Using BV to Describe Causal Quantum Evolution

In this note I describe how to capture the kinematics of quantum causal evolution using a logic called BV developed by the Calculus of Structures group at Dresden. The setting is discrete quantum mechanics. We imagine a finite “web” of spacetime points. The points are events in spacetime. They are viewed as verhtees in a directed acyclic graph (DAG); the edges of the DAG represent causal links mediated by the propagation of matter. The fact that the graph is acyclic captures a basic causality requirement: there are no closed causal trajectories. The DAG represents a discrete approximation to the spacetime on which a quantum system evolves. The graph is technically a dangling graph; there is a set of half edges in addition to the ordinary edges divided into two disjoint subsets: the incoming edges and the outgoing edges. An incoming edge has no initial point but has a terminal point, and dually for outgoing edges.