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.
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.
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 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.
Fourier transforms of the von Neumann density operators could be regarded as functions on a Lie group whose infinitesimal generators correspond to dynamical variables characterizing a given quantum system. Called ``characteristic functions'' in this paper, they are discussed with a view to reformulating quantum statistical mechanics in a group-theoretical formalism. It is shown that these characteristic functions allow a universal differentiating procedure for calculating averages of arbitrary ordered noncommuting observables and satisfy the same criterion of non-negative definiteness as the Fourier transforms of the everywhere non-negative probability distribution (the characteristic functions in classical probability theory). The group-theoretical characteristic-function formalism is especially useful in providing easy reductions of the dynamics of complicated quantum systems to the evolutions of a set of quantum variables that are of particular interest. As examples of applications, the following results are derived: (a) a hierarchy of equations of motion for subsystems of a given N-body quantum system (a quantum analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy), (b) an equation of motion for centers of mass of two interacting quantum systems, and (c) the behavior of spin fluctuations of an ideal spin gas in the thermodynamic limit.
We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.
In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.