论文标题
量子几何，逻辑和概率
Quantum geometry, logic and probability
论文作者
论文摘要
离散集上的量子几何形状是指定义量子度量的每个箭头的重量的有向图。但是，这些“晶格间距”重量不必独立于箭头的方向。 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 $\partial_+ f=(-Δ_θ+ q-p)f$ for the graph Laplacian $Δ_θ$, potential functions $q,p$ built from the probabilities, and finite difference $\partial_+$ in the time direction.以这种新的观点的启发，我们将“离散的schroedinger过程”引入了$ \ partial_+ψ= \ imath（-Δ+v）ψ$，用于与双模模连接相关的laplacian，以使离散的进化是统一的。我们为2态图明确地解决了此类连接的1参数家族，并针对$ f = |ψ|^2 $找到了1参数的“广义Markov进程”，其中有一个来自$ψ$的附加源电流。我们还讨论了我们最近在字段上以“数字”形式以$ \ bbb f_2 = \ {0,1 \} $在内的逻辑几何形状的量子几何的工作，包括de Morgan二元性及其可能的概括。
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 $\partial_+ f=(-Δ_θ+ q-p)f$ for the graph Laplacian $Δ_θ$, 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 $\partial_+ψ=\imath(-Δ+V)ψ$ 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 $\Bbb F_2=\{0,1\}$, including de Morgan duality and its possible generalisations.
