It is shown here and in the preceding paper (Bartlett S D, Rowe D J and Repka J 2002 J. Phys. A: Math. Gen. 35 ) that vector coherent state theory, the theory of induced representations and geometric quantization provide alternative but equivalent quantizations of an algebraic model. The relationships are useful because some constructions are simpler and more natural from one perspective than another. More importantly, each approach suggests ways of generalizing its counterparts. In this paper, we focus on the construction of quantum models for algebraic systems with intrinsic degrees of freedom. Semi-classical partial quantizations, for which only the intrinsic degrees of freedom are quantized, arise naturally out of this construction. The quantization of the SU (3) and rigid rotor models are considered as examples.
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).
We derive a generalization of the well-known theorem for the square integrability of a unitary irreducible representation of a locally compact group. The generalization covers the case of representations admitting vector coherent states. The result is illustrated by an example drawn from the isochronous Galilei group. The construction yields a wide variety of coherent states, labeled by phase space points, which satisfy a resolution of the identity condition, and incorporate spin degrees of freedom.
Rotor coherent state constructions are given for the Wigner supermultiplet SU(4) contains SU(2)xSU(2) and for the special irreducible representations (N0) of the SO(5) contains SO(3) contains SO(2) group chain in exact parallel with the rotor coherent state construction for the SU(3) contains SO(3) contains SO(2) case given by Rowe, LeBlanc,, and Repka. Matrix elements of the coherent state realizations of the group generators are given in all cases by very simple expressions in terms of angular momentum Wigner coefficients involving intrinsic projection labels K. The K-matrix technique of vector coherent state theory is used to effectively elevate these K labels to the status of good quantum numbers. Analytic expressions are given for the (K K*)-matrices for many of the more important irreducible representations.
"The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".
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.