This project is mathematical research in functional analysis applicable to theoretical physics. The idea of a Feynman - Dyson algebra, the main object of study, is that it permits the lifting of two non-commuting operators to a time-ordered commutative state, in which one can perform analytic operations and then return to the original non-commutative situation. Professor Gill will extend earlier work to include the general theory of linear and non-linear evolutions, with the weakest possible constraints. A second objective is to extend the connection between his approach and path integrals by finding methods to construct the transition kernel directly from the evolution generator. A final objective is to refine the integration theory to allow for the fact that many path integrals associated with physics and probability theory need not generate a measure on a generalized Wiener space.
