The current-algebra scheme, not including equal-time commutators of two space components, is formulated entirely in terms of covariant dispersion relations (i.e., over the laboratory energy for fixed mass) where the number of subtractions is chosen according to the usual Regge picture. To do this, one expresses the equal-time commutators and retarded commutators directly in terms of the covariant dispersion integrals, using an identity between covariant and noncovariant dispersion integrals of Lorentz-invariant distributions which is proved here, by means of the Jost-Lehmann representation, from the locality of the currents. This last result is a generalization of previous works of Schroer and Stichel and of Le Bellac and the author. In the scheme discussed here, the equal-time commutators and retarded commutators are rigorously defined as a consequence of the physical high-energy behavior, contrary to the usual approach where they are only formally written as product of the commutator with $\delta \left(x_0\right)$ and $\theta \left(x_0\right)$, respectively. In fact, we show that the properties usually derived formally (as, e.g., by partial integration) are rigorously true in our scheme. The quantities which in the usual approach are not well defined, and/or are model-dependent—such as the Schwinger term, the seagull term, and the equal-time commutator of the current and the divergence—are given in this approach by the subtraction constants of the dispersion relations introduced. They are shown to exhibit the properties which usually are more or less assumed, or only obtained in models; e.g., we prove that the divergence of the seagull term is equal to the Schwinger term. Only matrix elements averaged over spin and/or relative momenta, with the vacuum matrix element subtracted, are considered. No explicit value of the equal-time commutator is assumed at the beginning, so as to clearly show the respective roles of the internal symmetry group and of the analyticity and high-energy behavior. This paper also completes a previous work of the author on the infinite-momentum limit based on the same approach.

• Received 11 June 1968

DOI:https://doi.org/10.1103/PhysRev.177.2182