Duplex numbers, diffusion systems, and generalized quantum mechanics

TL;DR

This paper explores the algebraic structure of duplex numbers to clarify the relationship between quantum mechanics and diffusion processes, proposing a generalized quantum framework using normed algebras.

Contribution

It introduces duplex numbers as an algebraic tool to analyze quantum-diffusion relations and proposes a generalized quantum mechanics replacing complex numbers with normed algebras.

Findings

Algebraic relation between Schrödinger equation and diffusion processes clarified

Duplex numbers reveal the non-reducibility of quantum mechanics to diffusion theory

A generalized quantum mechanics framework using normed algebras proposed

Abstract

We show that the relation between the Schr\"odinger equation and diffusion processes has an algebraic nature and can be revealed via the structure of "duplex numbers." This helps one to clarify that quantum mechanics cannot be reduced to diffusion theory. Also, a generalized version of quantum mechanics where $\C$ is replaced by a normed algebra with a unit is proposed.