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Complex Boosts: A Hermitian Clifford Algebra Approach

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Abstract

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.

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Authors and Affiliations

  1. School of Technology and Management, Polytechnical Institute of Leiria, 2411-901, Leiria, Portugal

    Milton Ferreira

  2. Center for Research and Development in Mathematics and ApplicationsDepartment of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Milton Ferreira

  3. Department of Mathematical Analyis, Clifford Research Group, Ghent University, Galglaan 2, B-9000, Ghent, Belgium

    Frank Sommen

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  1. Milton Ferreira
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  2. Frank Sommen
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Correspondence to Milton Ferreira.

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Ferreira, M., Sommen, F. Complex Boosts: A Hermitian Clifford Algebra Approach. Adv. Appl. Clifford Algebras 23, 339–362 (2013). https://doi.org/10.1007/s00006-012-0377-x

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