Duplex numbers, diffusion systems, and generalized quantum mechanics

Abstract

We show that the relation between the Schrödinger 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.

10 pages. Term "hyperbolic numbers" would be a better choice than "duplex numbers" used in the article. The formalism presented became known as "hyperbolic quantum mechanics"