Racah's method for general subalgebra chains: Coupling coefficients of SO(5) in canonical and physical bases

TL;DR

This paper extends Racah's method to systematically compute reduced coupling coefficients for subalgebra chains, exemplified by SO(5)>SO(4), enabling transformations between canonical and physical bases.

Contribution

It formulates a general approach for calculating coupling coefficients for any subalgebra chain using Racah's method, including transformation procedures.

Findings

Successfully applied to SO(5)>SO(4) coupling coefficients

Provides a complete solution for irreps of SO(5) with respect to subalgebra chains

Enables basis transformations between canonical and physical bases

Abstract

It is shown that the method of infinitesimal generators ("Racah's method") can be broadly and systematically formulated as a method applicable to the calculation of reduced coupling coefficients for a generic subalgebra chain G>H, provided the reduced matrix elements of the generators of G and the recoupling coefficients of H are known. The calculation of SO(5)>SO(4) reduced coupling coefficients is considered as an example, and a procedure for transformation of reduced coupling coefficients between canonical and physical subalegebra chains is presented. The problem of calculating coupling coefficients for generic irreps of SO(5), reduced with respect to any of its subalgebra chains, is completely resolved by this approach.