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Although it is well known that Kaluza-Klein and Yang-Mills theories defineequivalent structures on principal bundles, the general conditions forequivalence of their Lagrangians have not been explicitly stated. In this paperwe address the conditions for equivalence. The formulation of these conditionsis based on previous work in which the Dirac and Einstein equations wereunified in a tetrad formulation of the Kaluza-Klein model. This Kaluza-Kleinmodel is derived from mapping a bispinor field to a set of SL(2,R) x U(1) gaugepotentials and a complex scalar field. (A straightforward derivation of thismap using Hestenes' tetrad for the spin connection in a Riemannian space-timeis included in this paper.) Investigation of this Kaluza-Klein model revealstwo general conditions for establishing an exact equivalence betweenKaluza-Klein and Yang-Mills theories. The first condition is that onlyhorizontal vector fields occur in the Kaluza-Klein Lagrangian. The second isthat the scalar curvature be restricted to a sum over horizontal sectionalcurvatures. We conclude that all known fields (including fermion fields) canthen be represented as components of a Kaluza-Klein metric together with scalarfields.

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