Relating Theories via Renormalization

The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is the outgrowth of one hundred and fifty years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different be-havior separated by abrupt phase transitions. These qualitative differ-ences seem to be present in experimentally observed condensed-matter systems. However, the “extended singularity theorem ” in statistical mechanics shows that sharp changes can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters. ∗Copies of this paper will be published on arXiv and on the author’s web site: jfi.uchicago.edu\ ∼ leop\. 1 Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales. However, no renormalization gives a fully satisfactory formulation of field theory. The renormalization (semi-)group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson’s theory is the first approach to phase transitions to agree with the extended singularity theorem. Some of the history of the study of these correlations and singu-larities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications, particularly to condensed-matter physics and particle physics, are summarized