Complex Boosts: A Hermitian Clifford Algebra Approach

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.