Loop quantum gravity coupled to a scalar field

We consider the model of gravity coupled to the Klein-Gordon time field. We do not deparametrize the theory using the scalar field before quantization, but quantize all degrees of freedom. Several new results for loop quantum gravity are obtained: (i) a Hilbert space for the gravity-matter system and a nonstandard representation of the scalar field thereon is constructed, (ii) a new operator for the scalar constraint of the coupled system is defined and investigated, (iii) methods for solving the constraint are developed. Commutators of the new quantum constraint operators correspond to the quantization of the Poisson bracket. This, however, poses problems for finding solutions. Hence the states we consider-and perhaps the whole setup-still needs some improvement. As a side result we describe a representation of the gravitational degrees of freedom in which the flux is diagonal. This representation is related to the BF theory vacuum of Dittrich and Geiller.