Equal-Time and Null-Plane Commutators of Conserved Currents on the Light Cone

Abstract

The usual manifestly conserved expansions of current densities on the light cone do not lead to the correct commutation relations at equal times or on a lightlike plane. Therefore a piece must be added which is compatible with current conservation only for a specific light-cone singularity, namely, ${({z}^{2}\ensuremath{-}i\ensuremath{\epsilon}{z}_{0})}^{\ensuremath{-}1}$, which obeys ${\ensuremath{\square}}^{(z)}{({z}^{2}\ensuremath{-}i\ensuremath{\epsilon}{z}_{0})}^{\ensuremath{-}1}=0$. A close relation exists between such a nonmanifestly conserved addition to the operator expansion and the well-known phenomenon of a noncausal part in the invariant function of the matrix element between one-particle states.