Matter fields and metric deformations in multidimensional unified theories

Abstract

This paper describes a first study of the effects due to including matter fields in generalized Kaluza-Klein (KK) theories with nonabelian compact gauge group G and nontrivial fibres V K. The approach is based on the first-order Einstein-Cartan (EC) general relativity in (4 + K) dimensions. In the EC theory there are two basic mechanisms which can lead to a spontaneously compactified KK background geometry R 4 × V K: (A) a particular kind of energy-momentum density matter condensate in the quantized ground state, or (B) a particular kind of spin-density matter condensate. If (A) or (B) are operating, the inconsistencies usually found between the KK ansatz and the matter-free EC theory are avoided. Mechanism (B) works only when V K is parallelizable. It is shown that the expansion of matter fields in normal modes on V K implies that one must include deformations of the Yang-Mills (YM) potentials contained in the usual KK metrics. We discuss and characterize one class of such deformations. As a case study, we consider fibres V K ∼ G′, where G′ is a semisimple compact Lie group. We allow for the “maximal” YM gauge group G L ′ × G R ′. We carry out the harmonic analysis for spinor fields and study the mass spectrum and YM quantum numbers of the normal modes. We rely on mechanism (B) to provide a curvature-free connection (“parallelization”) on V K G′ by means of a suitable vertical constant torsion. Minimal YM couplings are of size l L ≡ g , where l is the Planck length and L is the length of the fibre; nonminimal YM couplings are of size L. Nonzero masses are of size L −1. The massless modes are found and discussed. There would be no massless modes if the parallelizing vertical torsion were absent. This torsion also implies the vanishing of the cosmological constant. When the theory is restricted to massless modes, the YM deformations disappear and the dimensional reduction to four dimensions yields an effective YM theory, which is renormalizable at energies far below L −1: the effective theory is obtained by letting L → 0 with g ⪡ 1 fixed and by neglecting all masses of order L −1; g corresponds to the bare YM coupling constant. The surviving effective YM gauge group is G L ′ and the matter fields are in a particular representation of G L ′ × G R ′, corresponding to the zero mass eigenvalue. Explicit examples are discussed for G′ = SU(2) and G′ = SU(3).