The role of the internal metric in generalized Kaluza–Klein theories

Abstract

The dimensional reduction of a Weyl space WN of N=4+n dimensions to a principal fiber bundle P̃(W4,G̃n) over a four-dimensional space–time W4 with structural group G̃n of dimension n arising from the existence of n conformal Killing vector fields of the original N-metric is studied. The framework of a Weyl geometry is adopted in order to investigate conformal rescalings of the metric on the bundle P̃(W4,G̃n) obtained. The Weyl symmetry is then, finally, broken again by choosing a particular Weyl gauge in which the internal, i.e., fiber metric, is of constant Cartan–Killing form. This choice of gauge, yielding a Riemannian theory, forces the internal metric to play no dynamical role in the theory, as is usually assumed to be the case in non-Abelian gauge theories. However, this gauge induces a conformal transformation of the metric in the space–time base of P̃ compared to the space–time metric obtained by the ordinary Kaluza–Klein reduction of a Riemannian space VN. The role of vector torsion in this dimensional reduction by isometries and scale transformations is also investigated.