The present series of papers deals with various realizations of the dynamical group š®š c(2d,R) of microscopic collective states for an A nucleon system in d (=1,2, or 3) dimensions, when these collective states are assumed to be invariant under the orthogonal group O(n) associated with the n=Aā1 relative Jacobi vectors. In this paper, we further study the BarutāGirardello representation proposed in the first two papers of the present series to show that it may be reformulated in terms of some coherent states by generalizing to š®š c(2d,R) a class of Sp(2,R) coherent states introduced by Barut and Girardello. For such purpose, our starting point is another coherent state representation, namely the Perelomov one, previously considered by Kramer. We also propose a third, new class of coherent states leading to HolsteināPrimakoff representation in a straightforward way. We review various properties of these three classes of coherent states, such as their reproducing kernel and measure explicit forms, their generating function properties, and the representations they lead to for both the collective states and their dynamical group.
The present series of papers deals with various realizations of the dynamical group š®š c(2d,R) of microscopic collective states for an A nucleon system in d (=1,2, or 3) dimensions, when these collective states are assumed to be invariant under the orthogonal group O(n) associated with the n=Aā1 relative Jacobi vectors. In this paper, we further study the BarutāGirardello representation proposed in the first two papers of the present series to show that it may be reformulated in terms of some coherent states by generalizing to š®š c(2d,R) a class of Sp(2,R) coherent states introduced by Barut and Girardello. For such purpose, our starting point is another coherent state representation, namely the Perelomov one, previously considered by Kramer. We also propose a third, new class of coherent states leading to HolsteināPrimakoff representation in a straightforward way. We review various properties of these three classes of coherent states, such as their reproducing kernel and measure explicit forms, their generating function properties, and the representations they lead to for both the collective states and their dynamical group.
The present series of papers deals with various realizations of the dynamical group š®š c(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=Aā1 Jacobi vectors. In the present paper, we derive two boson representations of š®š c(2d,R), namely the Dyson representation and the HolsteināPrimakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ā±c of analytic functions in Ī½ =(1/2)d(d+1) complex variables. Basis functions in ā±c, classified according to the chain š®š c(2d,R)āš°c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain š(Ī½)āš(d), in a Bargmann Hilbert space ā¬ of analytic functions in Ī½ complex variables representing Ī½-dimensional boson states. By equating the complex variables of ā±c and their conjugate momenta with those of ā¬, we get the non-Hermitian Dyson representation of š®š c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the Ī½ boson creation and annihilation operators, it is possible to build the generators of a š(Ī½) group, which in the physical three-dimensional case reduces to š(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.
In the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an A-nucleon system, defined as those A-particle states invariant under the orthogonal group O(n) associated with the n = Aā1 Jacobi vectors. The present paper discusses the case of a hypothetical one-dimensional space. Simple invariance considerations show that the dynamical group of collective states is then the group š®š c(2,R), which is the restriction to the collective subspace of the group š®š (2,R) of linear canonical transformations in n dimensions conserving the O(n) symmetry. In addition to the well-known realization of the dynamical group in the Schrƶdinger representation based upon the DzublikāZickendraht transformation, two new realizations are proposed. The first acts in a Barut Hilbert space, which is the subspace of a Bargmann Hilbert space of analytic functions left invariant by O(n). A unitary mapping is established between the ordinary Hilbert space of collective states and the Barut Hilbert space and coherent collective states are defined in the latter. The second is carried out in terms of one boson creation and one boson annihilation operator through a generalized HolsteināPrimakoff representation. The generator of a U(1) group, which is the one-dimensional analog of the U(6) group of the interacting boson model (IBM), can then be expressed in terms of the generators of š®š c(2,R). Finally the generalization of the preceding analysis to a d-dimensional space is outlined in the cases where d = 2 or 3. The dynamical group of collective states becomes š®š c(2d,R).
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.
In the present paper, we introduce partially coherent states for the positive discrete series irreducible representations ćĪ»d+n/2,...,Ī»1+n/2ć of Sp(2d,R), encountered in physical applications. These states are characterized by both continuous and discrete labels. The latter specify the row of the irreducible representation [Ī»1Ī»2ā ā ā Ī»d] of the maximal compact subgroup U(d), while the former parametrize an element of the factor space Sp(2d,R)/H, where H is the Sp(2d,R) subgroup leaving the [Ī»1Ī»2ā ā ā Ī»d] representation space invariant. We consider three classes of partially coherent states, respectively, generalizing the Perelomov and BarutāGirardello coherent states, as well as some recently introduced intermediate coherent states. We prove that each family of partially coherent states forms an overcomplete set in the representation space of ćĪ»d+n/2,...,Ī»1+n/2ć, and study its generating function properties. We show that it leads to a representation of the Sp(2d,R) generators in the form of differential operator matrices. Finally, we relate the latter to a boson representation, namely a generalized Dyson representation in the cases of Perelomov and Barut-Girardello partially coherent states, and a generalized HolsteināPrimakoff representation in that of the intermediate partially coherent states.
The ClebschāGordan problems of the BarutāGirardello, and Perelomov coherent states of SL(2,R) are studied using the associated Hilbert spaces as the respective carrier spaces of the representations of the group. For the BarutāGirardello coherent states this Hilbert space is a subspace of the BargmannāSegal Hilbert space B(C2) called the āāreduced Bargmann space.āā The generators of the group in this realization are essentially the boson operators of Holman and Biedenharn which provide a convenient starting point of the problem. For the Perelomov coherent states the associated Hilbert space turns out to be Bargmannās canonical carrier space for the realization of the discrete series of representations, namely, the Hilbert space of functions analytic inside the open unit disc. The scalar product, the principal vector, and a complete orthonormal set in these Hilbert spaces are constructed and used for the explicit evaluation of the ClebschāGordan coefficients. For each of the coherent state systems the product state turns out to be the principal vector and, therefore, the coupled state itself is the ClebschāGordan coefficient. For the BarutāGirardello coherent states this is, apart from normalization, the product of a Bessel function and d-function. For the Perelomov coherent states, on the other hand, this closely resembles the ClebschāGordan coefficient of the SU(2) coherent states.
In the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations ćĪ»d+n/2,...,Ī»1+n/2ć encountered in physical applications, are analyzed in detail with special emphasis on those of Sp(4,R) and Sp(6,R). The present paper discusses the unitary-operator coherent states, as defined by Klauder, Perelomov, and Gilmore. These states are parametrized by the points of the coset space Sp(2d,R)/H, where H is the stability group of the Sp(2d,R) irreducible representation lowest weight state, chosen as the reference state, and depends upon the relative values of Ī»1,...,Ī»d, subject to the conditions Ī»1ā„Ī»2ā„ ā ā ā ā„Ī»dā„0. A parametrization of Sp(2d,R)/H corresponding to a factorization of the latter into a product of coset spaces Sp(2d,R)/U(d) and U(d)/H is chosen. The overlap of two coherent states is calculated, the action of the Sp(2d,R) generators on the coherent states is determined, and the explicit form of the unity resolution relation satisfied by the coherent states in the representation space of the irreducible representation is obtained. The Hilbert space of analytic functions arising from the coherent state representation is studied in detail. Finally, some applications of the formalism developed in the present paper are outlined. In particular, its relevance to the study of boson realizations of the Sp(2d,R) algebra is stressed.