Deparametrization and Quantization of Scalar-Tensor Gravity and Its Cosmological Model

TL;DR

This work develops a nonperturbative quantization of scalar-tensor gravity by deparametrizing the Hamiltonian constraint with the scalar field $\phi$ as internal time and applying loop quantum gravity techniques. It constructs a connection-dynamical formulation, implements polymer quantization for both geometry and the scalar field, and derives a Schrödinger-like evolution in $\phi$ for the deparametrized theory. In Brans-Dicke cosmology, the quantum dynamics yield discrete time evolution and a quantum bounce that resolves the classical big bang singularity. The results extend LQG methods to non-GR theories, offering a concrete framework for quantum cosmology in scalar-tensor gravity and highlighting open issues such as full-theory solutions and conformal constraints. Overall, the paper demonstrates the viability and consequences of deparametrization-based loop quantization for scalar-tensor gravity and its cosmological applications, while pointing to future refinements and tests.

Abstract

The degree of freedom of the scalar field in scalar-tensor gravity is employed as 'time' to deparametrize the Hamiltonian constraint of the theory. The deparametrized system is then non-perturbatively quantized by the approach of loop quantum gravity. This results in a discrete time evolution of the physical states with respect to the gravitational degree of freedom in the quantum theory. In the corresponding Brans-Dicke cosmological model, the physical solutions to the quantum Hamiltonian constraint is obtained in the light of the deparametrization. The quantum dynamics indicates that the classical big bang singularity is replaced by a quantum bounce.

Figures (9)

• Figure 1: The plot of $|e_k^{s}(v)|^2$ for an symmetric eigenstate $e_k^{s}(v)$ of operator $\hat{G}$ with the parameter is chosen as $k=100$: it shows that this eigenstate is not normalized.
• Figure 2: The plot of the local graph of $|e_k^{s}(v)|$ for an symmetric eigenstate $e_k^{s}(v)$ of operator $\hat{G}$ with the parameter $k=100$: the values of $e_k^{s}(v)$ vary significantly between two adjacent lattice points, and hence it is not possible to approximate this eigenstate with a smooth analytical function.
• Figure 3: The plot of $|e_k^{s}(v)|$ for an symmetric eigenstate $e_k^{s}(v)$ of operator $\hat{G}$ with into two sectors and with the parameter $k=100$: it shows that the eigenstate $e_k^{s}(v)$ tends to two distinct differentiable functions at large $v$.
• Figure 4: The real part of the symmetric eigenstate $e_k^{s}(v)$ in the sector supported on $v=4n$ (denoted by blue point) compare with the real part of its asymptotically approximate function (denoted by red solid line) with the parameter $k=100$: this demonstrates a very good agreement between the two functions at large $v$.
• Figure 5: The image part of the symmetric eigenstate $e_k^{s}(v)$ in the sector supported on $v=4n$ (denoted by blue point) compare with the image part of its asymptotically approximate function (denoted by yellow solid line) with the parameter $k=100$: this demonstrates a very good agreement between the two functions at large $v$.