Conditions for exact equivalence of Kaluza-Klein and Yang-Mills theories

TL;DR

This paper establishes the precise conditions under which Kaluza-Klein theories are exactly equivalent to Yang-Mills theories, focusing on the structure of the Lagrangian and the geometric restrictions involved.

Contribution

It explicitly states the conditions for Lagrangian equivalence between Kaluza-Klein and Yang-Mills theories, extending previous work on their geometric and field-theoretic unification.

Findings

Horizontal vector fields are essential in the Kaluza-Klein Lagrangian.

Scalar curvature must be restricted to horizontal sectional curvatures.

All known fields, including fermions, can be represented within this framework.

Abstract

Although it is well known that Kaluza-Klein and Yang-Mills theories define equivalent structures on principal bundles, the general conditions for equivalence of their Lagrangians have not been explicitly stated. In this paper we address the conditions for equivalence. The formulation of these conditions is based on previous work in which the Dirac and Einstein equations were unified in a tetrad formulation of the Kaluza-Klein model. This Kaluza-Klein model is derived from mapping a bispinor field to a set of SL(2,R) x U(1) gauge potentials and a complex scalar field. (A straightforward derivation of this map using Hestenes' tetrad for the spin connection in a Riemannian space-time is included in this paper.) Investigation of this Kaluza-Klein model reveals two general conditions for establishing an exact equivalence between Kaluza-Klein and Yang-Mills theories. The first condition is that only horizontal vector fields occur in the Kaluza-Klein Lagrangian. The second is that the scalar curvature be restricted to a sum over horizontal sectional curvatures. We conclude that all known fields (including fermion fields) can then be represented as components of a Kaluza-Klein metric together with scalar fields.