On the Geometry of the Hyperbolic Scator Space in 1+2 Dimensions

Abstract

Abstract.We consider the scator space in 1+2 dimensions-a hypercomplex, non-distributive hyperbolic algebra introduced by Fernández-Guasti and Zaldívar. We find a method for treating scators algebraically by embedding them into a distributive and commutative algebra. A notion of dual scators is introduced and discussed. We also study isometries of the scator space. It turns out that 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 arise some open questions, e.g., whether hypothetical tachyons can be considered as usual particles possessing time-like trajectories.Mathematics Subject Classification. 30G35, 20M14.

Geometric and Differential Features of Scators as Induced by Fundamental Embedding

“…Definition 1. Scators with positive square modulus will be referred to as time-like, scators with negative square modulus will be referred to as space-like, while scators with zero square modulus will be referred to as light-like [10].…”

“…The main ideas developed in the present paper follow directly from the distributive interpretation of the scator algebra introduced in [10]. Concepts appearing further will be of both geometric and differential nature.…”

“…since it is norm of different element, what is more, it is not even belonging to S set, on which we base our intuitions. Recall two important equations from [10]; firstly, we have…”

“…Algebraic properties of scators suffer from many inconvenient perplexities including, most importantly, lack of distributivity. In [10], we introduced so-called fundamental embedding (see also [11]), here engaging the map in making the scators more tractable in calculation and interpretable, and, last but not least, making them not as prone to mistakes due to the counter-intuitive algebraic features of these objects.…”

“…Section 3 is concerned with extended scator space automorphisms, as generated by its basis. We give a systematic study of dualities [10] of arbitrary dimension and parity, briefly characterizing their influence in causal realms appearing in the scator space. Sections 4 and 5 are centered around the derivative of scator function with respect to a scalar argument and derivation of formulas, enabling faster calculations of such derivatives in the 1 + 2 dimensional case.…”