Lesson from a soluble model of quantum electrodynamics

We study the model which results from quantum electrodynamics if all photons are replaced by their longitudinal parts -k1/4 k1/4(k2)2. The model is formulated in terms of a Lagrangian which features the higher-order or "grandfather" potential S which becomes a local field after quantization and from which the vector potential is derived by A1/4=S. The grandfather potential provides a convenient control over ultraviolet and infrared divergences, for its free propagator -(12)i(k1/4)[k1/4ln(-a2k2-iu)(-k2-iu)2] already contains the parameter a-1 with dimensions of mass which otherwise only appears in the theory, after regularization, as a renormalization mass. It produces much needed states, for we prove that there is no charged state satisfying the Gupta-Bleuler condition in DFO(A,), the closure of the space obtained from the vacuum by applying polynomials in A and (the charged spinor field), but such states are easily found in D(S,). As the lesson from the model, a new Lagrangian for quantum electrodynamics is exhibited, which features the grandfather or Hertz potential 1/41/2=- 1/21/4, from which the vector potential is derived by A1/4=1/4 1/2 1/3, and which is expected to yield corresponding advantages.