  We perform a reduced phase space quantization of gravity using four
Klein-Gordon scalar fields as reference matter as an alternative to the
Brown-Kuchar dust model in [1], where dust scalar fields are used. We compare
our results to an earlier model by Domagala et. al. [2] where only one
Klein-Gordon scalar field is considered as reference matter for the Hamiltonian
constraint but the spatial diffeomorphism constraints are quantized using Dirac
quantization. As a result we find that the choice of four conventional
Klein-Gordon scalar fields as reference matter leads to a reduced dynamical
model that cannot be quantized using loop quantum gravity techniques. However,
we further discuss a slight generalization of the action for the four
Klein-Gordon scalar fields and show that this leads to a model which can be
quantized in the framework of loop quantum gravity. By comparison of the
physical Hamiltonian operators obtained from the model by Domagala et. al. [2]
and the one introduced in this work we are able to make a first step towards
comparing Dirac and reduced phase space quantization in the context of the
spatial diffeomorphism constraints.
