Generally covariant geometro-stochastic quantum gravity

Abstract

A geometro-stochastic formulation of quantum gravity is presented which is covariant with respect to Poincare gauge transformations as well as to internal gauges related to local coordinate changes in the base manifold of quantum-gravitational fibre bundles. In this formulation the graviton is a massless geometro-stochastic exciton of spin 2 displaying a stringlike mean stochastic extension of the order of the Planck length. A quantum pregeometry is obtained by constructing from the states of such gravitons Fock-type fibres over the general linear frame bundleGL(M) of a base differential manifoldM. These fibres are then soldered toM by means of soldering forms supplied by various pseudo-Riemannian/Lorentzian geometries in suitable regions of that manifold. Graviton propagation is governed by a second-quantized connection∇ invariant under external as well as internal gauge transformations, which in combination give rise to a quantum-gravitational gauge group. The parallel transport of related gauge frames involves geometrized versions of Faddeev-Popov fields derived from Maurer-Cartan structural equations for that gauge group in a supergauge context. The quantum-gravitational connection∇ gives rise to a geometro-stochastic propagation of quantum-gravitational states which is interrelated to the evolution of various Lorentzian geometries in the base manifoldM. For each set of suitable initial conditions this mutual interaction reduces the quantum pregeometry overGL(M) to a quantum space-time represented by gravitational quantum (super)fibre bundles over supermanifolds based on mean Lorentzian geometries (M,ḡ) inM.