Dynamical Triangulation Induced by Quantum Walk

We present the single-particle sector of a quantum cellular automaton, namely a quantum walk, on a simple dynamical triangulated 2 - manifold. The triangulation is changed through Pachner moves, induced by the walker density itself, allowing the surface to transform into any topologically equivalent one. This model extends the quantum walk over triangular grid, introduced in a previous work, by one of the authors, whose space-time limit recovers the Dirac equation in (2+1)-dimensions. Numerical simulations show that the number of triangles and the local curvature grow as t α e − β t 2 , where α and β parametrize the way geometry changes upon the local density of the walker, and that, in the long run, flatness emerges. Finally, we also prove that the global behavior of the walker, remains the same under spacetime random fluctuations.

A toy model of discretized gravity in two dimensions and its extensions

We propose a toy model of quantum gravity in two dimensions with Euclidean signature. The model is given by a kind of discretization which is different from the dynamical triangulation. We show that there exists a continuum limit and we can calculate some physical quantities such as the expectation value of the area, that is, the volume of the two-dimensional Euclidean spacetime. We also consider the extensions of the model to higher dimensions.

Quantum gravity without vacuum dispersion

A generic prediction of quantum gravity is the vacuum dispersion of light, and hence that a photon’s speed depends on its energy. We present further numerical evidence for a scale-dependent speed of light in the causal dynamical triangulation (CDT) approach to quantum gravity. We show that the observed scale-dependent speed of light in CDT can be accounted for by a scale-dependent transformation of geodesic distance, whose specific functional form implies a discrete equidistant area spectrum. We make two nontrivial tests of the proposed scale transformation: a comparison with the leading-order quantum correction to the gravitational potential and a comparison with the generalized uncertainty principle. In both cases, we obtain the same functional form. However, contrary to the widespread prediction of vacuum dispersion in quantum gravity, numerous experiments have now definitively ruled out linear vacuum dispersion beyond Planckian energy scales [Formula: see text], and have even constrained quadratic dispersion at the level [Formula: see text]. Motivated by these experimental constraints, we seek to reconcile quantum gravity with the absence of vacuum dispersion. We point out that given a scale-dependent geodesic distance, a scale-dependent time interval becomes essential to maintaining an invariant speed of light. We show how a particular scale-dependent time interval allows a photon’s speed to remain independent of its energy.