Positive Discrete Series Research Articles

On the Wigner and the Racah coefficients of suq(2) and suq(1,1)

An explicit expression for the Wigner coefficients coupling two positive (or negative) discrete series of the representations is obtained and the Racah coefficients for these cases are found with a single sum.

Generalized vector coherent state theory of sp(2N,R) vector operators and of sp(2N,R) contains/implies u(N) reduced Wigner coefficients

sp(2N,R) contains/implies u(N) reduced Wigner coefficients coupling a positive discrete series unitary irreducible representation with the 2N-dimensional non-unitary irreducible representation characterized by (1) are evaluated by making use of generalized vector coherent state techniques. General formulae are given in terms of the K-matrix elements of vector coherent state theory and of some u(N) 6j or 9j-type recoupling coefficients. Explicit analytic expressions are obtained in some important special cases. Such results should allow a determination of wsp(2N,R) approximately=w (N)(+)p(2N,R) and osp(P/2N,R) matrix representations to be completed, because both the w(N) generators and the osp(P/2N,R) odd generators transform under the sp(2N,R) irrep(1).

Vector coherent state theory of the noncompact orthosymplectic superalgebras. I. General theory

Vector coherent states are defined for the positive discrete series irreducible representations of the non-compact orthosymplectic superalgebras osp(P/2N,R), where P=2M or 2M+1. An orthonormal Bargmann-Berezin basis, symmetry-adapted to osp(P/2N,R)so(P)(+)sp(2N,R)so(P)(+)u(N), is constructed and used to develop the K-matrix theory for osp(P/2N,R). A general method is provided for determining the conditions of existence of star representations (and of grade star representations in the osp(2/2N,R) case), and the branching rule for their decomposition into a direct sum of so(P)(+)sp(2N,R) irreducible representations. As a by-product, it also enables the matrix elements of the odd generators between basis states of lowest-weight so(P)(+)u(N) irreducible representations to be calculated in a straightforward way.

Vector coherent state theory of the noncompact orthosymplectic superalgebras. II. Some selected examples

For pt.I see ibid. vol.23, p.5383 (1990). The techniques, presented in the first paper of the present series, for determining the conditions for the existence of star or grade star positive discrete series irreducible representations of osp(P/2N,R) (P=2M or 2M+1), and the branching rule for their decomposition into a direct sum of so(P)(+)sp(2N,R) irreducible representations, as well as for constructing explicit matrix realizations, are illustrated with a few selected examples. The latter include the most general irreps of osp(1/2N,2), osp(2/2,R), osp(3/2,R), osp(4/2,R), osp(2/4,R), and the most degenerate irreps of osp(2/2N,R). In addition, all the information necessary for dealing with other cases amenable to a full analytic treatment is provided.

Vector coherent state theory of the semidirect sum Lie algebras wsp(2N,R)

Positive discrete series irreducible representations of the semidirect sum Lie algebras wsp(2N,R)=w(N)+sp(2N,R) are given in wsp(2N,R) contains/implies sp(2N,R) contains/implies u(N) bases. The branching rule for the reduction of a wsp (2N,R) irreducible representation into a direct sum of multiplicity-free sp(2N,R) irreducible representations is accounted for by the explicit construction of a wsp(2N,R) contains/implies sp(2N,R) contains/implies u(N) basis of vector-valued holomorphic functions.

Positive discrete series representations of the noncompact superalgebra Osp(4/2,R)

Some physically realizable positive discrete series representations of the noncompact orthosymplectic superalgebra Osp(4/2,R) are considered. The decomposition of these Osp(4/2,R) representations on reduction to Sp(2,R)×SO(4) is studied in detail, and the corresponding state vectors are explicitly constructed by acting with the generators on a general lowest weight state. Some examples are given to illustrate these results for particular single-particle spaces.

