Phase space Feynman path integrals of parabolic type with smooth functional derivatives

Abstract:  We give two general sets of functionals for which the phase space path integrals of parabolic type have a mathematically rigorous meaning.  More precisely, for any functional belonging to each set, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the final point of position paths and to the initial point of momentum paths.

Each set of functionals is closed under addition, multiplication, translation, invertible linear transformation and functional differentiation.

Therefore, we can produce many functionals which are phase space path integrable.

Furthermore, though we need to pay attention for the uncertain principle, we ensure that the following operations are valid for the phase space path integrals: 
(a) interchange of the order with some integrals, 
(b) interchange of the order with some limits, 
(c) invariance under orthogonal transformations of paths, 
(d) invariance under translations with respect to momentum paths,
(e) integration by parts formula with respect to momentum paths.