Splitting of 3D Quaternion Dimensions into 2D-Sells and a "World Screen Technology"

Yefremov, Alexander P.

doi:10.1166/asl.2012.1943

Cited by 6 publications

(5 citation statements)

References 3 publications

(4 reference statements)

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“…One directly verifies the fact composing all possible products of the units. Moreover, it is shown in the paper [12] that the representation of the Q-units through quadratic combinations of spinors is unique since only two linear combinations of nilpotent matrices of the type ψ ± ϕ ∓ and only two linear combinations of idempotent matrices of the type ψ ± ϕ ± exist; namely these four combinations form four Q-units. This means that any pair of orthogonal normalized spinor functions, not only those given by Eqs.…”

Section: Theory Of Matrices and Spinor Basement Of Q-numbersmentioning

confidence: 99%

“…This fact means that all eigenvectors of any Q-triad are functionally dependent. A special example of respective relations is adduced in [12]; below the relations are obtained in the universal procedure and in the general form. Let the spinors ρ ± , ξ ± be left and right eigenvectors of q1 and η ± , θ ± be left and right eigenvectors of q2 similarly to ϕ ± , ψ ± , eigenvector of q3.…”

Section: Eigenvectors Of Different Q-units and Cyclic Recurrent Formulaementioning

confidence: 99%

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Fundamental properties of quaternion spinors

Yefremov

2012

Gravit. Cosmol.

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Interior structure of arbitrary sets of quaternion units is analyzed using general methods of the theory of matrices. It is shown that the units are composed of quadratic combinations of fundamental objects having dual mathematical meaning as spinor couples and dyads locally describing 2D-surfaces. A detailed study of algebraic interrelations between the spinor sets belonging to different quaternion units is suggested as an initial step aimed to produce a self-consistent geometric image of spinor-surface distribution on the physical 3D space background.1 This vector notations relate to traditional Hamilton's notations (still used in literature) as q 1 = i, q 2 = j, q 3 = k.

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Section: Theory Of Matrices and Spinor Basement Of Q-numbersmentioning

confidence: 99%

Section: Eigenvectors Of Different Q-units and Cyclic Recurrent Formulaementioning

confidence: 99%

Fundamental properties of quaternion spinors

Yefremov

2012

Gravit. Cosmol.

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“…Considering direct (tensor) products of the dyad vectors with mixed components [8], we can construct only four such products (2 Â 2 matrices): two idempotent matrices…”

Section: Fractal Space Underlying Physical Spacementioning

confidence: 99%

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

Yefremov¹

2018

Space Flight

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An original mathematical instrument matching two different operational procedures aimed to change orientation and velocity of a spacecraft is suggested and described in detail. The tool's basements, quaternion algebra with its square-root (pregeometric) image, and fractal surface are represented in a parenthetical but in a sufficient format, indicating their principle properties providing solution to the operational task. A supplementary notion of vector-quaternion version of relativity theory is introduced since the spacecraft-observer mechanical system appears congenitally relativistic. The new tool is shown to have a simple pregeometric image of a fractal pyramid whose tilt and distortion evoke needed changes in the spacecraft's motion parameters, and the respective math procedures proved to be simplified compared with the traditionally used math methods.

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“…Second, it is proved [7] that each 2 × 2-matrix-vector Qunit has a set of right (2D column ψ ± ) and left (2D row ϕ ± ) eigenvectors with respective eigenvalues ±i, and with the orthogonality and normalization conditions satisfied…”

Section: Quaternions and Pre-geometric Spinor-planementioning

confidence: 99%

Pre-geometric structure of quantum and classical particles in terms of quaternion spinors

Yefremov

2013

Gravit. Cosmol.

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Analysis of fundamental structures underlying algebra of quaternion numbers leads to equations equivalent to those of quantum and classical mechanics. A short description of quaternion algebra is given, its units represented in terms of spinors forming vector basis on a complexnumber valued 2D space (spinor-plane). Demand of the algebra stability under rotations followed by conformal deformations of the spinor-plane yields a differential equation that in physical space at micro-scale becomes the Schrödinger equation. At macro-scale the equation acquires the Hamilton-Jacobi form, and a geometric interpretation of mechanical 'minimal action principal' is given. In the presence of a vector field in 3D quaternion space the stability condition takes the form of the Pauli equation for a charged quantum-mechanical particle in magnetic field.

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Splitting of 3D Quaternion Dimensions into 2D-Sells and a "World Screen Technology"

Cited by 6 publications

References 3 publications

Fundamental properties of quaternion spinors

Fundamental properties of quaternion spinors

Fractal Pyramid: A New Math Tool to Reorient and Accelerate a Spacecraft

Pre-geometric structure of quantum and classical particles in terms of quaternion spinors

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