A Tutorial Course in Constructive Field Theory

Abstract

Prior to 1970, the major focus of constructive field theory was the mathematical framework required to establish the existence of quantum fields [1,2]. Since 1970, the emphasis has gradually shifted, first toward verifying physical properties of the known models, and more recently toward bringing constructive field theory closer to the mainstream of physics [3]. In fact, by 1973 it had become more or less clear that the mathematical framework developed to give the first examples in d = 2, 3 space-time dimensions would be adequate to study d = 4. However, it was also clear that to solve the ultraviolet problem in d = 4, it would be necessary to incorporate into mathematical physics a deeper physical understanding of the questions being studied. In particular ideas of scaling and of critical behavior may be useful to select and analyze a suitable nontrivial critical point as the first step in dealing with the d = 4 ultraviolet problem. An infinite scaling transformation connects the problem of removing the ultraviolet cutoff with the problem of existence of scaling behavior at the critical point.