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International Journal of Modern Physics E

APPLICATION OF THE GENERALIZED WICK THEOREM IN THE DESCRIPTION OF THE NUCLEAR STRUCTURE IN THE BOSON SPACE

EDWARD KWAŚNIEWICZ

Published Apr 1, 2010• 0 citations
Publisher

Abstract

The generalized Wick theorem for coupled operators is applied for a description of a system of N interacting bosons. The matrix elements of the N-bosons Hamiltonian containing one- and two-body terms are expressed in terms of overlaps between N-boson states spanning a boson space. A possibility of transfering of the presented formalism for the description of 2N-fermion systems is also considered.

DOI
10.1142/s0218301310014959

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The generalized Wick theorem for coupled operators is applied for a description of a system of N interacting bosons. The matrix elements of the N-bosons Hamiltonian containing one- and two-body terms are expressed in terms of overlaps between N-boson states spanning a boson space. A possibility of transfering of the presented formalism for the description of 2N-fermion systems is also considered.
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A new formalism of the nuclear many-body problem is established in which one ean work with various physical state vector spaces consisting of both Fermions and Bosons, all being equivalent to the original one consisting only of Fermions. With the help of the usual commutation relations and anticommutation relations between the annihilation and creation operators of Boson and Fermion, a generalized state vector space is established, which contains all the physical spaces each being equivalent to the original physical space in terms of pure Fermion operators. Transformation between the state vectors in various equivalent physical spaces are constructed. Basic operators in the original state space are transformed into effective operators in the new physical spaces.
The introduction of an adequate auxiliary space allows us to replace the problem of the treatment of coupled phonons, which is very complicated and voluminous, with the problem of the treatment of usual matrices. As applications of this approach, the formulation of a Wick's theorem for coupled phonons and recursion formulas which can be used to calculate the overlap matrix elements of multiphonon states built with n coupled phonons of different types and collectivities are given. \textcopyright{} 1996 The American Physical Society.
I derive in a novel fashion exact formulas for the calculation of general matrix elements, including the overlap (norm) matrix, between states constructed from fermion pairs. Mapping the fermion pairs to bosons, I show how to construct finite and exact (in the sense of preserving matrix elements) boson representations of the norm operator and one- and two-fermion operators. This may lead to a microscopic basis for the Interacting Boson Model, as well as new truncation schemes for the nuclear shell model.
The new ccc-number calculus for functions of noncommuting operators, developed in Paper I and employed in Paper II to formulate a general phase-space description of boson systems, deals with situations involving equal-time operators only. In the present paper extensions are presented for the treatment of problems involving boson operators at two or more instants of time. The mapping of time-ordered products onto ccc-number functions is studied in detail. The results make it possible to evaluate time-ordered products of boson operators by phase-space techniques. The usual Wick theorem for boson systems is obtained as a special case of a much more general theorem on time ordering. Our method of derivation appears to provide the first direct proof of Wick's theorem as well as a clear insight into its true meaning. A closed expression is also obtained for the time-evolution operator in terms of the solution of the ccc-number differential equation for the phase-space equivalent of this operator. The new calculus is also applied to the problem of evaluating normally ordered time-ordered, and also the antinormally ordered time-ordered, correlation functions.