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Journal of Mathematical Physics

Novel correspondence of the fixed-seniority and the fixed-isospin averages in their reduction formulas

Masao Nomura

Published Feb 1, 1990• 2 citations
Publisher

Abstract

It is shown that known reduction formulas for partial averages of two different kinds, the fixed-seniority and the fixed-isospin averages, are transcribed into each other by replacement of arguments. The average here is that of a general k-body operator in a finite space of fermions (bosons). Description of the formulas in a common form in terms of eigenvalues of Casimir operators is also exploited.

DOI
10.1063/1.529032

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It is shown that known reduction formulas for partial averages of two different kinds, the fixed-seniority and the fixed-isospin averages, are transcribed into each other by replacement of arguments. The average here is that of a general k-body operator in a finite space of fermions (bosons). Description of the formulas in a common form in terms of eigenvalues of Casimir operators is also exploited.
The seniority scheme introduced by Racah is generalized in the case of degenerate levels. Many formulae for the pure configuration j n are modified in order to calculate matrix elements in this generalized seniority scheme. It is especially the most interesting result that the diagonal matrix element of two-body operators between the states with the same seniority v can be expressed in terms of the matrix elements in the v -particle system. This result, of course, holds in the case of the pure configuration.
The particle-hole symmetry (equivalence) of the full shell-model Hilbert space is straightforward and routinely used in practical calculations. In this work I show that this symmetry is preserved in the subspace truncated up to a certain generalized seniority and give the explicit transformation between the states in the two types (particle and hole) of representations. Based on the results, I study particle-hole symmetry in popular theories that could be regarded as further truncations on top of the generalized seniority, including the microscopic interacting boson (fermion) model, the nucleon-pair approximation, and other models.