GENERAL RENORMALIZATION THEORY

Abstract

We saw in the previous chapter that calculations in quantum electrodynamics involving one-loop graphs yield divergent integrals over momentum space, but that these infinities cancel when we express all parameters of the theory in terms of ‘renormalized’ quantities, such as the masses and charges that are actually measured. In 1949 Dyson sketched a proof that this cancellation would take place to all orders in quantum electrodynamics. It was immediately apparent (and will be shown here in Sections 12.1 and 12.2) that Dyson's arguments apply to a larger class of theories with finite numbers of relatively simple interactions, the so-called renormalizable theories, of which quantum electrodynamics is just one simple example. For some years it was widely thought that any sensible physical theory would have to take the form of a renormalizable quantum field theory. The requirement of renormalizability played a crucial role in the development of the modern ‘standard model’ of weak, electromagnetic, and strong interactions. But as we shall see here, the cancellation of ultraviolet divergences does not really depend on renormalizability; as long as we include every one of the infinite number of interactions allowed by symmetries, the so-called non-renormalizable theories are actually just as renormalizable as renormalizable theories. It is generally believed today that the realistic theories that we use to describe physics at accessible energies are what are known as ‘effective field theories.’ As discussed in Section 12.3, these are low-energy approximations to a more fundamental theory that may not be a field theory at all.