A double rotor vector coherent state construction of the SU(4) contains/implies SU(2)*SU(2) Wigner supermultiplet leads to a simple matrix representation of this scheme. Matrix elements of the vector coherent state realizations of the group generators are given by very simple expressions in terms of ordinary spin (S) and isospin (T) Wigner coefficients with intrinsic projection labels K S and K T . The K-matrix technique is used to effectively elevate these labels to the status of good quantum numbers. The KK matrices are given in analytic form for a number of important irreducible representations.
The introduction of a set of intrinsic coordinates to give an explicit construction of the intrinsic states of vector coherent state theory has greatly simplified earlier attempts to generalize this theory to include the construction of vector coherent state realizations of operators other than the group generators. The group U(3)⊇ U(2)×U(1) is used as a prototype. The construction of irreducible tensor operators with specific shift properties is illustrated with a number of examples. These show how the Wigner calculus for a higher symmetry group can be expressed solely in terms of the recoupling coefficients of the core subgroup and the simple K‐matrix elements of vector coherent state theory.
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.
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation
T of a real algebraic Lie group
G. This requires defining an inner product on the Hilbert space
mathbbH that carries the representation
T. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of
T, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (
mathfrakFmathbbHD,
mathbbHD,
mathfrakFmathfrakHD) of Bargmann spaces. Coherent state wave functions -- the elements of
mathfrakFmathbbHD and of
mathfrakFmathbbHD -- are used to define the inner product on
mathbbHD in a way that simplifies the calculation of the overlaps. This inner product and the group action
Gamma of
G on
mathfrakFmathbbHD are used to formulate expressions for the matrix elements of
T with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for
Gamma relies on matrix factorizations in the complexification of
G even though the representation
T does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and
Gamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in
mathbbH. The scalar and vector coherent state methods will both be applied to Sp(
n) and Sp(
n,mathbbR).
Bases for SU(3) irreps are constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions. The Weyl group is represented as the symmetric group of permutations of the particle coordinates of these spaces. Wigner functions for SU(3) are expressed as products of SU(2) Wigner functions and matrix elements of Weyl transformations. The constructions make explicit use of dual reductive pairs which are shown to be particularly relevant to problems in optics and quantum interferometry.
We study Bohr-Sommerfeld states in the context of the irreducible representations of SU(2). These states offer a precise bridge between the classical and quantum descriptions of angular momentum. We show that they recover the usual basis of angular momentum eigenstates used in physics, and give a self-contained proof in this setting of the formula of Bothwick, Paul and Uribe for the asymptotics of the inner product of arbitrary coherent loop states. As an application, we use these states to derive Littlejohn and Yu's geometric formula for the asymptotics of the Wigner matrix elements.