Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Abstract

The scator space, introduced by Fernández-Guasti and Zaldívar, is endowed with a product related to the Lorentz rule of addition of velocities. The scator structure abounds with definitions calculationally inconvenient for algebraic operations, like lack of the distributivity. It occurs that situation may be partially rectified introducing an embedding of the scator space into a higher-dimensonal space, that behaves in a much more tractable way. We use this opportunity to comment on the geometry of automorphisms of this higher dimensional space in generic setting. In parallel, we develop commutative-hypercomplex analogue of differential calculus in a certain, specific low-dimensional case, as also leaned upon the notion of fundamental embedding, therefore treating the map as the main building block in completing the theory of scators.

Geometric Algebra Framework Applied to Circuits with Non-sinusoidal Voltages and Currents

“…with the so-called scator algebra structure [15,16], where any number of commuting imaginary units can appear (in the elliptic case).…”

Group Structure and Geometric Interpretation of the Embedded Scator Space

“…Our main tool to understand the scator product and scator geometry is the so-called fundamental embedding, introduced in [9]; see also [10,13]. The fundamental embedding F maps the scator space S 1+n into A 1,n , where the space A 1,n is the algebra over R generated (using addition and multiplication, which is assumed to be commutative, associative, and distributive over addition) by elements e e e 1 , .…”

Powers of Elliptic Scator Numbers

“…The scator product provides an interesting route for a unified mathematical description of quantum dynamics, encompassing the quantum system time evolution and its reduction to observed states [11]. Scator algebra has also been successfully applied to other problems, such as a time-space description in deformed Lorentz metrics [12][13][14] and three dimensional fractal structures [15,16].…”