Investigation of hyperbolic spinors based on split quaternions and hyperbolic spinor representations of lorentz transformations

This thesis investigates the relationship between split quaternions and hyperbolic spinors and explores the role of hyperbolic spinors in the representation of Lorentz transformations. The primary objective is to establish hyperbolic spinor representations associated with split quaternion algebra construct the corresponding matrix representations and formulate Lorentz transformations in terms of hyperbolic spinors. The first chapter introduces the aim, significance, assumptions and limitations of the study. The second chapter presents the fundamental concepts of real quaternions, spinors, split quaternions and split quaternion matrices. The third chapter introduces hyperbolic numbers, discusses their algebraic properties and reviews the spinor formulation of Lorentz transformations. The original contributions of the thesis are presented in the fourth chapter. Hyperbolic spinor representations corresponding to split quaternions are first constructed and the resulting hyperbolic spinors are classified and investigated. Subsequently, left and right hyperbolic spinor matrices are defined, the eigenvalues of the fundamental hyperbolic spinor matrices are computed and their algebraic properties are examined. Lorentz transformations are then expressed in terms of hyperbolic 4-spinors within the Weyl and Dirac representations and the associated generator matrices are derived together with their properties. The relationships between the hyperbolic 4-spinor representations in the Weyl and Dirac representations are also established. Finally, Lorentz transformations are represented by means of hyperbolic 4-spinors associated with split quaternions yielding single-matrix representations that provide a more direct hyperbolic spinor formulation. The final chapter summarizes the main findings and presents suggestions for future research.