Exact solution of a relativistic quantum field theory

Abstract

The relativistic quantum field theory with derivative coupling £′ = λΨ̅γ^μΨ^∂_μΦ between a Fermi field of spin ⅔ and a neutral Bose field of spin zero is solved exactly by an expplicit solution of the Tomonaga-Schwinger equation for Dyson's U-operator. It is shown that the theory can be renormalised by means of a single divergent renormalization constant. The most remarkable feature of the theory is the occurrence of an essential singularity on the light cone in the fermion Green's function which makes it a non-tempered distribution, as a consequence of which the theory is not amenable to analytic continuation techniques or to Källén-Lehmann spectral representation technique. The S-matrix works out to be unity if the asymptotic states are constructed by the "adiabatic switching off" procedure. It is, however, different from unity, giving rise to finite scattering, if the asymptotic states are constructed by the LSZ procedure.