An sl(4,R) Lie algebraic approach to the Bargmann functions and its application to the second Poschl-Teller equation

The sl(4,R) Lie algebraic treatment of the Winger SU(2) matrices (1959) is extended by analytical continuation to the Bargmann SU(1,1) matrices (1947) corresponding to the positive discrete series irreducible representations. It is then used to obtain an sl(4,R) dynamical potential algebra for the negative-energy solutions of the second Poschl-Teller equations (1933).

So(3, 1) versus sp(4,R) as dynamical potential algebra of the symmetrical Poschl-Teller potentials

The results of a previous paper (ibid., vol.21 p.4487, 1988), concerned with the first family of two-parameter Poschl-Teller potentials, are used to obtain potential and dynamical potential algebras for the subfamily of one-parameter symmetrical Poschl-Teller potentials. They are respectively identified with so(3) and so(3, 1) algebras, and shown to be subalgebras of the so(4) potential and sl(4, R) dynamical potential algebras of the two-parameter potentials. All the Hamiltonian eigenstates, corresponding to the subfamily of potentials with quantised potential strengths mu differing by an integer, are proved to belong to a single degenerate continuous unitary irreducible representation of so(3, 1), which may be labelled by generalised Young pattern labels ( omega 0), where omega =- 1+1/4i eta , and eta is some real parameter. An sp(4, R) approximately=so(3, 2) algebra is also constructed and shown to provide an alternative choice for the dynamical potential algebra of the symmetrical potentials. All the above-mentioned eigenstates belong to a single sp(4, R) unitary irreducible representation of the positive discrete series, which may be labelled by its lowest weight (1/2 1/2). Such an alternative choice for the dynamical potential algebra, however, does not lead to a unified treatment of the one- and two-parameter potentials as does the first one.

The nuclear collective wsp(6, R) model

By including the electric dipole operator among the generators, the sp(6, R) algebra of the nuclear symplectic model is extended to the semidirect sum algebra wsp(6, R) = w(3) ⨮ sp(6, R), where w(3) is a Heisenberg-Weyl algebra spanned by three pairs of boson creation and annihilation operators and the unit operator. The mathematics of the extended model is studied in detail and techniques are provided to perform numerical applications of the latter. The wsp(6, R) irreducible representations realized in the model are shown to be positive discrete series ones, which can be labelled by their lowest weight. The branching rule and the raising operators for the wsp(6, R) ⊃ sp(6, R) chain are obtained. Various bases are constructed in the carrier space of a wsp(6, R) irreducible representation and the corresponding matrix representations of the generators are determined. For this purpose, use is made of a Dyson boson realization of wsp(6, R). Finally, the choice of a physically relevant wsp(6, R) irreducible representation and of a phenomenological collective Hamiltonian in the enveloping algebra of wsp(6, R) is reviewed.

Matrix representation of the generators of symplectic algebras. II. The general case with explicit results for sp(6, R)

Pt. I. not available. One of the important problems in Lie algebras is to determine the matrix representation of their generators in a basis associated with a given irrep of the corresponding group. For the unitary and orthogonal groups these representations were obtained by Gelfand and Zetlin. In the present paper the authors give results of the same degree of generality for the symplectic Lie algebras sp(2d, R), where d is any integer, when the irreps are in the positive discrete series. The basis is constructed from powers of the raising generators acting on Gelfand states. Using the Dyson boson realisation of the generators of sp(2d, R), and the result of Gelfand for unitary Lie algebras, the matrix representation of these generators was obtained in closed form. Explicit results are given for sp(6, R) and the authors sketch their application to the symplectic model of the nucleus.

