Fundamental properties of quaternion spinors

Yefremov, Alexander P.

doi:10.1134/s0202289312030103

Cited by 3 publications

(3 citation statements)

References 6 publications

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“…(32) can be obtained from a single unit, say, q 3 by a transformation (34). Then, all vector units have same eigenvalues AEi, and the eigenfunctions of the derived units are linear combinations of the eigenfunctions of the initial unit [9]. This also means that the mapping (34) is a secondary one, but the primary one is SL 2; C ðÞ transformation of dyad vectors, thus forming a set of spinors from the viewpoint of the 3D space described by the triad vectors q k .…”

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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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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“…(4) and (6) mean that ϕ ± , ψ ± are SL(2, C)-spinors transformed asψ ± = U ψ ± ,φ ± = ϕ ± U −1 , while Eqs. (5) state that the couple ψ ± forms a vector basis in a 2D space with the metric g = ϕ + ϕ + + ϕ − ϕ − , the couple ϕ ± being a reciprocal basis of co-vectors [8]. The 2D space named the "spinor-plane" is said locally forming "pre-geometry" [9] whose each dimension directed by ψ + or ψ − , due to Eqs.…”

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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“…Quaternions were used in mechanics since they were introduced by Hamilton in 1843 [7]. The intensive application of the quaternions and octonions and their derivatives could be observed in electrodynamics, cosmology, quantum mechanics and special relativity [8][9][10][11][12]. Moreover, the investigation of hypercomplex fractal sets may be helpful in the defining of dynamic systems.…”

Section: Introductionmentioning

confidence: 99%

On the symmetry of bioctonionic Julia sets

Katunin¹

2013

J. Appl. Math. Comput. Mech.

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Abstract. The hypercomplex fractals obtained from generalizations of J-and M-sets, apart from their visual aesthetics, play an important role in the mathematical description in various fields of physics. The generalizations of J-and M-sets to the four-dimensional Euclidean space are well known and well described. However, very few studies were done for the higher-dimensional generalizations. The paper discusses the J-sets generalization to the hypercomplex algebra of bioctonions and completes the previous studies in this domain. The symmetry properties were studied for quadratic mapping of the bioctonionic J-sets. The discussion of limitations of the further generalizations of J-sets to higher hypercomplex spaces was also provided.

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Fundamental properties of quaternion spinors

Cited by 3 publications

References 6 publications

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

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

On the symmetry of bioctonionic Julia sets

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