Equal-Time Commutators Involving Divergences of Currents

Abstract

A study is made on the equal-time commutators between the time and space components of vector and axial-vector currents and also between the time components and the divergences of these currents. The only assumption made is that these commutators consist of terms proportional to the $\ensuremath{\delta}$ function and at most the first-order space derivatives of the $\ensuremath{\delta}$ function. Lorentz covariance and consistency among the commutators lead to partial determination of the above commutators, including the result that the Schwinger terms are symmetric in the $\mathrm{SU}(3)$ indices. The resulting set of commutators is sufficient to demonstrate that the off-mass-shell strong matrix element, in which two of the bosons are made off the mass shell by means of the divergences of the axial-vector currents, maintains manifest symmetry with respect to the boson variables even after partial integrations are done. Conservation of some of the currents leads to further determination of the commutators. Thus, for example, when $\mathrm{SU}(2)$ symmetry is exact, all the components and also the divergences of these currents have the same $\mathrm{SU}(2)$ transformation properties as the corresponding time components.