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 $\partial_+\psi=\imath(-\Delta+V)\psi$ 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 $\Bbb F_2=\{0,1\}$, including de Morgan duality and its possible generalisations.