An essay on the dynamics and statistics of critical field theories

1. Introduction. This article is even more provisional and premature than its title suggests. Its author is entitled to no pretensions. Field theories and especially conformally invariant field theories are becoming familiar to mathematicians, largely because of their influence on the study of Lie algebras and above all on topology. Nonetheless, in spite of the progress in constructive quantum field theory during recent decades, many analytic problems, especially the existence of the scaling limit, are given short shrift. These problems are difficult and fascinating and merit more attention. Some important basic concepts, due to physicists, are available but, so far as I can see, need substantial mathematical elaboration.