Reduced Loop Quantization with four Klein-Gordon Scalar Fields as Reference Matter

TL;DR

This paper explores reduced phase space quantization of gravity with four Klein-Gordon scalar fields as reference matter, comparing models that can and cannot be quantized using loop quantum gravity techniques.

Contribution

It introduces a generalized model with four Klein-Gordon fields that can be quantized in loop quantum gravity, and compares it to previous models using Dirac quantization.

Findings

Four Klein-Gordon fields lead to a model not directly quantizable by loop quantum gravity.

A slight generalization of the action enables loop quantum gravity quantization.

Comparison of physical Hamiltonians advances understanding of Dirac vs. reduced phase space quantization.

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 [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.