Abstract
Lattice formulations of gravity can be used to study non-perturbative aspects of quantum gravity. Causal Dynamical Triangulations (CDT) is a lattice model of gravity that has been used in this way. It has a built-in time foliation but is coordinate-independent in the spatial directions. The higher-order phase transitions observed in the model may be used to define a continuum limit of the lattice theory. Some aspects of the transitions are better studied when the topology of space is toroidal rather than spherical. In addition, a toroidal spatial topology allows us to understand more easily the nature of typical quantum fluctuations of the geometry. In particular, this topology makes it possible to use massless scalar fields that are solutions to Laplace’s equation with special boundary conditions as coordinates that capture the fractal structure of the quantum geometry. When such scalar fields are included as dynamical fields in the path integral, they can have a dramatic effect on the geometry.
Highlights
Causal Dynamical Triangulations (CDT) is an attempt to formulate a non-perturbative lattice theory of quantum gravity
While dynamical triangulations” (DT) represented a completely coordinate-free lattice formulation of the path integral of quantum gravity, the CDT formulation should be viewed as coordinate-free in the spatial directions, while a gauge fixing of the lab-function has been made in the time direction
The picture of an emergent background geometry around which there are relatively small quantum fluctuations is corroborated by the results obtained using toroidal spatial topology
Summary
Causal Dynamical Triangulations (CDT) is an attempt to formulate a non-perturbative lattice theory of quantum gravity (see [1,2] for reviews). It provided a lattice formulation where the functional integration over the intrinsic worldsheet metric was represented as a summation over two-dimensional triangulations [4,5,6,7] These triangulations were constructed by gluing together equilateral triangles, which were considered flat in the interior, such that they formed a piecewise linear manifold with a fixed topology. After the rotation from Lorentzian triangulations TL to Euclidean triangulations TE, we can view the path integral as a partition function for a statistical theory of random geometries, but note that the set of Euclidean triangulations we obtain this way is different from the set of Euclidean triangulations one constructs in DT, since they have a time foliation. One can use a sequence of such configurations to calculate expectation values of spacetime “observables”, or one can try to study specific features of the configurations
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