Noncommutative Riemannian geometry can be applied in principle to any bidirected graph, with the metric viewed as assigning weights to each arrow. We completely solve for a quantum Levi-Civita connection for any metric with un-directed edge weights on a square graph. We find a 1-parameter family of quantum-Levi-Civita connections and a proposal for an Einstein-Hilbert action that does not depend on the parameter. The minimum of this action or `energy' is precisely the rectangular case where parallel edges have the same weight. We also allow negative weights corresponding to a Minkowski signature time direction and we look at the eigenvalues of the quantum-geometric graph Laplacian in both signatures.
Quantum gravity on a square graph
Tue, 07/11/2017 - 14:00