A solution to the SU(n) external state labelling problem based upon a U(n-1,n-1) group. I. General theory

The state labelling problem arising in the reduction of the direct product of a p positive-row U(n) irreducible representation (h1. . .hp0) with a q negative-row one (0-h'q. . .-h'1) into a sum of mixed U(n) irreducible representations (k1. . .kp0-k'q. . .-k'1) is solved by using the complementarity between U(n) and U(p,q) within some positive discrete series irreducible representations of U(pn,qn). This complementarity allows analysis of the problem in terms of the group chain U(p,q) contains/implies U(p)*U(q) instead of U(n)*U(n) contains/implies U(n). For the most general SU(n) irreducible representations corresponding to p=q=n-1, the relevant group chain is therefore U(n-1,n-1) contains/implies U(n-1)*U(n-1). In such a case, the additional labels include those of an intermediate U(n-1) irreducible representation (h1s. . .hn-1s), as well as the additional labels solving the state labelling problems for the products (k1. . .kn-1)*(h1s. . .hn-1s) and (k'1. . .k'n-1)*(h1s. . .hn-1s) of U(n-1) irreducible representations. Hence the proposed solution reflects in a direct way the operation of King's branching rule for the chain U(n)*U(n) contains/implies U(n), supplemented, whenever necessary, with King's modification rule.

Matrix representation of the generators of symplectic algebras. I. The case of sp(4,R)

Because of their many physical applications, there has been considerable interest lately in the basis for irreducible representations (irreps) of symplectic groups and in the matrix elements of the generators of these groups with respect to this basis. In the present paper the authors carry out this programme for the irreps in the positive discrete series of sp(4,R) by building up this basis from powers of the raising generators applied to the lowest weight state. By using the Dyson boson realisation of the generators, their matrix representation can be obtained by direct differentiation, while the overlap of the states of the basis can be determined with the help of coherent states. The extension of the analysis to symplectic groups of arbitrary dimension is straightforward and will be implemented explicitly for sp(6,R) in a later paper.

Complementarity relation between the U(p,q) and U(n) lie groups and some applications to atomic physics

The complementarity relation between the unitary groups U(d) and U(n) within the symmetrical irreducible representations of the larger unitary group U(dn) is extended to non-compact groups. It is proved that the pseudo-unitary group U(p,q) is complementary with respect to U(n) within some positive discrete series irreducible representations of the larger pseudo-unitary group U(pn,qn). The latter arise when reducing the metaplectic irreducible representations ((1/2)dn) and ((1/2)dn-13/2) of the real symplectic group Sp(2dn, R), where d=p+q, and they are characterised by a single label, the eigenvalue of the first order Casimir operator. Some applications of the U(p,q)-U(n) complementarity to atomic physics are outlined. For such purposes, the isomorphism between the Lie algebras of SU(2,2) and SO(4,2) is used extensively. The mathematical framework underlying the Kibler-Negadi approach (1983,4) of the hydrogen atom dynamical group is extended to the independent-electron dynamical group of intrashell many-electron states as well as to the correlated electron dynamical group of intrashell doubly excited states.

Hilbert spaces of analytic functions and representations of the positive discrete series of Sp(6, R)

Hilbert spaces of analytic functions are constructed which carry irreducible representations of the positive discrete series of Sp(6, R).

Boson realization of sp(4, R). II. The generating kernel formulation

In a previous paper of this series, the matrix elements were discussed with respect to boson states of an operator K2 required for the boson realization of the sp(4, R) Lie algebra. In the present paper, it is shown that these matrix elements can be obtained from a generating kernel given by the overlap of sp(4, R) coherent states. The results have relevance for the determination of the matrix elements of the generators of the sp(4, R) Lie algebra with respect to the basis of irreps of the positive discrete series for the corresponding group, and are, in principle, generalizable to symplectic algebras of higher dimensions.

Coherent states of the real symplectic group in a complex analytic parametrization. I. Unitary-operator coherent states

In the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations 〈λd+n/2,...,λ1+n/2〉 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 λ1,...,λd, subject to the conditions λ1≥λ2≥ ⋅ ⋅ ⋅ ≥λd≥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.

