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If one replaces the constraints of the Ashtekar-Barbero $SU(2)$ gauge theoryformulation of Euclidean gravity by their $U(1)^3$ version, one arrives at aconsistent model which captures significant structure of its $SU(2)$ version.In particular, it displays a non trivial realisation of the hypersurfacedeformation algebra which makes it an interesting testing ground for(Euclidean) quantum gravity as has been emphasised in a recent series of papersdue to Varadarajan et al. In this paper we consider a reduced phase spaceapproach to this model. This is especially attractive because, after acanonical transformation, the constraints are at most {\it linear} in themomenta. In suitable gauges, it is therefore possible to find a closed andexplicit formula for the physical Hamiltonian which depends only on thephysical observables. Not surprisingly, that physical Hamiltonian isgenerically neither polynomial nor spatially local. The corresponding reducedphase space quantisation can be confronted with the constraint quantisation dueto Varadarajan et al to gain further insights into the quantum realisation ofthe hypersurface deformation algebra.

Reduced Phase Space Approach to the $U(1)^3$ model for Euclidean Quantum Gravity