Speaker: Ekaterina Nathanson, Department of Mathematics, University of Iowa
ABSTRACT
One of the key elements of Feynman’s formulation of non-relativistic quantum mechanics is a so-called Feynman path integral. It plays an important role in the theory, but appears as a postulate based on intuition rather than a well-defined object. Unfortunately, all attempts to supply Feynman’s theory with rigorous mathematics have not been satisfactory. The difficulty comes from a need to define a measure on infinite dimensional space of paths and to create an integral that would possess all of the properties requested by Feynman. I will introduce a new approach to define the Feynman’s path integral. It is based on the theory developed by P. Muldowney. Muldowney uses the Henstock integration technique and non-absolute integrability of Fresnel integral to obtain a representation of the Feynman's path integral as a functional. We will show how the new approach fixes the main problems in earlier attempts and what role the nonabsolute integrability of Fresnel integrals plays in establishing mathematical rigor supporting Feynman’s intuitive derivations.