Since the first Erice course on Constructive field theory, in 1973, considerable progresses have been done in understanding the exact mathematical meaning of field theory. In particular the renormalization problem is now fully understood as well at the perturbative level than at the constructive one, at least for the so-called asymptotically free theories. It is now a fact that there are no big differences between a super renormalizable theory and an asymptotically free one and I will briefly argue on this point. To construct a field theory one introduces cutoff: space and momentum cutoff; the aim of the games is then to prove the existence of some object of interest, for instance the generating functional, when one removes these cutoff. The removal of the space cutoff is done through an expansion, the Cluster Expansion, which was proposed in 1973 by Glimm, Jaffe and Spencer . No big changes have been introduced since then. The general mechanism which control the removal of the momentum cutoff is the convergence of diagrams in perturbation theory. This idea was already understood by many mathematicians or physicists in the beginning of the seventies.
KeywordsCluster Expansion External Line Internal Line Fermi Field Fermion Propagator
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