R. Stora: Källén’s Constant M

Abstract

In his Handbook Article [1] G. Källén states the asymptotic condition in quantum electrodynamics where the arrow stands for the “weak” asymptotic limit a la LSZ and M is a finite computable constant expressible in terms of the Källén-Lehmann weight function Π(a) for the photon two point function.

All the time, there were conflicting points of view between Källén and LSZ, the former insisting on the canonical formulation (which is sick for coupled fields) and the latter who insist on the asymptotic condition.

Amusingly enough followers of LSZ in the framework of LSZ assume [2] which turns out to be inconsistent with perturbative renormalizability.

Now Källén’s asymptotic condition is perfectly compatible with LSZ since the free vector field A ⁱⁿ _μ (x) is associated with a reducible representation of the Poincaré group.

The best way to sort this out is to look at massive QED (add a mass term to Källén’s Lagrangian in the Handbook Article, see below).

The constant M produces in the canonical commutation relations anomalous additional gradient terms, later called Schwinger terms because Schwinger showed [3] generally that such terms were required by Lorentz covariance, in even more general contexts (including gauge theories). In connection with M, Källén quotes Goto and Imamura.