  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.
