Coordinate space modification of Fock's theory. Harmonic tensors in the quantum Coulomb problem

We consider Fock's fundamental theory of the hydrogen atom in momentum space, which allows a realization of the previously predicted rotation group of a three-dimensional (3D) sphere in four-dimensional (4D) space. We then modify Fock's theory and abandon the momentum-space description. To transform and simplify the theory, we use invariant tensor methods of electrostatics in 3D and 4D spaces. We find a coordinate 4D space where the Schr$\ddot {\rm o}$dinger equation becomes the 4D Laplace equation. The transition from harmonic 4D polynomials to the original 3D physical space is algebraic and involves derivatives with respect to a coordinate that is interpreted as time. We obtain a differential equation for eigenfunctions in the momentum space and find its solutions. A concise calculation of the quadratic Stark effect is given. The Schwinger resolvent is derived by the method of harmonic polynomials. Ladder operators are also considered.