Derivation of theSU(3)×SU(3)Space-Time Local Current Commutators

The commutators of the charge with the current density of vector and axial-vector currents are derived, and restrictions are placed on the Schwinger terms present in the charge-density-current-density equal-time commutators. In order to prove these results, the commutator of the time component of a current with the energy density is derived. The following assumptions are made: (1) Equal-time commutation relations between time components of vector and axial-vector currents satisfy the local $\mathrm{SU}\left(2\right)\mathrm{}$, $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$, or $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ algebra; (2) the transformation properties of the divergence of the axial current are assumed to be known. The second assumption is shown to be necessary as well as sufficient. It is shown that the Schwinger terms involve at most one derivative of a $\delta$ function and have definite symmetry properties. Symmetry properties frequently conjectured for the Schwinger terms are examined in the context of the present investigation, and the consequences of these conjectures are explored. The current-density-current-density equal-time commutation is also studied with the present techniques, and it is found that only very mild restrictions can be imposed in a model-independent fashion.