Realization of Dirac quantization in loop quantum gravity

TL;DR

This paper demonstrates how to implement Dirac quantization within loop quantum gravity using a gravity system coupled to dust, leading to a well-defined physical Hamiltonian and explicit solutions at the quantum level.

Contribution

It provides a complete Dirac quantization scheme in loop quantum gravity for a gravity-dust system, including the construction of the physical Hamiltonian and solutions.

Findings

Scalar constraint promoted to a symmetric operator

Explicit solutions to the quantum scalar constraint derived

Physical observables constructed on the solution space

Abstract

The system of gravity coupled to the non-rotational dust field is studied at both classical and quantum levels. The scalar constraint of the system can be written in the form of a true physical Hamiltonian with respect to the dust time. In the framework of loop quantum gravity, the scalar constraint is promoted to a well-defined operator in a suitable Hilbert space of the coupled system, such that the physical Hamiltonian becomes a symmetric operator. By the deparametrized form, a general expression of the solutions to the quantum scalar constraint is obtained, and the observables on the space of solutions can be constructed. Moreover, the Dirac quantization procedure can be fully carried out in loop quantum gravity by this system.