Advances in Applied Clifford Algebras

, Volume 23, Issue 2, pp 339-362

First online:

Complex Boosts: A Hermitian Clifford Algebra Approach

  • Milton FerreiraAffiliated withSchool of Technology and Management, Polytechnical Institute of LeiriaCenter for Research and Development in Mathematics and ApplicationsDepartment of Mathematics, University of Aveiro Email author 
  • , Frank SommenAffiliated withDepartment of Mathematical Analyis, Clifford Research Group, Ghent University

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The aim of this paper is to study complex boosts in complex Minkowski space-time that preserves the Hermitian norm. Starting from the spin group Spin\({^+(2n, 2m, \mathbb{R})}\) in the real Minkowski space \({\mathbb{R}^{2n,2m}}\) we construct a Clifford realization of the pseudo-unitary group U(n,m) using the space-time Witt basis in the framework of Hermitian Clifford algebra. Restricting to the case of one complex time direction we derive a general formula for a complex boost in an arbitrary complex direction and its KAK-decomposition, generalizing the well-known formula of a real boost in an arbitrary real direction. In the end we derive the complex Einstein velocity addition law for complex relativistic velocities, by the projective model of hyperbolic n-space.


Pseudo-unitary group complex boosts Hermitian Clifford algebra Complex Einstein velocity addition