Advances in Applied Clifford Algebras

, Volume 25, Issue 3, pp 553–567 | Cite as

Extended Lorentz Transformations in Clifford Space Relativity Theory

  • Carlos Castro


Some novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) were explored recently. In particular, generalized photon dispersion relations allowed for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, but breaking the extended Lorentz symmetry in C-spaces. In this work we analyze in further detail the extended Lorentz transformations in Clifford Space and their physical implications. Based on the notion of “extended events” one finds a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. A generalized Weyl-Heisenberg algebra, involving polyvector-valued coordinates and momenta operators, furnishes a realization of an extended Poincare algebra in C-spaces. In addition to the Planck constant ħ, one finds that the commutator of the Clifford scalar components of the Weyl-Heisenberg algebra requires the introduction of a dimensionless parameter which is expressed in terms of the ratio of two length scales : the Planck and Hubble scales. We finalize by discussing the concept of “photons”, null intervals, effective temporal variables and the addition/subtraction laws of generalized velocities in C-space.


Clifford algebras Extended Relativity in Clifford Spaces Doubly Special Relativity Quantum Clifford-Hopf algebras 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. Castro and M. Pavsic, The Extended Relativity Theory in Clifford-spaces. Progress in Physics vol. 1 (2005) 31; Phys. Letts B 559 (2003) 74; Int. J. Theor. Phys 42 (2003) 1693.Google Scholar
  2. 2.
    C. Castro, The Extended Relativity Theory in Clifford Phase Spaces and Modifications of Gravity at the Planck/Hubble scales. To appear in Advances in Applied Clifford Algebras.Google Scholar
  3. 3.
    M. Pavsic, Found. of Phys. 33 (2003), 1277. M. Pavsic, The Landscape of Theoretical Physics : A Global View, from point particles to the brane world and beyond, in search of a Unifying Principle. (Fundamental Theories of Physics, vol. 19, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001).Google Scholar
  4. 4.
    C. Castro, Novel Physical Consequences of the Extended Relativity in Clifford Spaces. To appear in Advances in Applied Clifford Algebras.Google Scholar
  5. 5.
    C. Castro, Superluminal particles and the Extended Relativity Theories. Foundations of Physics vol 42 issue 9 (2012), 1135.Google Scholar
  6. 6.
    M. Pavsic, J. Phys. A41 332001, (2008).Google Scholar
  7. 7.
    K. Becker, M. Becker and J. Schwarz, String Theory and M-Theory : An Introduction. Pages 543-545 (Cambridge University Press, 2007)Google Scholar
  8. 8.
    G. Amelino-Camelia, Int. J. Mod. Phys D 11 (2002) 35. Int. J. Mod. Phys D 11 (2002, 1643. G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, The principle of relative locality. : 1101.0931.
  9. 9.
    C. Castro, Progress in Clifford Space Gravity. Advances in Applied Clifford Algebras vol 23 (1) (2013).Google Scholar
  10. 10.
    A. Cayley, Cambridge Math. J. 4 (1845), 193.Google Scholar
  11. 11.
    I. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, Resultants and Determinants. (Birkhauser 1994).Google Scholar
  12. 12.
    E. Guendelman and A. Kaganovich, Physical Consequences of a Theory with Dynamical Volume Element. (Plenary talk given at the Workshop Geometry, Topology, QFT and Cosmology, Paris, 28-30 May 2008) arXiv : 0811.0793.

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Center for Theoretical Studies of Physical SystemsClark Atlanta UniversityAtlantaUSA

Personalised recommendations