Pseudo-Cartesian coordinates in a model of Causal Dynamical Triangulations

Abstract

Causal Dynamical Triangulations is a non-perturbative quantum gravity model, defined with a lattice cut-off. The model can be viewed as defined with a proper time but with no reference to any three-dimensional spatial background geometry. It has four phases, depending on the parameters (the coupling constants) of the model. The particularly interesting behavior is observed in the so-called de Sitter phase, where the spatial three-volume distribution as a function of proper time has a semi-classical behavior which can be obtained from an effective minisuperspace action. In the case of the three-sphere spatial topology, it has been difficult to extend the effective semi-classical description in terms of proper time and spatial three-volume to include genuine spatial coordinates, partially because of the background independence inherent in the model. However, if the spatial topology is that of a three-torus, it is possible to define a number of new observables that might serve as spatial coordinates as well as new observables related to the winding numbers of the three-dimensional torus. The present paper outlines how to define the observables, and how they can be used in numerical simulations of the model.

Scalar fields in causal dynamical triangulations

“…frames and to define coordinates by geodesic distances from them [21]. Such a proposal has some drawbacks as the coordinates are in general dependent on the position of nonphysical boundaries, but it led nevertheless to a better understanding of generic CDT geometries, which in phase C can be described as a semiclassical torus with a number of quantum fractal outgrowths; see figure 3.…”

“…We will now discuss how to implement the jump and solve the corresponding discretized Laplace equation. Suppose we have a given oriented boundary or hypersurface (again, see [21,22] for explicit constructions), defined as a non-contractible (in a given spatial or time direction) connected subset of 3D tetrahedral faces of four-simplices or, equivalently, as a subset of links on the dual lattice. The field φ i in a simplex i adjacent to the boundary will perceive the value of the field φ j in a simplex j on the other side of the boundary as shifted by ±δ (the sign depends on the orientation of the boundary); see figure 4 for a 2D illustration.…”

“…We will address these questions in section 4. Assuming that the issues mentioned have satisfactory answers, let us return to our original problem: for a given toroidal triangulation we have defined in some way (see [21,22]) four independent non-contractible boundaries which we can label with x, y, z, t, and we want to use the corresponding classical solutions φμ i , μ = x, y, z, t as coordinates, but without any explicit reference to the chosen boundaries and the specific range of these solutions. We have managed to do that by introducing the coordinate system (α x , α y , α z , α t ) where α μ ∈ [0, 1] and the corresponding scalar fields φμ i (α μ ) are characterized by being solutions to the Laplace equations that jump from 0 to 1 at the α μ -hypersurface.…”

“…The sign depends on the flow of the winding number, i.e. whether the four-simplex is on the positive or negative side of the oriented boundary 21. Even though the equation (19) formally looks like a Poisson equation, we will call it the Laplace equation since b is not a source term when we view the field as a field with values in S 1 .…”

Cosmic voids and filaments from quantum gravity

“…Geometric information provided by the model is local, in the form of the neighborhood relations between the elements of the geometry. We may determine the geodesic distance between the simplices, but capturing the global properties of the system without a good choice of coordinates is difficult [13][14][15]. It is not a priori clear if such a choice is at all possible for a locally highly fluctuating geometry.…”