Commutative Fuzzy Geometry and Quantum Particle Dynamics

TL;DR

This paper proposes fuzzy geometry as a new mathematical framework for reformulating quantum mechanics, showing that fuzzy points can model quantum states and their evolution aligns with Schrödinger dynamics.

Contribution

It introduces fuzzy geometry as a geometric foundation for quantum mechanics, connecting fuzzy points to Hilbert space vectors and quantum evolution.

Findings

Fuzzy points correspond to quantum states in Hilbert space.

Uncertainty in fuzzy geometry reproduces quantum uncertainty.

Quantum dynamics emerge from fuzzy geometric structures.

Abstract

Fuzzy geometry considered as the possible mathematical framework for reformulation of quantum-mechanical formalism in geometric terms. In this approach the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. In 1-dimensional case, due to manifold ultraweak (fuzzy) topology, m space coordinate x acquires principal uncertainty dx and described by positive, normalized density w(x,t). Analogous uncertainties appear for fuzzy point on 3-dimensional manifold. It's shown that m states on such manifold are equivalent to vectors (rays) on complex Hilbert space, their evolution correspond to Shroedinger dynamics of nonrelativistic quantum particle.