Multiplicative Representation of a Hyperbolic non Distributive Algebra

Fernández‐Guasti, M.; Zaldívar, Felipe

doi:10.1007/s00006-014-0454-4

Cited by 11 publications

(21 citation statements)

References 8 publications

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“…there are additive and multiplicative (or polar) representations of scators in a one to one correspondence [11]. The restricted space conditions are implicit in the hyperbolic trigonometric functions involved in the multiplicative representation.…”

Section: Restricted Subsetmentioning

confidence: 99%

Hyperbolic Superluminal Scator Algebra

Fernández‐Guasti

Zaldívar

2014

Adv. Appl. Clifford Algebras

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An alternate time-space framework is proposed by means of a hyperbolic scator algebraic formalism where the deformed Lorentz transformations are an immediate consequence of the formal properties of the algebra. We survey the group properties of the hyperbolic product in the restricted, super-restricted and zero magnitude subsets as well as in the complete scator set. Some properties of the magnitude are derived together with a decomposition of 1 + n dimensional scators in n products of scator elements with scalar component and only one non-vanishing director component. These results are used to describe the scator properties in 1 + n dimensions. The super-restricted subset is associated with superluminal frames or tachyons. The various compositions of subluminal and superluminal events in the same direction follow the canonical superluminal Lorentz transformations interpretation. Composition of orthogonal components produce novel results. Products of scators with components within and outside the light pyramid yield sub or super luminal scators as well as scators belonging to a mixed state.

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Section: Restricted Subsetmentioning

confidence: 99%

Hyperbolic Superluminal Scator Algebra

Fernández‐Guasti

Zaldívar

2014

Adv. Appl. Clifford Algebras

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“…Note the base difference between what is proposed above and what was done in [8]: in the cited paper classification of scators was performed using components of the objects and quite sensible assumption that nothing observable can happen outside of the light bipyramid. As opposite, here we perform a classification in terms of the deformed metric (1.6) only, what has its formal consequences in the appearance of some new causal realms, with potential physical interpretations.…”

Section: Vol 27 (2017)mentioning

confidence: 95%

“…By the way, the scator appearing on the right-hand side of (1.5) will be referred to as dual to o c (see Definition 2.1 below). Many properties of the scator product (1.2) were widely investigated in many contexts [1,8,9], also physical [4]. To gain some insight in possible physical interpretation of the scator algebra, we recall here some basic terminology from [1] The classification proposed above is analogous to what is well known from special relativity.…”

Section: Vol 27 (2017)mentioning

confidence: 99%

“…The first three elements span the scator space S. The basis {1 1 1, i i i 1 , i i i 2 } is related to {1,ê 1 ,ê 2 } used in papers [1,8]. As in these papers, we demand that…”

Section: Some Further Algebramentioning

confidence: 99%

“…while the authors of [1,8] make different assumptions:ê 1ê2 =ê 2ê1 = 0. The last equalities can be neatly interpreted in our framework because…”

Section: Some Further Algebramentioning

confidence: 99%

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

Kobus

Cieśliński

2016

Adv. Appl. Clifford Algebras

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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.

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A Non-distributive Extension of Complex Numbers to Higher Dimensions

Fernández‐Guasti

2015

Adv. Appl. Clifford Algebras

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Multiplicative Representation of a Hyperbolic non Distributive Algebra

Cited by 11 publications

References 8 publications

Hyperbolic Superluminal Scator Algebra

Hyperbolic Superluminal Scator Algebra

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

A Non-distributive Extension of Complex Numbers to Higher Dimensions

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