Equal-Time Commutators and Electromagnetic Mass Splittings

Abstract

The relation between electromagnetic mass shifts (or splittings) to order $\alpha$ and equal-time commutators is analyzed. If the invariants in the forward spin-averaged Compton amplitude go as $k^{-2\mathrm{}}$ for $k^2\to \infty$, it is possible to relate the quadratic and logarithmic divergences in $\delta m^2$ to certain Lorentz scalars formed from the commutators. These results are illustrated by models with currents bilinear or linear in canonical fields. The Jost-Lehmann representation is then used to analyze more general behavior. It is shown that the spectral function may generate a higher-rank tensor character of the amplitude than that expected from its kinematic factors. The relation is then more complicated even if the commutators still exist. Finally, detailed criteria for the existence of equal-time commutators are developed.