State vectors in higher-dimensional gravity with kinematic quantum numbers of quarks and leptons

Abstract

We examine the question of whether fundamental fermions can be regarded as topological solitons (geons) of a theory of pure quantum gravity in n + 4 dimensions. In particular, we consider the quantum gravitational field for ( n + 4)-dimensional space-times which have asymptotic Kaluza-Klein behavior: asymptotic topology of the form R 4 × G/H and a metric that is asymptotically the direct sum of the Minkowski metric on R 4 and the natural symmetric metric on G/H. We argue that if there is a nonsingular theory of pure quantum gravity (a theory that, below Planck energy, involves only the space-time metric), then there are stable ground states with the kinematical quantum numbers of fundamental fermions: that is, states which have spin 1 2 and belong to the fundamental representation of G . Multi-valued representations of the asymptotic symmetry group can arise from diffeomorphisms (diffeos) of the ( n + 3)-dimensional space that are not isotopic to the identity. In a path integral approach, the creation of fundamental fermions (topological geons) requires a space-time in which the topology of space-like hypersurfaces changes and in which the relevant diffeos on the final ( n + 3)-space (containing the geons) are not extendible to diffeos of the ( n + 4)-space-time. We prove that such pair-creation geometries occur in a lorentzian as well as euclidean framework and that the diffeos are not extendible. Although zero-mass modes of finite-size fermions need not belong to a real representation of the internal group, it is not difficult to prove that, independent of the fermion size, one cannot avoid mirror fermions. There is, however, an interesting loophole: A nonzero vacuum expectation value of a Higgs-like vector constructed from the metric can break chiral invariance without changing the lowest-order Einstein-Yang-Mills form of the gravitational lagrangian.