On geometry of the scator space

TL;DR

This paper explores the geometric structure and symmetries of the scator space, a hypercomplex algebra with unique properties, discussing its algebraic treatment, physical interpretation, and open questions about tachyons.

Contribution

It provides an algebraic approach to isometries in the scator space and discusses the implications of zero divisors within this hypercomplex algebra.

Findings

Identified algebraic methods for treating isometries

Highlighted the role of zero divisors in the algebra

Discussed potential physical interpretations and open questions

Abstract

We consider the scator space - a hypercomplex, non-distributive hyperbolic algebra introduced by Fern\'andez-Guasti and Zald\'ivar. We discuss isometries of the scator space and find consequent method for treating them algebraically, along with scators themselves. It occurs that introduction of zero divisors cannot be avoided while dealing with these isometries. The scator algebra may be endowed with a nice physical interpretation, although it suffers from lack of some physically demanded important features. Despite that, there arises some open questions, e.g., whether hypothetical tachyons can be considered as usual particles possessing time-like trajectories.