Finite path integrals on stochastic branched structures

Abstract

In this paper, we present a statistical model of spacetime trajectories based on a finite collection of paths organized into a branched manifold. For each configuration of the branched manifold, we define a Shannon entropy. Given the variational nature of both the action in physics and the entropy in statistical mechanics, we explore the hypothesis that the classical action is proportional to this entropy. Under this assumption, we derive a Wick-rotated version of the path integral that remains finite and exhibits both quantum interference at the microscopic level and classical determinism at the macroscopic scale. In effect, this version of the path integral differs from the standard one because it assigns weights of non-uniform magnitude to different paths. The model suggests that wave function collapse can be interpreted as a consequence of entropy maximization. Although still idealized, this framework provides a possible route toward unifying quantum and classical descriptions within a common finite-entropy structure.

Figures (4)

• Figure 1: We depict a 1+1-spacetime with just one branch, which is decomposed into several simplices. The value of the field $\varphi$ is not shown in this diagram. The dashed line represents where two simplices meet. The left simplex has branch weight $w_a$, and the right simplex has branch weight $w_b$. The branch weight conservation constraint implies that $w_a=w_b$ in this case.
• Figure 2: We show a branched 0+1-dimensional spacetime manifold in which branches intersect and subsequently diverge. The 1-simplices along these branches carry branch weights $w_1$ through $w_6$. The 1-simplices intersect at 0-simplices (i.e., vertices).
• Figure 3: We see two examples of 0+1-dimensional branched manifolds. (a) A branched manifold with high branch cohesion enables frequent intersections. This case has a higher-dimensional null space of branch weights, corresponding to greater entropy. (b) A branched manifold with little cohesion has infrequent intersections. This case has less entropy.
• Figure 4: Gray lines indicate possible field configurations corresponding to two distinct measurement outcomes. After the measurement event, the field values diverge significantly. Thin lines represent branches of the spacetime manifold. The entropy constraint requires that branches remain close to each other. Since they cannot simultaneously remain close to both outcomes, the manifold collapses to a single result (here, the right-hand outcome) to maximize entropy.