A good introduction to the q-deformed oscillators and its representation can be found in ref [4] Macfarlane and Biedenharn defined the q-deformed harmonic oscillator algebra or the q-boson algebra as that generated by the operators {a, a†, N} obeying the commutation relations [N, a†] = a†, (1a) [N, a] = −a, (1b) aa† − qa†a = q−N . (1c) The casimir operator for the algebra, C1, is given by C1 = q −N(a†a− [N ]). (2) In the undeformed situation, namely in the limit q → 1, this central element becomes trivial since its eigenvalues can be shifted to zero and by von Neumann theorem, all representations of the oscillators are unitarily equivalent to each other. This is not the case for the q-deformed oscillator. Different values of the central element can label different representations. The representation theory for a class of q-deformed oscillator algebras defined in terms of arbitrary function of the number operator N has also been developed by Quesne and Vansteenkiste [5]. More recently, Irac-Astaud and Rideau [6, 7] have constructed Bargmann representations corresponding to these generalized q-deformed oscillator algebras and showed that Bargmann representations can exist for some deformed harmonic oscillators which admit non-Fock represntations. The Macfarlane-Biedenharn q-deformed oscillator does not seem to possess a Hopf structure. Hong Yan [8] has proposed another q-deformed oscillator that can admit a Hopf structure. The commutation relation for this q-oscillator are given by [N, a†] = a†, (3a) [N, a] = −a, (3b) [a, a†] = [N + 1]− [N ] (3c) where the bracket [x] denotes q − q−x q − q−1 . The casimir C2 for this algebra is given by C2 = a †a− [N ]. (4) This q-deformed oscillator algebra can be contracted from the q-deformed su(2) algebra by a generalized Inoue-Wigner transformation [9]. Nevertheless if we impose positive norm requirement for the states, then at the representation level, the identification of Hong Yan q-oscillator algebra with the q-deformed su(2) algebra can break down for some values of |q| = 1. In fact, the positive norm requirement [10] is in conflict with the truncation condition [11] imposed on the states of the oscillator so as to get finite multiplets for su√q(2). In short, for |q| = 1 (q = e, ǫ arbitrary) Hong Yan oscillator algebra is different from su√q(2) algebra. Note that the Macfarlane-Biedenharn and Hong Yan algebra are distinct, but in the usual q-Fock representation, they become equivalent. The Macfarlane-Biedenharn oscillator can be written in several equivalent forms. For Arik and Coon [1] q-deformed oscillator, the commutation relation for the operators a and a† in eq(1c) is expressed as aa† − qa†a = 1 (5) For Chaturvedi and Srinivasan [12] q-oscillator, they define the commutation relation above as aa† − a†a = q−N (6) and for Chakrabarti and Jagannathan [13] q-oscillator, they modify the commutation relation so that it accommodates two parameters, q1 and q2, giving the relation aa† − q1a †a = q−N 2 . (7) Many other generalizations have been constructed [5, 6, 7]. The purpose of this letter is to devise a new q-oscillator algebra such that it not only reduces to the Hong Yan algebra for a specific choice of parameters but also becomes the Macfarlane-Biedenharn q-oscillator when one of the two parameters vanishes. Thus our general q-oscillator algebra straddles across the Macfarlane-Biedenharn algebra and the Hong Yan algebra. This feature is, to our knowledge, not present in any previously known generalized q-oscillator algebras. In the following we first write down the generalized Macfarlane-Biedenharn algebra and show that it can be reduced to previously known forms of the Macfarlane-Biedenharn algebra. We then proceed to present our general oscillator. Following Duc [14], we may generalize the Macfarlane-Biedenharn q-oscillator algebra [9] by introducing two additional parameters α and β. The generalized commutation relations for the Macfarlane-Biedenharn algebra are [N, a†] = a†, (8a) [N, a] = −a, (8b) aa† − qαa†a = q , (8c) C3 = q −αN(a†a− [N ]α,β), (8d) where [x]α,β = q − q qα − qβ is a generalized q-bracket. Despite its complexity, this algebra is not a new one. It can be reduced to the usual Macfarlane-Biedenharn q-oscillator algebra. To see this, one can define new operators A = q− (α+β)N 4 a, A† = a†q− (α+β)N 4 and map the generalized commutation relations eq(8) to [N,A†] = A†, (9a) [N,A] = −A, (9b) AA† − q′A†A = q′−N . (9c) where q′ = q α−β 2 . Clearly, one gets the form of eq(1) by identifying qβ−α as (q′)−2. Indeed, with this generalized form of the Macfarlane-Biedenharn q-oscillator algebra, one can see easily that some of the well-known q-oscillator algebras are essentially the same as the Macfarlane-Biedenharn algebra and can be mapped from one to another by invertible transformation. To be specific, we note that the Arik-Coon oscillator [1], Chaturvedi-Srinivasan q-oscillator [12] and the Chakrabarti-Jaganathan [13] oscillator correspond to the cases when α = 1, β = 0, α = 0, β = 1 and α, β arbitrary respectively. We now propose a new general q-oscillator algebra in which the operators {a, a†, N} satisfy the following relations [N, a†] = a†, (10a) [N, a] = −a, (10b) [a, a†] = [N + 1]α,β − [N ]α,β (10c) C4 = q α+β 2 (a†a− [N ]α,β). (10d) Note