BUNDLE METRICS IN KALUZA-KLEIN THEORIES AND THE CONSISTENCY PROBLEM

In this work, bundle metrics which allow a unified description of gravity and the other interactions, a la Kaluza-Klein theory, are considered. After recalling some basic notions of fiber bundles and Riemannian geometry, it is shown that the transition functions which give the global structure of a fiber bundle play an important role in the construction of the metric. In particular, they allow the metric on the vertical subspaces to vary from fiber to fiber thus giving rise to Brans-Dicke-like scalars. The invariance properties of these metrics, under general gauge transformations are also discussed. It is found that the usual requirement of a gauge-invariant metric leads to constraints on the gauge fields. To avoid them, it is shown that the metric on the typical fiber should instead be covariant under the action of the structure group. Furthermore, to implement the full isometry group of the internal space, as the gauge group, a fiber bundle which is associated to a 'spliced' principal bundle is constructed. The horizontal vector fields which show the presence of all the gauge potentials are explicitly evaluated. The consistency problem, inherent in the usual Kaluza-Klein scheme is also discussed. In particular, a consistency condition which requires the gauge group to be restricted to a subgroup of the full isometry group is obtained. The problem is also considered in the context of fiber bundles and the importance of a global isometry that is transitive on the fibers is pointed out. This ls explicitly demonstrated for the principal bundle. However, for the case of a homogeneous bundle with G/H as the typical fiber, it is shown that the right action which is canonically given on the principal bundle is not well defined. Here, the problem is resolved by requiring the automorphisms to be the isometries. Finally the isometries are analysed from the geometrical point of view by using the Bochner-Yano type formula. In particular, it is shown by generalizing the Bochner-Yano formula, that the constraints that arise in the 5-dimensional Kaluza-Klein scheme can be averted by introducing torsion.

Author dc:creator

• KULDIP SINGH