Properties of dynamical fractal geometries in the model of causal dynamical triangulations

Scalar fields in causal dynamical triangulations

“…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.…”

“…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. Another way of analyzing such geometric structures was proposed in [22], where the boundaries were used to define the shortest loops (starting at any four-simplex) with nontrivial winding numbers in all three spatial directions and in the time direction. The length of such loops measured in a given geometry (triangulation) is 'topological' as it does not depend on the position of the boundaries.…”

“…Let us choose one of them and a closed hypersurface that intersects the loop only once. For a description of how to actually choose such hypersurfaces for our CDT triangulations, we refer to [21,22]. Let the field φ i jump by δ = 1 when crossing the hypersurface.…”

“…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.…”

“…One of these is that it makes it difficult to relate the results obtained in the lattice theory to more analytical approaches where coordinate systems are used (even if physics of course should be independent of a specific coordinate system). The issue of reintroducing suitable coordinate systems in the lattice theory of gravity has been extensively studied recently by our group [21,22], and in this article we will discuss a new promising way of doing it by using scalar fields-see section 3. (c) The formulation of the EDT lattice field theory of (Euclidean) quantum GR is simple.…”