Composition of velocities and momentum transformations in a scator-deformed Lorentz metric

Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of holomorphicity. Until now, only a few particular cases of scator holomorphic functions have been found. In this paper we obtain all solutions of the generalized Cauchy–Riemann system which describes analogues of holomorphic functions in the (1+2)-dimensional scator space.

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“…where p 0 = const. Hence, u is proved to be of the form (5). Note that now the equations from the first line of (12) are identically satisfied.…”

Section: Components Exponential Functionsmentioning

confidence: 93%

“…In the hyperbolic case, scators have potential physical applications related to generalizations of the special theory of relativity (breaking the Lorentz symmetry) [4,5]. The elliptic case is an interesting new example of (non-distributive) hypercomplex numbers [6].…”

Section: Introductionmentioning

confidence: 99%

Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

Cieśliński

Zhalukevich

2020

Symmetry

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Powers of Elliptic Scator Numbers

Fernández‐Guasti

2021

Axioms

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Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

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“…In general, scator multiplication is not distributive over addition [4]. Scator algebra has been successfully applied to several problems: time-space description in a deformed Lorentz metric [5], a wave-function evolution and collapse unified description in quantum mechanics [3] and three dimensional fractal structures [6]. Explicit formulae for scator holomorphic functions recently published, will very likely expand applications to other areas [7].…”

Section: Introductionmentioning

confidence: 99%

Roots of Second Order Polynomials with Real Coefficients in Elliptic Scator Algebra

Fernández‐Guasti

2021

Journal of New Theory

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The roots of second order polynomials with real coefficients are obtained in the S 1+2 scator set. Explicit formulae are computed in terms of the polynomial coefficients. Although the scator product does not distribute over addition, the lack of distributivity is surmountable in order to find the zeros of the polynomial. The structure of the solutions and their distribution in 1+2 dimensional scator space are illustrated and discussed. There exist six, two, or eight solutions, depending on the value of polynomial coefficients. Four of these roots only exist in the hypercomplex S 1+2 \ S 1+1 set.

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Composition of velocities and momentum transformations in a scator-deformed Lorentz metric

Cited by 5 publications

References 21 publications

Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

Powers of Elliptic Scator Numbers

Roots of Second Order Polynomials with Real Coefficients in Elliptic Scator Algebra

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