Complex Analytic Realizations for Quantum Algebras

A method for obtaining complex analytic realizations for a class of deformed algebras based on their respective deformation mappings and their ordinary coherent states is introduced. Explicit results of such realizations are provided for the cases of the q-oscillators (q-WeylHeisenberg algebra) and for the suq(2) and suq(1, 1) algebras and their co-products. They are given in terms of a series in powers of ordinary derivative operators which act on the Bargmann-Hilbert space of functions endowed with the usual integration measure. In the q → 1 limit these realizations reduce to the usual analytic Bargmann realizations for the three algebras. Supported by DGICYT, Spain. *azcarrag @ evalvx.ific.uv.es. **ellinas @ evalvx.ific.uv.es. 1 I.-Introduction The representation theory of quantum algebras and groups [1, 2, 3, 4], constitutes an open field of research. Some of the features specific to quantum algebras, for example those appearing when the deformation parameter q becomes a root of unity, have been studied already [5, 6]. It will be shown here for some of the simplest types of deformed algebras that also for q real there are some interesting realizations which can elucidate the relation between deformation and non-linearity. We will confine our scope in this paper to three of the simplest quantum algebras i.e., the so called q-oscillator and the suq(2) and suq(1, 1) algebras [7, 8, 9, 10, 11]. We will seek complex analytic realizations of the above algebras which, unlike the ones existing in the literature which are based on the so called q-coherent states [12, 13, 14] and involve q-deformed (Jackson) derivatives [13, 15], will be given instead in terms of a series of higher powers of ordinary derivatives. The representation spaces of these deformed realizations will be the ordinary Hilbert spaces of square-integrable analytic functions L2( H , dμ(ζ)), built on the corresponding cosets of the non-deformed Lie groups, i.e G/H = Weyl−Heisenberg U(1) , SU(2) U(1) and SU(1,1) U(1) . The invariant measure of integration dμ(ζ), the so-called Bargmann measure, will be explicitly given below for each case. One feature of the obtained realizations of the quantum algebra generators is that they constitute a deformation of the ordinary Lie algebra generators in the sense that they involve a series in powers of ordinary derivatives (the coefficients of which depend on the deformation parameter q) that reproduces in the ’classical’ q → 1 limit the Lie algebra vector field generators. The outline the paper is as follows: in Section II, the required coherent states (CS) formulae for each of the non-deformed Weyl-Heisenberg (wu), su(2) and su(1, 1) algebra are given. Also the deformation mappings relating the quantum versions of the above algebras to the generators of the respectiveΥ non-deformed ones will be provided, as they will be important in the next Section for the analytic realizations. Section IV will extend the method of obtaining the generator realizations, described in Section III, to the coproducts realized on complex functions depending on two arguments. Finally 2 some conclusions are given in Section V. II.Coherent states and deforming mappings Let G+, G− and G0 be the generic expressions for the generators of the three-dimensional algebras G = wh, su(2) and su(1, 1). The (unnormalized, see below) coherent states (see [16, 17, 18, 19, 20] to which we refer for details on the general group definition of coherent states) can be defined generically as: |ζ) = e+ |φ > , (ζ | =< φ|e−, (1) where |φ > is the corresponding lowest weight state of the different algebras, i.e. |φ >≡ |n = 0 > is the Fock vacuum state for the oscillator; |φ > is given by |j,m = −j > for su(2) with j = 1/2, 1, 3/2, ... and |φ >≡ |k, l = 0 > for su(1, 1) with k = 1, 3 2 , 2, 5 2 , .... The round ket indicates that the CS are unnormalized; the normalized ones are given by |ζ >= 1 √ (ζ|ζ) |ζ). The complex variables ζ (ζ = α, z, ξ) label the CS; ζ and ζ̄ are the projective coordinates of the respective coset spaces G/H where H is the isotropy group of the vacuum state |φ >, namely G/H = WH/U(1) ≈ R, SU(2)/U(1) ≈ S;SU(1, 1)/U(1) ≈ S, i.e. the two-dimensional plane, sphere and hyperboloid respectively. Moreover G+ stands for the generic creation operator which together with the two other generators G−, G0, close into the respective Lie algebras G. One of the generic relations that will be extensively used below reads [G±, f(G0)] = (f(G0 ∓ 1)− f(G0))G± (2) for any analytic function f of G0; this relation is common to the three algebras considered and is a consequence of the first commutator in eqs. (4),(5) and (6) below. It also follows from the definition (1) that G+|ζ) = ∂ζ̄ |ζ), (ζ |G− = ∂ζ(ζ |. Let us now turn to the deforming mapping by which the generators of the quantum q-oscillator, suq(2) and suq(1, 1) are written uniquely in terms of their non-deformed counterparts. Generically (see eg. [21, 22, 23, 24, 11]), 3 G± = G±F (G0) (3a) Gq0 = G0 (3b) where Gq0 ≡ N, J3, K3 and G± ≡ aq , J ± , K ± and F(G0) is given for each algebra by whq suq(2) suq(1, 1) F(G0) √