Quantum geometry, logic and probability

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these lattice spacing' weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form $\partial_+ f=(-\Delta_\theta+ q-p)f$ for the graph Laplacian $\Delta_\theta$, potential functions $q,p$ built from the probabilities, and finite difference $\partial_+$ in the time direction. Motivated by this new point of view, we introduce a discrete Schroedinger process’ as ${\mathrm{\partial }}_{+}\psi =\mathrm{ı}(-\mathrm{\Delta }+V)\psi$ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced generalised Markov process' for $f=|\psi|^2$ in which there is an additional source current built from $\psi$. We also discuss our recent work on the quantum geometry of logic in digital’ form over the field ${\mathbb{F}}_2=\{0,1\}$, including de Morgan duality and its possible generalisations.