Riemann surface and quantization

TL;DR

This paper explores a unified view of classical and quantum mechanics through complex analysis, interpreting quantization via Riemann surfaces and providing visualizations of quantum trajectories and path integrals.

Contribution

It introduces a novel interpretation of quantization using Riemann surfaces of the multivalent logarithm of the wave function, linking complex analysis with quantum mechanics.

Findings

Quantization interpreted through Riemann surfaces of log(Ψ)

Visual representations of quantum trajectories and path integrals

Derived a magnetic dipole satisfying Dirac quantization rule

Abstract

This paper proposes an approach of the unified consideration of classical and quantum mechanics from the standpoint of the complex analysis effects. It turns out that quantization can be interpreted in terms of the Riemann surface corresponding to the multivalent $\operatorname{Ln}\Psi $ function. A visual interpretation of "trajectories" of the quantum system and of the Feynman's path integral is presented. A magnetic dipole having a magnetic charge that satisfies the Dirac quantization rule was obtained.