The extreme-eigenstate approach to the solution of dynamic group representation is formulated systematically based on the continuous representation of the group generators and their intrinsic counterparts by virtue of two complete sets of commuting operators of first and second kinds (CSCO I and CSCO II). The high-order partial differential equations for the general representation eigenstates in the conventional approach are replaced by a set of first-order partial differential equations for the extreme eigenstate in this approach. This set of first-order partial differential equations is tractable for analytical solutions. By applying the ladder operators, any representation eigenstates can be obtained by standard differential operation. In application, the Wigner SU(2) D function and the Elliott SU(3) irreducible representation basis are obtained by this approach as illustrations. The approach is proved to be simple, useful, and efficient for physicists.
The extreme-eigenstate approach to the solution of dynamic group representation is formulated systematically based on the continuous representation of the group generators and their intrinsic counterparts by virtue of two complete sets of commuting operators of first and second kinds (CSCO I and CSCO II). The high-order partial differential equations for the general representation eigenstates in the conventional approach are replaced by a set of first-order partial differential equations for the extreme eigenstate in this approach. This set of first-order partial differential equations is tractable for analytical solutions. By applying the ladder operators, any representation eigenstates can be obtained by standard differential operation. In application, the Wigner SU(2) D function and the Elliott SU(3) irreducible representation basis are obtained by this approach as illustrations. The approach is proved to be simple, useful, and efficient for physicists.
The continuous-variable representation of the dynamic group is discussed in general terms, based on the differential of the group manifold. The unified formalism of various continuous-variable representations of the dynamic group is presented, and the essential common features of different continuous-variable representation theories are exploited from the general mathematical-physical point of view. The right-hand isomorphic representation and the left-hand anti-isomorphic representation of the dynamic group, and their relationship and physical implication are discussed in detail. Specialization of the general formalism to the function spaces of the group, based on both the Hilbert space of quantum-mechanical states and the von Neumann space of the quantum-statistical states (density matrices), leads to the results of both the generator coordinate approach to the dynamic-group representation and the generalized quantum characteristic function method. Techniques for calculating the continuous-variable representation of the group algebra are proposed and illustrated with the Lipkin SU(2), the Elliott SU(3), and its extension Sp(6) symmetries as examples. The results obtained for these dynamic groups are instructive for both physicists and mathematicians.
This paper discusses an application to the study of dynamics of the typical overcomplete, non-independent sets of unit vectors that characterize continuous-representation theory. It is shown in particular that the conventional, classical Hamiltonian dynamical formalism arises from an analysis of quantum dynamics restricted to an overcomplete, nonindependent set of vectors which lie in one-to-one correspondence with, and are labeled by, points in phase space. A generalized ``classical'' mechanics is then defined by the extremal of the quantum-mechanical action functional with respect to a restricted set of unit vectors whose c-number labels become the dynamical variables. This kind of ``classical'' formalism is discussed in some generality, and is applied not only to simple single-particle problems, but also to finite-spin degrees of freedom and to fermion field oscillators. These latter cases are examples of an important class of problems called exact, for which a study of the classical dynamics alone is sufficient to infer the correct quantum dynamics.
We generalize our modified spectral method for the solution of the coupled real partial differential equations in phase space for the stationary Wigner function of an energy eigenstate. This generalization allows us to apply our algorithm to arbitrary high-order partial derivatives without increasing the numerical costs. This is possible since we can derive a sum factorization formula converting a multiple sum into a simple product. We apply our method to evaluate the Wigner function of the Morse oscillator and an asymmetric double-well potential, and compare our results with the exact solution when it is known.
The method of self-adjoint ladder operators, developed in Parts I and II, is applied to the solution of the generalized angular-momentum problem. This reveals many interesting aspects of this approach to eigenvalue problems and, in particular, its relationship to addition of angular momentum. The complete set of irreducible unitary representations of the underlying algebra is obtained and also the corresponding Clebsch-Gordan (Wigner) coefficients for the addition of spin and angular momentum in a space of arbitrary dimension.
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.
The generator coordinate method approach to the dynamic group representation is discussed in general. In various cases, either in group space or in coset space, representations of the dynamic group can readily be obtained with the generator coordinate method. Boson representation is just one form of the generator coordinate method approach to the dynamic group representation. Various representations of the dynamic group are described by the generator coordinate method approach to the dynamic group representation in a unified way. Not only is the algebraic structure of generators preserved, but also conditions imposed by (a) the Pauli principle, (b) symmetry properties, (c) dynamic properties, and (d) other concrete properties of nuclear systems are well incorporated in these representations, so that the original fermion representation is faithfully realized. The generator coordinate method approach to the dynamic group representation is strictly a transformation theory from the fermion representation to continuous variable or boson representations of the dynamic group. Examples are given for showing the essence and the prospects of applications of the generator coordinate method approach to the dynamic group representation.