The polymer quantization in LQG: massless scalar field

The polymer quantization of matter fields is a diffeomorphism invariant framework compatible with Loop Quantum Gravity. Whereas studied by itself, it is not explic- itly used in the known completely quantizable models of matter coupled to LQG. In the current paper we apply the polymer quantization to the model of massless scalar field coupled to LQG. We show that the polymer Hilbert space of the field degrees of freedom times the LQG Hilbert space of the geometry degrees of freedom admit the quantum constraints of GR and accommodate their explicit solutions. In this way the quantization can be completed. That explicit way of solving the quantum constraints suggests interesting new ideas.