Reduced phase space approach to the U(1)3 model for Euclidean quantum gravity

Abstract

If one replaces the constraints of the Ashtekar–Barbero SU(2) gauge theory formulation of Euclidean gravity by their U(1)3 version, one arrives at a consistent model which captures significant structure of its SU(2) version. In particular, it displays a non trivial realisation of the hypersurface deformation algebra which makes it an interesting testing ground for (Euclidean) quantum gravity as has been emphasised in a recent series of papers due to Varadarajan et al. In this paper we consider a reduced phase space approach to this model. This is especially attractive because, after a canonical transformation, the constraints are at most linear in the momenta. In suitable gauges, it is therefore possible to find a closed and explicit formula for the physical Hamiltonian which depends only on the physical observables. The corresponding reduced phase space quantisation can be confronted with the constraint quantisation due to Varadarajan et al to gain further insights into the quantum realisation of the hypersurface deformation algebra.