2020/02/27 by Majid, Shahn
#FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2002.11851
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 ∂+ f=(-Δθ+ q-p)f for the graph Laplacian Δθ, potential functions q,p built from the probabilities, and finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a `discrete Schroedinger process' as ∂+ψ=\imath(-Δ+V)ψ 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=|ψ|2 in which there is an additional source current built from ψ. We also discuss our recent work on the quantum geometry of logic in `digital' form over the field \Bbb F2=\0,1\, including de Morgan duality and its possible generalisations.