Nonperturbative loop quantization of scalar-tensor theories of gravity

The Hamiltonian formulation of scalar-tensor theories of gravity is derived from their Lagrangian formulation by Hamiltonian analysis. The Hamiltonian formalism marks off two sectors of the theories by the coupling parameter $\omega (\varphi )$. In the sector of $\omega (\varphi )=-\frac{3}{2}$, the feasible theories are restricted and a new primary constraint generating conformal transformations of spacetime is obtained, while in the other sector of $\omega (\varphi )\ne -\frac{3}{2}$, the canonical structure and constraint algebra of the theories are similar to those of general relativity coupled with a scalar field. By canonical transformations, we further obtain the connection-dynamical formalism of the scalar-tensor theories with real $su(2)$ connections as configuration variables in both sectors. This formalism enables us to extend the scheme of nonperturbative loop quantum gravity to the scalar-tensor theories. The quantum kinematical framework for the scalar-tensor theories is rigorously constructed. Both the Hamiltonian constraint operator and master constraint operator are well defined and proposed to represent quantum dynamics. Thus the loop quantum gravity method is also valid for general scalar-tensor theories.