Group Structure and Geometric Interpretation of the Embedded Scator Space

• Main Authors: Jan L. Cieśliński, Artur Kobus
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Symmetry
• Subjects: scators; fundamental embedding; orthogonal groups; quadrics; Lorentz velocity addition formula
• Online Access: https://www.mdpi.com/2073-8994/13/8/1504

The set of scators was introduced by Fernández-Guasti and Zaldívar in the context of special relativity and the deformed Lorentz metric. In this paper, the scator space of dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>+</mo><mi>n</mi></mrow></semantics></math></inline-formula> (for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>) is interpreted as an intersection of some quadrics in the pseudo-Euclidean space of dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mi>n</mi></msup></semantics></math></inline-formula> with zero signature. The scator product, nondistributive and rather counterintuitive in its original formulation, is represented as a natural commutative product in this extended space. What is more, the set of invertible embedded scators is a commutative group. This group is isomorphic to the group of all symmetries of the embedded scator space, i.e., isometries (in the space of dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mi>n</mi></msup></semantics></math></inline-formula>) preserving the scator quadrics.