Coupling and Recoupling Coefficients for Wigner's U(4) Supermultiplet Symmetry

TL;DR

This paper introduces a new, efficient method for calculating Wigner coupling and Racah recoupling coefficients for U(4) symmetry, applicable to nuclear physics and other systems with this symmetry group.

Contribution

A novel procedure that simplifies and accelerates the computation of U(4) coupling coefficients, avoiding binomial coefficients and series summations.

Findings

Provides a fast, accurate computational method for U(4) coefficients.

Applicable to systems with U(4) symmetry, including nuclear structure.

Enables development of a C++ library for coefficient calculations.

Abstract

A novel procedure for evaluating Wigner coupling coefficients and Racah recoupling coefficients for U(4) in two group-subgroup chains is presented. The canonical U(4)->U(3)->U(2)->U(1) coupling and recoupling coefficients are applicable to any system that possesses U(4) symmetry, while the physical U(4)->SU_S(2)xSU_T(2) coupling coefficients are more specific to nuclear structure studies that utilize Wigner's Supermultiplet Symmetry concept. The procedure that is proposed sidesteps the use of binomial coefficients and alternating sum series, and consequently enables fast and accurate computation of any and all U(4)-underpinned features. The inner multiplicity of a (S,T) pair within a single U(4) irreducible representation is obtained from the dimension of the null space of the SU(2) raising generators; while the resolution for the outer multiplicity follows from the work of Alex et al. on U(N). It is anticipated that a C++ library will ultimately be available for determining generic coupling and recoupling coefficients associated with both the \textit{canonical} and the \textit{physical} group-subgroup chains of U(4).