Advances in Applied Clifford Algebras

, Volume 23, Issue 2, pp 339–362

Complex Boosts: A Hermitian Clifford Algebra Approach


    • School of Technology and ManagementPolytechnical Institute of Leiria
    • Center for Research and Development in Mathematics and ApplicationsDepartment of MathematicsUniversity of Aveiro
  • Frank Sommen
    • Department of Mathematical AnalyisClifford Research Group, Ghent University

DOI: 10.1007/s00006-012-0377-x

Cite this article as:
Ferreira, M. & Sommen, F. Adv. Appl. Clifford Algebras (2013) 23: 339. doi:10.1007/s00006-012-0377-x


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 groupcomplex boostsHermitian Clifford algebraComplex Einstein velocity addition

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© Springer Basel 2012