Dynamical Triangulation Induced by Quantum Walk
This work addresses the challenge of simulating quantum gravity or geometric evolution in physics, but it appears incremental as it extends a previous quantum walk model to include dynamical triangulation.
The paper tackles the problem of modeling quantum dynamics on a dynamically changing geometric surface by introducing a quantum walk on a triangulated 2-manifold where triangulation changes are induced by the walker's density, with numerical simulations showing that the number of triangles and local curvature evolve as t^α e^{-β t^2} and flatness emerges in the long run.
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 $t^αe^{-βt^2}$, 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.