Coherent states of the real symplectic group in a complex analytic parametrization. II. Annihilation-operator coherent states

In the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations 〈λd+n/2,..,λ1+n/2〉, 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 annihilation-operator coherent states, i.e., the eigenstates of the noncompact lowering generators corresponding to complex eigenvalues. These states generalize the coherent states introduced by Barut and Girardello for Sp(2,R), and later on extended by Deenen and Quesne to the Sp(2d,R) irreducible representations of the type 〈(λ+n/2)d〉. When λ1,...,λd are not all equal, it was shown by Deenen and Quesne that the eigenvalues do not completely specify the eigenstates of the noncompact lowering generators. In the present work, their characterization is completed by a set of continuous labels parametrizing the (unitary-operator) coherent states of the maximal compact subgroup U(d). The resulting coherent states are therefore of mixed type, being annihilation-operator coherent states only as regards the noncompact generators. A realization in a subspace of a Bargmann space of analytic functions shows that such coherent states satisfy a unity resolution relation in the representation space of 〈λd +n/2,...,λ1+n/2〉, and therefore may be used as a continuous basis in such space. The analytic functions and the differential operators representing the representation space discrete bases and the Sp(2d,R) generators, respectively, are found in explicit form. It is concluded that the annihilation-operator coherent state representation provides the mathematical foundation for the use of differentiation operators with respect to the noncompact raising generators in symbolic expressions of the Sp(2d,R) generators. This is to be compared with the habit of replacing a boson annihilation operator by a symbolic differentiation with respect to the corresponding creation operator, accounted for by the Bargmann representation of such operators.

A class of unitary representations of the Lie group Sp(3, R), its coherent states, and its map to a symplectic realization on sp*(3, R)

Unitary representations from the positive discrete series of Sp(3, R) with lowest weight w={w0w0w0} are considered. By use of new relations on the enveloping algebra, the generators are constructed as differential operators acting on functions of six real variables. Coherent states for these representations are constructed with the help of the Iwasawa decomposition and used to map the representation space to a symplectic realization on the dual sp*(3, R).

Boson representations of the real symplectic group and their applications to the nuclear collective model

Both non-Hermitian Dyson and Hermitian Holstein–Primakoff representations of the Sp(2d,R) algebra are obtained when the latter is restricted to a positive discrete series irreducible representation 〈λd +n/2,...,λ1+n/2〉. For such purposes, some results for boson representations, recently deduced from a study of the Sp(2d,R) partially coherent states, are combined with some standard techniques of boson expansion theories. The introduction of Usui operators enables the establishment of useful relations between the various boson representations. Two Dyson representations of the Sp(2d,R) algebra are obtained in compact form in terms of ν=d(d+1)/2 pairs of boson creation and annihilation operators, and of an extra U(d) spin, characterized by the irreducible representation [λ1⋅⋅⋅λd]. In contrast to what happens when λ1=⋅⋅⋅=λd=λ, it is shown that the Holstein–Primakoff representation of the Sp(2d,R) algebra cannot be written in such a compact form for a generic irreducible representation. Explicit expansions are, however, obtained by extending the Marumori, Yamamura, and Tokunaga method of boson expansion theories. The Holstein–Primakoff representation is then used to prove that, when restricted to the Sp(2d,R) irreducible representation 〈λd +n/2,...,λ1+n/2〉, the dn-dimensional harmonic oscillator Hamiltonian has a U(ν)×SU(d) symmetry group. Finally, the results are applied to the Sp(6,R) nuclear collective model to demonstrate the existence of a hidden U(6)×SU(3) symmetry in this model.

Partially coherent states of the real symplectic group

In the present paper, we introduce partially coherent states for the positive discrete series irreducible representations 〈λd+n/2,...,λ1+n/2〉 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 [λ1λ2⋅⋅⋅λd] 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 [λ1λ2⋅⋅⋅λd] representation space invariant. We consider three classes of partially coherent states, respectively, generalizing the Perelomov and Barut–Girardello 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 〈λd+n/2,...,λ1+n/2〉, 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 Holstein–Primakoff representation in that of the intermediate partially coherent states.