Modification of Kaluza-Klein theory

A modification of the traditional formulation of Kaluza-Klein theory is proposed in which the internal structure is described by a noncommutative geometry based on a semisimple algebra. The classical theory of Yang-Mills fields and of Dirac fermions is developed in the resulting geometry. The generalized connection is written down which should describe the unification of the Yang-Mills fields with gravity, but the gravitational sector is not described in detail. As an example the mass and charge spectrum of the Yang-Mills-Dirac theory are given for a specific internal structure.