Necessity of Additional Unitary-Antisymmetric<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>q</mml:mi></mml:math>-Number Terms in the Commutators of Spatial Current Components

Abstract

It is proved that to maintain consistency of the commutation relations among spatial current components with the Jacobi identity, a Schwinger term antisymmetric with respect to interchange of isotopic (or unitary) indices is needed. The proof is based on the use of the Jacobi identity for triple commutators and of the Lehman-K\"allen expression for the vacuum expectation value of a current commutator. Additional conditions to be satisfied by the new Schwinger term are derived from an analysis of the origin of a discrepancy between the Lee-Dashen-Gell-Mann and the Cabibbo-Radicati sum rules for magnetic moments, and a solution of the discrepancy is proposed.