Equal-Time Commutation Relations of the Isovector Current Densities

The one-particle matrix elements of the local equal-time commutators of the isovector currents are derived by applying the Dyson representation to the causal parts of the invariant absorptive parts. Assuming the equal-time commutation relation between the total charge and charge density and also assuming certain asymptotic limits for the Dyson spectral function, we can generate the local equal-time commutation relations between various components of the currents. It is shown that, for a reasonable asymptotic behavior of the spectral function, the charge-space component has no antisymmetric (in isospin) Schwinger terms, but involves two possible q-numer symmetric Schwinger terms, and that the space-space components can have no antisymmetric Schwinger terms. It is pointed out that as the asymptotic behavior becomes worse, we can no longer define the equal-time commutation relations uniquely.