A New Perspective on Kaluza-Klein Theories

Abstract

By assuming that the geometry of spacetime is uniquely determined by the energy momentum tensor of matter alone, i.e. without any interactions, enables us to construct the Lagrangian from which the metric of higher dimensional spacetime follows. From the geodesic equations that follow it becomes clear that the incorrect mass of elementary particles predicted by Kaluza-Klein theories arises from the assumption that in the absence of gravity the solution to the Einstein field equations reduces to the Minkowski metric. From construction of a consistent theory of $4\mathcal{D}$ electromagnetism, we find that this assumption does not only result in the incorrect mass of elementary particles, but also the incorrect value of the cosmological constant. This suggests that these incorrect predictions, which are often regarded as major flaws of Kaluza-Klein theories, just reflects the inconsistency of some postulates of general relativity and gauge theories. Abandoning this assumption results in modifications of general relativity. We show that the unified description of fundamental interactions naturally incorporates the Higgs mechanism. For non-Abelian gauge fields, we find that the manifold comprising the extra dimensions has to be a group manifold and show that the standard model is realised in 16$\mathcal{D}$ spacetime. We show that charge and spin are the same concept, but what makes them different is that the former follows from symmetry of $4\mathcal{D}$ spacetime while the latter follows from symmetry of the internal space.