that in the limit q → 1, this algebra reduces to the algebra for usual undeformed harmonic oscillator. § The general q-oscillator algebra is not equivalent to the Hong Yan algebra given by eq(3). In fact, it includes the generalized Macfarlane-Biedenharn algebra, eq(8), and the Hong Yan algebra, eq(3), as special cases. To see this, we first show that this general q-oscillator algebra eq(10) reduces to the Macfarlane-Biedenharn q-oscillator for α = 0 or β = 0. We note that the generalized q-bracket [N ]α,β can be rewritten as [N ]α,β = q (α+β)(N−1) 2 q α−β 2 N − q− α−β 2 N q α−β 2 − q− α−β 2 (11a) = q (α+β)(N−1) 2 [N ]q′ (11b) By considering q = q, we can actually rewrite the generalized q-bracket, [N ]α,β as q − q q − q′−1 , where k = β α . However, we find that it is sometimes more convenient to stick to the two-parameter deformation so that the special case of α = 0 which corresponds effectively to k → ∞ can be discussed in similar straightforward manner with the case of β = 0 which corresponds to k → 0. where q′ = q α−β 2 . With this result, we can recast the commutation relation in eq(10c) as aa† − a†a = q (α+β)N 2 [N + 1]q′ − q (α+β)(N−1) 2 [N ]q′ (12a) q− (α+β)N 4 aa†q− (α+β)N 4 − q− (α+β)N 2 a†a = [N + 1]q′ − q −(α+β) 2 [N ]q′ (12b) AA† − q− (α+β) 2 A†A = [N + 1]q′ − q −(α+β) 2 [N ]q′ (12c) Using the identity [N + 1]− q[N ] = q−N , (13) one immediately observes that the algebra in eq(10) reduces in the special case of α = 0 to the Macfarlane-Biedenharn q-oscillator algebra with q′ = q −β 2 . Similarly, when the other parameter β = 0, one observes eq(10) reduces to the Macfarlane-Biedenharn q-oscillator with parameter q′′ = q −α 2 . Note also that the casimir operator C4 in eq(10d) can be rewritten with the new operators A and A† as q α+β 2 N(A†A− [N ]q′). In the limit α, β → 0, this latter casimir reduces to the original casimir operator C1 in eq(2) as we would have expected. It is interesting to note that for α+β = 0, the general q-oscillator algebra eq(10) reduces to the usual Hong Yan q-oscillator. Further for α = β, one observes that the generalized q-bracket for the operator N can be rewritten as [N ]α,α = lim β→α q − q qα − qβ = lim β→α q(N−1)α + q(N−2)α+β + q(N−3)α+2β + · · ·+ q(N−1)β = Nq(N−1)α, (14a) so that the commutation relation in eq(10c) now reads aa† − a†a = q{1 +N(1− q−α)} (15a) ⇒ q− N 2 αaa†q− N 2 α − q−αa†q−Nαa = 1 +N(1− q−α) (15b) ⇒ AA† − q′′A†A = 1 +N(1− q′′) (15c) where q′′ = q−α and the operator A and A† are defined by the relations A = q− N 2 a and A† = a†q− N 2 α respectively. Eq(15) can be regarded as a generalized Arik-Coon algebra. It is well-known that the Macfarlane-Biedenharn q-oscillator and the Hong Yan q-oscillator cannot be mapped by invertible transformation to each other. Since the general q-oscillator eq(10) can respectively be reduced to these algebras for certain suitably chosen parameters, we can conclude that the general q-oscillator is not equivalent to the usual MacfarlaneBiedenharn or the Hong Yan q-oscillator. Indeed, we see that in the α− β parameter space shown in figure 1, we have an interesting and elegant picture in which the general q-oscillator algebra eq(10) seems to interpolate between the various q-oscillators. Furthermore, if we de........ . . . . . . ........... . . . . . .... ................................................................................................................ ................................................................................................................. ........................ ............................................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HYO Generalized Arik-Coon Macfarlane-Biedenharn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... Hong Yan .............................................. ....... Undeformed ...................................... . . . . . .... ................................... . . ..... Figure 1: Schematic picture of the α− β parameter space fine k as the ratio β α and q′ = q and recast eq(10c) as aa† − a†a = q′ N+1 − q′ q′ − q′ − q′ N − q′ q′ − q′ , (16) we can show that the general oscillator algebras corresponding to two arbitrary k-values, say k1 and k2, are not equivalent to each other. To see this, one defines operators A = ( q′ − q′1 q′ − q′2 )a and A† = ( q′ − q′1 q′ − q′2 )1/2a† and rewrites eq(16) for k = k1 as AA† − A†A = q′ − q′2 q′ − q′2 − q′ − q′2 q′ − q′2 + F(N, k1, k2) (17) where F(N, k1, k2) = (q′2 − q′1)− (q′2 − q′1 ) q′ − q′2 . Clearly F(N, k1, k2) 6= 0 and consequently the algebras corresponding to eq(16) with k = k1 and k = k2 respectively cannot be equivalent to each other. In particular, one also sees that the general oscillator corresponding to arbitrary k value is not equivalent to the Macfarlane-Biedenharn algebra which corresponds to k = 0 or k → ∞. We have already showed that the generalized Macfarlane-Biedenharn algebra in eq (8) can be related to the usual Macfarlane-Biedenharn oscillator, eq(1), by invertible transformations of the form A = q− (α+β)N 4 a, A† = a†q− (α+β)N 4 . It is interesting to explore whether there exists a similar transformation which can convert the general oscillator eq(10) to the MacfarlaneBiedenharn q-oscillator. To do this, one postulates existence of new operators