Nonperturbative Loop Quantization of Scalar-Tensor Theories of Gravity

TL;DR

This paper develops a nonperturbative loop quantum gravity framework for scalar-tensor theories of gravity, deriving their Hamiltonian formalism, and constructing quantum operators to describe their dynamics.

Contribution

It introduces a connection dynamical formalism for scalar-tensor theories and extends loop quantum gravity methods to these theories, including the construction of quantum operators.

Findings

Hamiltonian formalism derived from Lagrangian analysis.

Connection formalism with real su(2)-connections established.

Quantum Hamiltonian and master constraint operators constructed.

Abstract

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 $\omega(\phi)$. In the sector of $\omega(\phi)=-3/2$, the feasible theories are restricted and a new primary constraint generating conformal transformations of spacetime is obtained, while in the other sector of $\omega(\phi)\neq-3/2$, 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 non-perturbative 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 loop quantum gravity method is also valid for general scalar-tensor theories.