The isospin structure

The spin structure is irrelevant here, so let us simply denote by Jj a set of three Hermitian currents (j = 1,2,3) which transform like an isospin triplet under isotopic spin rotations. Thus from (1.2.18), since for SU 2) the are just the antisymmetric Cjki, we have 

Now consider the commutator of J3 with the isotopic spin raising operator T+ = Ti -I- iT2 which, in the Condon-Shortley phase convention (Condon and Shortley, 1963) has the following effect  

An immediate application connects part of the neutron /3-decay matrix element to the electromagnetic form factors of protons and neutrons. Taking for J+ the vector part of the hadronic weak current and using (1.2.11), (1.3.17) yields 

This relation, and many similar ones, are basically just applications of the Wigner-Eckart theorem (see, for example, Merzbacher, 1962), and can be obtained directly from it. 

Further useful relations follow from making a rotation of tt about the 1 axis in isospace. This has the effect of making J2 — — J2 and J3 — — J3. In particular therefore 

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