Quantum Field Theories at Small Distances

Abstract

The behaviour of the propagation functions of quantized field theories at small distances is investigated in connection with the problem of consistency of the perturbation theoretical renormalization scheme. An attempt is made to adapt the conventional Hamiltonian and S-matrix formalism to the renormalization concept in such a way that a finite theory of interacting physical (dressed) particles results which as a whole and at each step of approximation satisfies the axioms of the general structure theory of quantized fields, notably causality and positive definiteness. If the physical particles of a field theory with point interaction are extended objects owing to the extension of the cloud of virtual quanta in the physical states, then the theory admits of a finite formulation with the extent of the cloud introducing a natural, coupling-dependent built-in cutoff. The limiting case of a pointlike cloud corresponds to physical particles which because of their strong self-interaction do not interact with one another. This case, corresponding to the zero coupling limit, represents the most singular one given by the theory. An increase of interaction or coupling implies an increase of the size of the cloud and a corresponding charge spread. Conversely, there cannot be interaction if there is no extended cloud and charge spread. Cloud and charge structure come in via causal form factors related to vertex parts and electromagnetic form factors, which however a perturbation theoretical scheme cannot take into account.