Velocity Polytopes of Periodic Graphs and a No-Go Theorem for Digital Physics

TL;DR

This paper introduces velocity polytopes as geometric invariants of periodic graphs, revealing that their anisotropic nature prevents the emergence of isotropic space in discrete models, thus establishing a no-go theorem for digital physics.

Contribution

It defines velocity polytopes for periodic graphs and demonstrates their role as invariants, providing a geometric perspective on large-scale discrete structures in physics.

Findings

Velocity sets form polytopes in  space.

No periodic graph can produce an isotropic velocity set.

Supports a no-go theorem for isotropic space emergence.

Abstract

A periodic graph in dimension $d$ is a directed graph with a free action of $\Z^d$ with only finitely many orbits. It can conveniently be represented in terms of an associated finite graph with weights in $\Z^d$, corresponding to a $\Z^d$-bundle with connection. Here we use the weight sums along cycles in this associated graph to construct a certain polytope in $\R^d$, which we regard as a geometrical invariant associated to the periodic graph. It is the unit ball of a norm on $\R^d$ describing the large-scale geometry of the graph. It has a physical interpretation as the set of attainable velocities of a particle on the graph which can hop along one edge per timestep. Since a polytope necessarily has distinguished directions, there is no periodic graph for which this velocity set is isotropic. In the context of classical physics, this can be viewed as a no-go theorem for the emergence of an isotropic space from a discrete structure.