Reduced loop quantization with four Klein–Gordon scalar fields as reference matter

Abstract

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 Giesel and Thiemann (2010 Class. Quantum Grav. 27 175009), where dust scalar fields are used. We compare our results to an earlier model by Domagala et al (2010 Phys. Rev. D 82 104038) 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 (2010 Phys. Rev. D 82 104038) 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.