Gauge theories in Epstein-Glaser approach to renormalization theory

The traditional approach to renormalization theory starts from Bogolyubov axioms imposed on the S-matrix (or equivalently on the chronological products) and translates them into axioms on the Feynman amplitudes (the so-called Hepp axioms). Then one tries to find explicit solutions of these axioms using some regularization procedure and extracting in a consistent way the ultra-violet infinities. Already for a real scalar field this task is not very easy, but the theory becomes extremely complicated when one considers systems with gauge invariance. The rigorous analysis performed by Becchi, Rouet and Stora shows the complexity of the theory. In recent years, a new way to consider renormalization theory was advocated by prof. G. Scharf starting from the analysis of Epstein and Glaser. In this approach one also starts from Bogolyubov axioms on the chronological products, but tries to find out solutions in a purely recursive way using the support properties of various distributions appearing in the theory and a procedure called distribution splitting. More important, in this approach one can bring new light on the problem of gauge invariance, i.e., one can show that gauge theories are the only candidate for a consistent perturbative theory involving spin 1 Bosons. From the physical point of view, one should proceed in strict accordance with the principles of the second quantization of Fock-Cook as follows: one starts with an one-particle Hilbert space of the spin 1 Boson h which is usually some representation of the Poincare group. Then one chooses a statistics, which in this case should be Bose-Einstein and considers as a physical space the associated symmetric Fock space F+(h). On the other hand, it seems to be more convenient to work with non-physical particles i.e. the ghosts. In this way, the description of the theory takes place in a auxiliary Hilbert space H in which the existence of a supercharge Q is assumed. We have showed that the gauge invariance principle is a natural consequence of the description of spin-one particles in a factor Hilbert space: gauge invariance expresses the possibility of factoring the S-matrix to the physical space, which is constructed using according to the cohomologic-type formula: Hphys = Ker(Q)/Im(Q). This equality is not obvious, and in a previous work, we proved it for spin 1 massless Bosons, spin 1 massive Bosons and spin 2 massless Bosons, respectively. The first two cases are relevant for the construction of the Standard Model and the last one for the quantization of gravity. The obstructions to such a factorisation process are the well-known anomalies. As a byproduct, we have simpler analysis of the unitarity of the S-matrix. Earlier we have extended a result of Aste and Scharf concerning the uniqueness of the non-Abelian gauge theory describing the consistent interaction of r null-mass Bosons of spin 1. Also in previous paper we have generalized in the same way the case when the spin-one Bosons of non-null mass are considered. This case was previously studied by Aste, Duetsch and Scharf, for the concrete case of the electroweak interaction i.e. when the gauge group is exactly SU(2) x U(1). In a previous paper we extended the whole analysis to the case when Dirac Fermions are present and derived a modified (and weaker) obstruction condition of the axial anomaly type. The main problem in this approach is the fact that the factorisation of the S-matrix can be implemented only in the adiabatic limit so one has to solve simultaneously the ultra-violet and the infra-red problems. There is no easy solution to this problem up to now, so to avoid the infrared limit it is more convenient to consider the factorisation condition for the S-matrix as a heuristic idea and replace it by an 'infinitesimal' version which avoids completely the adiabatic limit. (author)