Geometry of Riccati equations over normed division algebras

TL;DR

This paper explores Riccati equations over normed division algebras, revealing their geometric structure, extending to octonionic cases, and applying quaternionic Riccati equations to quantum mechanics.

Contribution

It introduces a geometric framework for Riccati equations over division algebras and extends the theory to octonionic cases with physical applications.

Findings

Riccati equations over division algebras are conformal equations on Euclidean spaces.

Octonionic Riccati equations are extended to the octonionic projective line.

Quaternionic Riccati equations are applied to quaternionic Schrödinger equations.

Abstract

This work presents and studies Riccati equations over finite-dimensional normed division algebras. We prove that a Riccati equation over a finite-dimensional normed division algebra $A$ is a particular case of conformal Riccati equation on a Euclidean space and it can be considered as a curve in a Lie algebra of vector fields $V\simeq\mathfrak{so}(\dim A+1,1)$. Previous results on known types of Riccati equations are recovered from a new viewpoint. A new type of Riccati equations, the octonionic Riccati equations, are extended to the octonionic projective line $\mathbb{O}{\rm P}^1$. As a new physical application, quaternionic Riccati equations are applied to study quaternionic Schr\"odinger equations on 1+1 dimensions.