  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.
