James Fraser (Durham): “Understanding Ultraviolet Divergences”

Abstract: When physicists first tried to formulate quantum field theories they ran into problems with ultraviolet divergences, integrals which blow up in the region of large momenta. Initially, this was taken as a sign that a radically new theoretical framework was needed to unify quantum theory and relativity. The renormalisation techniques developed in the late 40s showed that these infinities could be systematically eliminated, apparently solving the problem without a revolution. Still, the question of why ultraviolet divergences occur in the first place was unanswered, and the suspicion remained that they point to the break down of quantum field theories at high energies. In this talk, I sketch the development of the causal perturbation theory programme, a lesser-known approach to perturbative quantum field theory which sheds light on this issue. Originating in the work of Stueckelberg and Bogoluibov in the 50s, this approach brings the machinery of distribution theory to bear on the offending terms in the perturbative expansion. The divergences are taken to result from an improper treatment of products of distributions occurring in the perturbation series, which can if care is taken, be given a proper definition. Rather than indicating a physical problem with the high energy behaviour of quantum field theories then, ultraviolet divergences end up being understood as a purely mathematical issue.