# Smooth functional derivatives in Feynman path integrals by time slicing approximation

by Naoto Kumano-go (GFMUL / Dept. Mathematics, Kogakuin University, Japan)

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid.

- N. Kumano-go, "Feynman path integrals as analysis on path space by time slicing approximation".
*Bull. Sci. Math***128**, issue 3, pp.197-251 (2004). - D. Fujiwara and N. Kumano-go, "Smooth functional derivatives in Feynman path integrals by time slicing approximation".
*Bull. Sci. Math.***129**, issue 1, pp 57-79 (2005).