Advances in Applied Clifford Algebras

, Volume 23, Issue 2, pp 339–362 | Cite as

Complex Boosts: A Hermitian Clifford Algebra Approach

Article

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.

Keywords

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

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References

  1. 1.
    Barut A.: Complex Lorentz group with real metric: Group structure. J. Math. Phys. 5, 1562–1656 (1964)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brackx F., Bureš J., De Schepper H., Eelbode D., Sommen F., Souček V.: Fundaments of Hermitean Clifford Analysis Part I: Complex Structure. Compl. Anal. Oper. Theory 1, 341–365 (2007)CrossRefMATHGoogle Scholar
  3. 3.
    Brackx F., Bureš J., De Schepper H., Eelbode D., Sommen F., Souček V.: Fundaments of Hermitean Clifford Analysis II: splitting of h-monogenic equations. Complex Var. Elliptic Equ. 52, 1063–1079 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. Research Notes in Mathematics vol. 76, MA: Pitman Publishers, Boston, 1982.Google Scholar
  5. 5.
    Brackx F., De Schepper H., Sommen F.: The Hermitian Clifford analysis toolbox. Adv. appl. Clifford alg. 18, 45–487 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. Delanghe, F. Sommen and V. Souček V, Clifford algebra and spinor valued functions - a function theory for the Dirac operator. Mathematics and its Applications vol. 53, Kluwer, Dordrecht, 1992.Google Scholar
  7. 7.
    Doran C. C, Lasenby A. A: Geometric Algebra for Physicists. Cambridge university Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Doran C., Hestenes D., Sommen F.: Lie groups as spin groups. J. Math.Phys. 34((8), 3642–3669 (1993)MathSciNetADSCrossRefMATHGoogle Scholar
  9. 9.
    D. Eelbode, Clifford analysis on the hyperbolic unit ball. PhD. Thesis University of Ghent, Belgium, 2004.Google Scholar
  10. 10.
    Ferreira M.: Spherical continuous wavelet transforms arising from sections of the Lorentz group. Appl. Comput. Harmon. Anal. 26, 212–229 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    M. Ferreira, Gyrogroups in Projective Hyperbolic Clifford Analysis. I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications - Trends in Mathematics, Springer, Basel (2010), 61–80.Google Scholar
  12. 12.
    Gilbert J., Murray M.: Clifford Algebra and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  13. 13.
    G. Kaiser, Quantum Physics, Relativity, and Complex Space-time: Towards a New Synthesis. North-Holland, Amsterdam, 1990.Google Scholar
  14. 14.
    Kaiser G.: Physycal wavelets and their sources: real physics in complex spacetime. J. Phys. A: Math. Gen. 36, 291–338 (2003)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    G. Kaiser and T. Hansen, Generalized Huygens principle with pulsed-beam wavelets. J. Phys. A: Math. Theor. 42 (2009), 475403.Google Scholar
  16. 16.
    Rudin W.: Function theory in the unit ball of Cn. Springer-Verlag, New York (1980)CrossRefGoogle Scholar
  17. 17.
    Sabadini I., Sommen F.: Hermitian Clifford analysis and resolutions. Math. Meth. Appl. Sci. 25((16-18), 451–487 (2002)MathSciNetGoogle Scholar
  18. 18.
    Smith J., Ungar A. A.: Abstract space–times and their Lorentz groups. J. Math. Phys. 37((6), 3073–3098 (1996)MathSciNetADSCrossRefMATHGoogle Scholar
  19. 19.
    Tarakanov A. N.: Real and complex “boosts” in arbitrary pseudo-Euclidean spaces. Theoret. and Math. Phys. 28((3), 838–842 (1976)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Ungar A. A.: The relativistic velocity composition paradox and the Thomas rotation. Found. Phys. 19, 1385–1396 (1989)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Ungar A. A.: The abstract Lorentz transformatin group. Am. J. Phys. 60, 815–828 (1992)MathSciNetADSCrossRefMATHGoogle Scholar
  22. 22.
    Ungar A. A.: The abstract complex Lorentz transformation group with real metric. I. Special relativity formalism to deal with holomorphic automorphism group of the unit ball in any complex Hilbert space. J. Math. Phys. 35((3), 1408–1426 (1994)MathSciNetADSCrossRefMATHGoogle Scholar
  23. 23.
    Ungar A. A.: The abstract complex Lorentz transformation group with real metric. II. The invariance group of the form \({\| t \|^{2} - \| x \|^{2}}\) . J. Math. Phys. 35((4), 1881–1913 (1994)MathSciNetADSCrossRefMATHGoogle Scholar
  24. 24.
    A. A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics. Found. Phys., vol. 27, no. 6 (1997), 881–951.Google Scholar
  25. 25.
    A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces. Dordrecht:Kluwer Acad. Publ., 2001.Google Scholar
  26. 26.
    Ungar A. A.: Analytic Hyperbolic Geometry - Mathematical Foundations and Applications. World Scientific, Singapore (2005)CrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.School of Technology and ManagementPolytechnical Institute of LeiriaLeiriaPortugal
  2. 2.Center for Research and Development in Mathematics and ApplicationsDepartment of MathematicsUniversity of AveiroAveiroPortugal
  3. 3.Department of Mathematical AnalyisClifford Research Group, Ghent UniversityGhentBelgium

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