Split-Quaternion Analyticity and (2 + 1)-Electrodynamics

Gogberashvili, M.

doi:10.22323/1.394.0007

Cited by 3 publications

(1 citation statement)

References 17 publications

(46 reference statements)

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“…In recent years, with the further research and development of four-dimensional algebras, split quaternions played important roles in quantum mechanics, rigid body rotation, electromagnetic, and other fields [2][3][4][5][6][7][8]. Split quaternionic structure of a space is similar to Minkowski space [9], making it possible to study interparticle interactions on Hamiltonian systems using the four-dimensional algebraic structure.…”

Section: Introductionmentioning

confidence: 99%

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Jiang,

Wang,

Guo

et al. 2024

Math Methods in App Sciences

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With the breakthroughs made by physicists in high‐dimensional mathematics, it has become possible to represent and solve a number of classical mathematical physics problems using the split quaternion algebra. In this paper, we study the least squares approximation of a class of Schrödinger equations in split quaternionic mechanics and propose two algebraic algorithms to the generalized right eigen‐problem for an i‐Hermitian split quaternion matrix pencil by using two isomorphic mappings. Numerical examples show the effectiveness of the proposed theories and algorithms.

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Section: Introductionmentioning

confidence: 99%

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Jiang,

Wang,

Guo

et al. 2024

Math Methods in App Sciences

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(2 + 1)-Maxwell Equations in Split Quaternions

Gogberashvili

2022

Physics

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The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two SO(2,1) subgroups of the full group of symmetry of the norms of split quaternions, SO(2,2). One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second SO(2,1) subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.

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Dirac and Maxwell Systems in Split Octonions

Gogberashvili¹,

Gurchumelia²

2023

JAMP

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The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4 + 4)-space. Split octonionic representation of SO(4, 4) and Spin(4, 4) groups and the trilinear invariant form are explicitly written and compared with Clifford algebraic matrix representation. It is noted that the complete algebra of split octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements. Lagrangians on split octonionic fields that generalize Dirac and Maxwell systems are constructed using group invariant forms. It is shown that corresponding equations are related to split octonionic analyticity conditions.

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Split-Quaternion Analyticity and (2 + 1)-Electrodynamics

Cited by 3 publications

References 17 publications

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

(2 + 1)-Maxwell Equations in Split Quaternions

Dirac and Maxwell Systems in Split Octonions

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