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Phase Transitions: Scaling, Universality and Renormalization

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In present-day physics, the renormalization method, as developed by Kenneth G. Wilson, serves as the primary means for constructing the connections between theories at different length scales.This method is rooted in both particle physics and the theory of phase transitions. It was developed to supplement mean field theories like those developed by van der Waals and Maxwell, followed by Landau.

Sharp phase transitions are necessarily connected with singularities in statistical mechanics, which in turn require infinite systems for their realization. (I call this result the extended singularity theorem.) A discussion of this point apparently marked a 1937 meeting in Amsterdam celebrating van der Waals.

Mean field theories neither demand nor employ spatial infinities in their descriptions of phase transitions. Another theory is required that weds a breaking of internal symmetries with a proper description of spatial infinities. The renormalization (semi-)group provides such a wedding. Its nature is described. The major ideas surrounding this point of view are described including especially scaling, universality, and the development of connections among different theories.

This talk is part of the Dirac Lecture series.

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