We provide general sets of functionals for which parabolic phase space Feynman path integrals on the torus Td ≡ (R/2πZ)d have a mathematically rigorous meaning. More exactly, for each functional belonging to each set, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of Td ×Zd to some function of the ending point of position paths and the starting point of momentum paths. Each set of functionals is closed under addition, multiplication, translation, invertible integer linear transformation, and functional differentiation. As a result, we can create a large number of path integrable functionals. Though we must exercise caution when using phase space path integrals, several properties comparable to those of conventional integrals are applicable.
Mathematics Subject Classification (2010) 81S40 · 35S05 · 35K25