Heat Kernels and Path Integrals

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The P.I. will study three problems. The first problem involves the study of heat kernel bounds by stochastic methods. The goal is to find estimates which are independent of dimension and hence applicable to infinite dimensional heat flows. In the case of hypoelliptic heat kernels, the P.I. is looking for precise heat kernel upper and lower bounds. The second problem is to study the quasi-invariance and smoothness properties of heat kernel measures on infinite dimensional Lie groups. The P.I. hopes to find examples of infinite dimensional subelliptic operators which are hypoelliptic. The proof of these results will rely on the results of the first problem. The third problem relates to the understanding of Euclidean Feynman path integrals by finite dimensional approximations. In particular, it is proposed to carry this program out for super-symmetric path integrals. These results and their generalizations should have ramifications to the Stolz-Teichner program of geometrizing elliptic cohomologies. In the 1940's, Feynman proposed a beautiful, but highly heuristic, theory for understanding quantum mechanics in terms of the more familiar concepts arising in classical mechanics. Feynman's heuristic ideas have had a significant influence on the mathematical theory of topology. (Topology is the study of the basic shape properties of surfaces and other more general "manifolds.") The P.I. will investigate a number of problems related to Feynman's model in the context of elementary particle physics. Despite being studied for 70 plus years, the mathematical foundations of these types of theories are still not well understood and many interesting questions remain open. For example, physicists believe that "quarks" should be confined. However, with the current state of the models, the theoretical derivation of quark confinement is out of reach. The specific goals of this proposal are: 1) to use probabilistic techniques to describe qualitative and quantitative behavior of the flow of heat in high dimensional spaces, and 2) to develop methods for understanding the "super-symmetric Feynman path integral." It is hoped that the results developed during this grant period will be applicable to the particle physics models and to mathematical topology. This proposal includes a training component as some of the problems will be given to graduate students.