Solution of Coulomb Path Integral in Momentum Space

The path integral for a point particle in a Coulomb potential is solved in momentum space. The solution permits us to give for the first time a negative answer to an old question of quantum mechanics in curved spaces raised in 1957 by DeWitt, whether the Hamiltonian of a particle in a curved space contains an additional term proportional to the curvature scalar $R$. We show that this would cause experimentally wrong level spacings in the hydrogen atom. Our solution also gives a first experimental confirmation of the correctness of the measure of integration in path integrals in curved space implied by a recently discovered nonholonomic mapping principle.