O. A. Ol'khov | Searcharxiv

arXiv2006

Geometrization of some quantum mechanics formalism

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.

hep-th↗

arXiv2005

Topological Interpretation of the Dirac Equation and Geometrical Foundations of the Gauge Invariance Idea

Soon after the Yang-Mills work, the gauge invariance became one of the basic principles in the elementary particles theory. The gauge invariance idea is that Lagrangian has to be invariant not only with respect to the coordinates transformations corresponding to the Lorentz group (external symmetry). It is supposed that Lagrangian has also to be invariant with respect to wave functions (not coordinates) transformations corresponding to some additional groups (so-called "internal symmetry groups"). Useful though this idea is, there is no satisfactory understanding of the above additional symmetries origin, and the gauge invariance is considered as an auxiliary theoretical hypotheses. We propose a new, topological interpretation of the basic quantum mechanical equation -- the Dirac equation, and within the framework of this interpretation the notions of internal symmetry and gauge invariance bear a simple geometrical meaning and are natural consequences of the basic principles of the proposed geometrical description. According to this interpretation the Dirac equation proves to be the group-theoretical relation that accounts for the symmetry properties of a specific 4-manifold -- localized microscopic deviation of the space-time geometry from the Euclidean one. This manifold covering space plays the role of the internal space, and the covering space automorphism group plays the role of the gauge group.

hep-th↗