Mathematics
Professor
Vancouver, British Columbia
db5d@math.ubc.ca
Office:
(604) 822-3620

A large part of theoretical physics is built around the “functional integral” formulation of quantum field theory. These functional integrals are defined in the sense of formal power series (renormalised perturbation theory). It is commonly thought by mathematicians, that no precise definition ...

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A large part of theoretical physics is built around the “functional integral” formulation of quantum field theory. These functional integrals are defined in the sense of formal power series (renormalised perturbation theory). It is commonly thought by mathematicians, that no precise definition that is useful for rigorous analysis is within sight. However the renormalization group (RG), as pioneered by Ken Wilson (Nobel prize in Physics, 1982), provides a good roadmap for defining functional integrals and studying the remarkable range of phenomena contained within them, in particular, renormalisation, scaling limits and the phase transitions of statistical mechanics. In these cases the theory of Gaussian measures on spaces of functions or distributions is the starting point and it is a good starting point for any problem which is "close to real Gaussian". This allows a surprising large range of applications. My interests in recent years have been in applications to self-avoiding walk in four dimensions. It is possible to express self-avoiding walk and other systems in terms of nearly Gaussian integrals with supersymmetry which can be studied by RG. My colleague Joel Feldman is also using RG but more in the context of condensed matter physics. His work leads to nearly Gaussian integrals with complex densities. These integrals are not so well understood but they appear in any problem where time has a direction, for example random walk in a random environment which is not symmetric.

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