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Journal of Mathematical Physics

A reduction relation for an n-body trace of an operator product consisting of coupled C+’s and C’s

Masao Nomura

Published Apr 1, 1985• 7 citations
Publisher

Abstract

A reduction relation for the trace of an operator product consisting of C+’s and C’s is derived in the truncated space of n fermions or n bosons being distributed over definite orbits. The state operator formalism (derived in a preceding paper) is fully used so as to perform normal ordering (or contraction) of coupled C+’s and C’s in the operator string with the least destruction of the preexisting coupling.

DOI
10.1063/1.526561

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A reduction relation for the trace of an operator product consisting of C+’s and C’s is derived in the truncated space of n fermions or n bosons being distributed over definite orbits. The state operator formalism (derived in a preceding paper) is fully used so as to perform normal ordering (or contraction) of coupled C+’s and C’s in the operator string with the least destruction of the preexisting coupling.
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.
Recurrence and explicit formulae for contractions (partial traces) of antisymmetric and symmetric products of identical trace class operators are derived. Contractions of product density operators of systems of identical fermions and bosons are proved to be asymptotically equivalent to, respectively, antisymmetric and symmetric products of density operators of a single particle, multiplied by a normalization integer. The asymptotic equivalence relation is defined in terms of the thermodynamic limit of expectation values of observables in the states represented by given density operators. For some weaker relation of asymptotic equivalence, concerning the thermodynamic limit of expectation values of product observables, normalized antisymmetric and symmetric products of density operators of a single particle are shown to be equivalent to tensor products of density operators of a single particle.
A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.