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Quaternion symmetry of the Dirac equation

  • T. Saue
  • H. J. Aa. Jensen
Part of the Lecture Notes in Chemistry book series (LNC, volume 74)

Abstract

Following van der Waerden, the Dirac equation is derived from linearization of the Klein-Gordon equation using the algebraic properties of the Pauli spin matrices. As the algebra of these matrices is identical to that of quaternions, the Dirac equation can be reformulated in terms of quaternion algebra and therefore without reference to a specific spin quantization axis. In this paper we consider the symmetry content of the Dirac equation. It is found that the basic binary symmetry operations in spin space map onto the unit vectors of complex quaternions. We argue that a consistent choice of the inversion operator in spin space is of order four. We furthermore show that quaternion algebra is the natural language for time reversal symmetry. These considerations lead to the formulation of a symmetry scheme that automatically provides maximum point group and time reversal symmetry reduction in the solution of the Dirac equation in the finite basis approximation.

Keywords

Dirac Equation Dirac Operator Symmetry Operation Spin Space Quaternion Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • T. Saue
    • 1
  • H. J. Aa. Jensen
    • 2
  1. 1.The Institute of ChemistryUniversity of TromsøTromsøNorway
  2. 2.Department of ChemistryUniversity of Southern Denmark - Main campus: Odense UniversityOdense MDenmark

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