Product associativity in scator algebras and the quantum wave function collapse

Fernández‐Guasti, M.

doi:10.32323/ujma.423045

Cited by 11 publications

(24 citation statements)

References 15 publications

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“…The scator product is always commutative as can be seen from the symmetry of the scator components in the product definition. The product is associative if all possible product pairs have a nonvanishing additive scalar component . However, the product operation is not bilinear in the scator components; therefore, the product does not distribute over addition but in some particular cases ,.…”

Section: Scator Algebra Preamblementioning

confidence: 99%

“…In a recent communication, where the scator product associativity conditions were established, it was suggested that scator algebra seemed a promising route to provide a unified description of quantum dynamics. The two distinct procedures U (unitary time evolution propagator) and R (reduction), required in order to describe a quantum system, are referred to as the quantum measurement problem .…”

Section: Final Remarksmentioning

confidence: 99%

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Components exponential scator holomorphic function

Fernández‐Guasti

2019

Math Methods in App Sciences

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The components exponential scator function, succinctly labeled "cexp," is a function of scator variable in 1 + n dimensional elliptic scator algebra. This function is introduced here, and its elementary properties are studied. In 1 + 1 dimensions, the cexp function becomes the usual complex exponential function of complex variable. The cexp function is shown to be scator holomorphic in the entire scator set according to the differential quotient criterion. The scator derivative of the cexp function is the cexp function itself. The relationship between the Cartesian and polar forms of the cexp function can be seen as a higher-dimensional extension of Euler formula. The mappings of grids in 1 + 2 dimensional space exhibit ellipses and Lissajous-like figures in addition to circles and radial lines. A cusphere surface is generated for the isometric condition in the function's image. An interesting application of the cexp function as propagators regarding the quantum measurement problem is outlined. KEYWORDS holomorphic functions, hypercomplex variables 2 SCATOR ALGEBRA PREAMBLE Elliptic scator algebra is a 1+n finite dimensional algebra. In the additive form, the elements of this algebra are represented by o = (z 0 ; z 1 , … , z n ) = z 0 + n ∑ =1 zě ,Math Meth Appl Sci. wileyonlinelibrary.com/journal/mma

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Section: Scator Algebra Preamblementioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

Components exponential scator holomorphic function

Fernández‐Guasti

2019

Math Methods in App Sciences

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“…In the additive representation, associativity is only insured if all possible product pairs have nonvanishing additive scalar component. 15 Nonetheless, it is always commutative as can be seen from the symmetry of the scator components in the product definition. The product operation is not bilinear in the scator components; therefore, the product does not distribute over addition but in some particular cases.…”

Section: Definition the Product Of Two Scatorsmentioning

confidence: 99%

“…The scator product provides a natural object to model the quantum wave function evolution and collapse with a single operation. 15 In Section 2, the structure of imaginary scator algebra is shortly presented. In Section 3.1, we extend the concept of differential quotients to define scator derivatives.…”

Section: Introductionmentioning

confidence: 99%

Differential quotients in elliptic scator algebra

Fernández‐Guasti

2018

Math Methods in App Sciences

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The quotient of the function difference over the variable increment is evaluated in 1+n finite dimensional imaginary scator algebra. A distinctive property of the scator product is that it is commutative albeit not distributive over addition. In S 1+n , a subset of R 1+n where the product is defined, all the elements have inverse provided that zero is excluded. The quotient of scators and their differential limit can thus be defined in this subset, establishing the notion of scator differentiability. The necessary conditions that a scator holomorphic function must satisfy are then derived in terms of an extended set of partial differential equations. It is shown that affine transformations involving a scaling and a translation are scator holomorphic. These results are compared with holomorphy in quaternion and C 0,n Clifford algebras. Functions, such as the quadratic mapping, are shown not to satisfy the scator holomorphic conditions. KEYWORDS differential calculus, hypercomplex analysis, hyperholomorphic functions, scator algebra On sabbatical leave from Universidad Autónoma Metropolitana -Iztapalapa.

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“…However, the scator product is not bilinear, thus, it cannot be represented as a matrix -matrix product. 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 [8]. Scator algebra has also been successfully applied to other problems, such as a time-space description in deformed Lorentz metrics and three dimensional fractal structures [9,10].…”

Section: Introductionmentioning

confidence: 99%

Powers of Elliptic Scator Numbers

Fernández‐Guasti¹

2021

Preprint

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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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Product associativity in scator algebras and the quantum wave function collapse

Cited by 11 publications

References 15 publications

Components exponential scator holomorphic function

Components exponential scator holomorphic function

Differential quotients in elliptic scator algebra

Powers of Elliptic Scator Numbers

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