Discrete space-time geometry and skeleton conception of particle dynamics

TL;DR

This paper explores a discrete space-time geometry framework that differs from Riemannian geometry, describing particle dynamics through world chains and linking stochastic motion to quantum mechanics.

Contribution

It introduces a nonaxiomatizable, multivariant discrete geometry described by a world function, providing a new geometric basis for particle motion and quantum behavior.

Findings

Discrete geometry is nonaxiomatizable and multivariant.

Particle motion is inherently stochastic in this geometry.

Statistical analysis yields the Schrödinger equation.

Abstract

It is shown that properties of a discrete space-time geometry distinguish from properties of the Riemannian space-time geometry. The discrete geometry is a physical geometry, which is described completely by the world function. The discrete geometry is nonaxiomatizable and multivariant. The equivalence relation is intransitive in the discrete geometry. The particles are described by world chains (broken lines with finite length of links), because in the discrete space-time geometry there are no infinitesimal lengths. Motion of particles is stochastic, and statistical description of them leads to the Schr\"{o}dinger equation, if the elementary length of the discrete geometry depends on the quantum constant in a proper way.