Geometrization of some quantum mechanics formalism

Abstract

There were many attempts to geometrize electromagnetic field and find out new interpretation for quantum mechanics formalism. The distinctive feature of this work is that it combines geometrization of electromagnetic field and geometrization of material field within the unique topological idea. According to the suggested topological interpretation, the Dirac equations for a free particle and for a hydrogen atom prove to be the group--theoretical relations that account for the symmetry properties of localized microscopic deviations of the space--time geometry from the pseudoeuclidean one (closed topological 4-manifolds). These equations happen to be written in universal covering spaces of the above manifolds. It is shown that "long derivatives" in Dirac equation for a hydrogen atom can be considered as covariant derivatives of spinors in the Weyl noneuclidean 4-space and that electromagnetic potentials can be considered as connectivities in this space. The gauge invariance of electromagnetic field proves to be a natural consequence of the basic principles of the proposed geometrical interpretation. Within the suggested concept, atoms have no inside any point-like particles (electrons) and this can give an opportunity to overcome the difficulties of atomic physics connected with the many-body problem.