Saturation of Equal-Time Charge Commutation Relations in Perturbation Theory

Abstract

The quark-model equal-time charge commutation relations are studied in a simple perturbation-theoretic model. The summations over the intermediate states are explicitly performed. The zeroth-order results, such as the fact that one-particle intermediate states saturate the chiral $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ commutators at infinite momentum, are reviewed. In second order it is shown that the $[V_0, V_0]=if{V}_0$ and $[A_0, V_0]=if{A}_0$ commutation relations are satisfied provided the renormalized charges are used. It is argued that the relation $[A_0, A_0]=if{V}_0$ is still best handled at infinite momentum. However, higher-mass intermediate states do not vanish in this order if the observable axial charge renormalization is finite.