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 tosolve 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.KeywordsIsing ModelScaling LimitCluster ExpansionSingle Phase RegionAsymptotic CompletenessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.