Fuzzy Topology, Quantization and Gauge Invariance

TL;DR

This paper explores fuzzy topology as a framework for quantum geometry, demonstrating that particle evolution on fuzzy manifolds aligns with Schrödinger and Dirac formalisms, and emphasizing gauge invariance in interactions.

Contribution

It introduces fuzzy topology as a mathematical basis for quantum geometry, linking fuzzy points to particle states and deriving fundamental quantum equations.

Findings

Fuzzy points correspond to particle states with inherent uncertainty.

Particle evolution follows Schrödinger and Dirac equations under this formalism.

Interactions are argued to be gauge invariant on fuzzy manifolds.

Abstract

Dodson-Zeeman fuzzy topology considered as the possible mathematical framework of quantum geometric formalism. In such formalism the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty dx. It's shown that m evolution with minimal number of additional assumptions obeys to schroedinger and dirac formalisms in norelativistic and relativistic cases correspondingly. It's argued that particle's interactions on such fuzzy manifold should be gauge invariant.