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

TL;DR

This paper offers a topological and geometrical interpretation of the Dirac equation, linking internal symmetries and gauge invariance to the properties of a specific 4-manifold and its automorphism group.

Contribution

It introduces a novel topological framework that explains gauge invariance and internal symmetries as natural geometric consequences of space-time manifold properties.

Findings

Dirac equation is a group-theoretical relation for a 4-manifold

Internal symmetry corresponds to automorphisms of a covering space

Gauge invariance arises from the geometry of space-time deviations

Abstract

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.