Dynamics of a scalar field in a polymer-like representation

In the last 20 years, loop quantum gravity, a background- independent approach to unify general relativity and quantum mechanics, has been widely investigated. We consider the quantum dynamics of a real massless scalar field coupled to gravity in this framework. A Hamiltonian operator for the scalar field can be well defined in the coupled diffeomorphism-invariant Hilbert space, which is both self-adjoint and positive. On the other hand, the Hamiltonian constraint operator for the scalar field coupled to gravity can be well defined in the coupled kinematical Hilbert space. There are one-parameter ambiguities due to scalar field in the construction of both operators. The results heighten our confidence that there is no divergence within this background- independent and diffeomorphism-invariant quantization approach of matter coupled to gravity. Moreover, to avoid possible quantum anomaly, the master constraint programme can be carried out in this coupled system by employing a self-adjoint master constraint operator on the diffectmorphism-invariant Hilbert space.