Independent-Cluster Parametrizations of Wave Functions in Model Field Theories: II. Classical Mappings and Their Algebraic Structure

Abstract

Independent-cluster methods (ICM) are, on the one hand, perturbative, diagrammatically formulated, field-theoretic approaches to the quantum many-body problem. Nevertheless, at the same time they are variational and nonperturbative and permit a complete study of the dynamics of the pure quantal state. The methods divide into three principal classes, namely the configuration-interaction method, the normal coupled cluster method, and the extended coupled cluster method, which differ from each other mainly in the degree to which the basic amplitudes which parametrize the quantum mechanical state are connected. Each of the methods introduces a particular differentiable ICM manifold which is endowed with a symplectic structure. The basic ICM amplitudes provide the local coordinates on these manifolds. In this paper we are concerned with the algebraic structure of these methods and derive a number of rigorous mathematical results, using the anharmonic oscillator with its infinite-dimensional Hilbert space as a nontrivial example. The principal tool in this analysis is the Bargmann Hilbert space representation, which maps the operator algebra in the bosonic Hilbert space into the algebra of differential operators acting on functions of a complex variable. We study in detail the "ICM Star product" which is a newly introduced algebraic structure on the ICM manifolds. It provides a classical map for the operator algebra, and can, for example, be used to map the quantum mechanical commutator into the classical Poisson bracket. Comparisons are also made with the Moyal star product and the more general deformation theory, which provide an autonomous route to quantization, namely the "star product quantization." They are naturally related to such topical theories as quantum groups. We also point out that the present methods allow a purely algebraic approach to deal with diagrammatic expansions, and we consider the structure of the various ICM diagrams from this algebraic point of view.