Regularization and renormalization

The analysis begun in Part I of the conditions under which the regularization, performed by the adoption of the modified integrals introduced there, acts as a renormalization is completed. The «conditions of the second type» announced in I are formulated and discussed; a quantitative analysis may give results different from the standard requirements for renormalizability: as an example, it is shown that the neutral scalar meson theory is not renormalizable, contrary to current belief. The Lie equations of the renormalization group can be derived without difficulty, and their integrability conditions investigated. Finally, it is shown that using our modified integrals amounts to solving the differential branching equations for kernels under the condition that the solutions belong to a certain well-defined mathematical class Open image in new window . In this way, ultraviolet infinities never appear, and the search for the renormalizability conditions becomes a search for the self-consistency of a theory, which need be made once for all and cannot cause inconvenience in computations. The result is a rigorous mathematical formulation of the renormalized theory, which avoids all mentions of «bare particles», is completely rid of ambiguities and is suited both for practical computations and for the study of fundamental questions. The unphysical splitting of processes into Feynman graphs is avoided; the troubles due to overlaps are shown to have a trivial origin and are altogether eliminated; all vertex-part contributions vanish with this method, at least in electrodynamics, since the cancellation of them against electron self-energy contributions occurs prior to actual computation. These criteria will be applied in a forthcoming work to an exhaustive study of electrodynamics; they are expected to play a relevant role in a search for consistent theories of elementary particles.