SU(1,1) quasi-spin formalism of the many-boson system in a spherical field

Abstract

A new approach is developed to treat the many boson system in a spherical field, such as, for example, the nuclear many surface phonon state. This approach is a generalization of the SU(2) quasi-spin formalism for fermion systems to the noncompact SU(1, 1) group for boson systems. It is shown that the pair creation, pasr destruction, and number operators form a Lie algebra of the SU(1, 1) group and that the wave-function | l n vLM> of n bosons with seniority v constitutes a two-valued projective representation of the SU(1, 1) group. The transformation properties of tensor operators in the SU(1, 1) quasi-spin space are examined in detail; the Wigner-Eckart theorem is established for the matrix elements of a tensor operator the unitary representations of the SU(1, 1) group. A new kind of Wigner coefficient of the SU(1, 1) group is obtained which couples unitary and nonunitary representations to get a unitary representation. Algebraic formulas of this Wigner coefficient are derived; this coefficient is proved to be an analytic continuation of the Clebsch-Gordan coefficient of the SU(2) group. Since the particle number, n, appears only in the projective quantum numbers of this formalism, the n-dependence of any matrix element can be separated entirely into that of the Wigner coefficient alone; thus, n-boson problem reduces directly to a ν-boson problem.