Scalar Fields in Four-Dimensional CDT

The model of Causal Dynamical Triangulations (CDT) is a non-perturbative and background-independent approach to the quantum theory of gravity. It provides a lattice regularization of the formal gravitational path integral and is manifestly coordinate-free. The lack of a coordinate system is an alluring property from the point of view of General Relativity (GR). Nevertheless, it is often convenient to have coordinates. In this Chapter, we define a coordinate system using classical scalar fields taking values in a target space with a topology matching the toroidal topology of the underlying spacetime manifold. These coordinates are equivalent to harmonic coordinates developed in the context of GR. Using Monte Carlo computer simulations of the four-dimensional CDT model and the introduced coordinates, we examine the properties of the quantum geometryQuantum geometry. Although a single configuration is not physical, its properties can be useful in understanding the details of geometric nature. Visualizations of geometries using scalar fields as coordinates reveal cosmic structures of voids and filaments surprisingly similar to those observed in the real Universe and show clear differences between the individual phases of the model. It is indeed nontrivial that these ideas can be applied to understanding structures that appear in highly irregular and fluctuating geometries. In the second part, we study the backreaction of dynamical matter fields on quantum geometries. The quantum universe has a spatial topology of a three-torus, and the matter fields are multicomponent scalar fields taking values in a torus with circumference δ in each spatial direction. For sufficiently large δ, the scalar field induces a phase transition in which the spacetime topology changes from toroidal to spherical. This discovery may have important implications for quantum universes with nontrivial topologies.