9706166 Brydges A major problem in equilibrium statistical mechanics is the calculation of critical exponents. The construction of Euclidean quantum field theory is a closely related problem. The standard tools in theoretical physics used for these calculations are the epsilon expansion and the 1/N expansion, but neither of these are known to be correct methods of calculation. The epsilon expansion is particularly perplexing, because it involves analytic continuation in the dimension of space from 4 to 4 - epsilon. This continuation is made term by term in perturbation theory, but it is not known that the resulting expansion is the perturbation theory of any model. This proposal considers a different way to introduce a parameter epsilon, where there clearly are models for epsilon not = 0, and yet the epsilon parameter plays the same role of moving four dimensional theories off a bifurcation. The objective is to study the properties of expansion in epsilon and also expansion in 1/N. It is believed that methods based on previous work by the author and his collaborators can be extended to prove that these expansions are correct for epsilon small and for 1/N small. One possible outcome of this study is the construction of quantum field theory in three dimensions using the 1/N expansion. This theory should obey all Wightman axioms, but the perturbation theory in the coupling constant will be non-renormalizable. Quantum field theory originated in elementary particle physics, but it is really a mathematical framework for systems with very large numbers of degrees of freedom. Important applications include: predicting the size and shape of polymers, the physics of sound waves and electrons in crystals, the physics of phase transitions. When quantum field theories were first proposed, it was not clear that they were free of internal contradictions. Examples are now known which are free of contradiction, but consistent approximations for prediction remain elusiv e. The purpose of this grant is to investigate two approximations proposed by Wilson and Fisher, to show that they are truly consistent approximations, and to delineate their range of validity.
