Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 017, 9 pages      math-ph/0511077

Subgroups of the Group of Generalized Lorentz Transformations and Their Geometric Invariants

George Bogoslovsky
Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia

Received October 06, 2005, in final form November 09, 2005; Published online November 15, 2005

It is shown that the group of generalized Lorentz transformations serves as relativistic symmetry group of a flat Finslerian event space. Being the generalization of Minkowski space, the Finslerian event space arises from the spontaneous breaking of initial gauge symmetry and from the formation of anisotropic fermion-antifermion condensate. The principle of generalized Lorentz invariance enables exact taking into account the influence of condensate on the dynamics of fundamental fields. In particular, the corresponding generalized Dirac equation turns out to be nonlinear. We have found two noncompact subgroups of the group of generalized Lorentz symmetry and their geometric invariants. These subgroups play a key role in constructing exact solutions of such equation.

Key words: Lorentz, Poincaré and gauge invariance; spontaneous symmetry breaking; Finslerian space-time.

pdf (187 kb)   ps (136 kb)   tex (13 kb)


  1. Kostelecký A., Samuel S., Spontaneous breaking of Lorentz symmetry in string theory, Phys. Rev. D, 1989, V.39, N 2, 683-685.
  2. Colladay D., Kostelecký A., CPT violation and the standard model, Phys. Rev. D, 1997, V.55, N 11, 6760-6774, hep-ph/9703464.
    Colladay D., Kostelecký A., Lorentz-violating extension of the standard model, Phys. Rev. D, 1998, V.58, 116002, hep-ph/9809521.
  3. Kostelecký A., Gravity, Lorentz violation, and the standard model, Phys. Rev. D, 2004, V.69, 105009, hep-th/0312310.
  4. Kostelecký A. (Editor), CPT and Lorentz symmetry III, Singapore, World Scientific, 2005.
  5. Allen R.E., Yokoo S., Searching for Lorentz violation, Nucl. Phys. Proc. Suppl. B, 2004, V.134, 139-146, hep-th/0402154.
  6. Cardone F., Mignani R., Broken Lorentz invariance and metric description of interactions in a deformed Minkowski space, Found. Phys., 1999, V.29, N 11, 1735-1783.
  7. Bogoslovsky G.Yu., A special relativistic theory of the locally anisotropic space-time, Nuovo Cim. B, 1977, V.40, N 1, 99-134.
  8. Bogoslovsky G.Yu., Theory of locally anisotropic space-time, Moscow, Moscow Univ. Press, 1992.
  9. Bogoslovsky G.Yu., From the Weyl theory to a theory of locally anisotropic space-time, Class. Quantum Grav., 1992, V.9, 569-575.
    Bogoslovsky G.Yu., Finsler model of space-time, Phys. Part. Nucl., 1993, V.24, 354-379.
    Bogoslovsky G.Yu., A viable model of locally anisotropic space-time and the Finslerian generalization of the relativity theory, Fortschr. Phys., 1994, V.42, N 2, 143-193.
  10. Bogoslovsky G.Yu., Goenner H.F., On the possibility of phase transitions in the geometric structure of space-time, Phys. Lett. A, 1998, V.244, 222-228, gr-qc/9804082.
    Bogoslovsky G.Yu., Goenner H.F., Finslerian spaces possessing local relativistic symmetry, Gen. Rel. Grav., 1999, V.31, N 10, 1565-1603, gr-qc/9904081.
  11. Arbuzov B.A., Muon g-2 anomaly and extra interaction of composite Higgs in a dynamically broken electroweak theory, hep-ph/0110389.
    Arbuzov B.A., Spontaneous generation of effective interaction in a renormalizable quantum field theory model, Teor. Mat. Fiz., 2004, V.140, N 3, 367-387 (and references therein).
  12. Winternitz P., Fris I., Invariant expansions of relativistic amplitudes and subgroups of the proper Lorentz group, Yadern. Fiz., 1965, V.1, N 5, 889-901.
  13. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental groups of physics. II. The similitude group, J. Math. Phys., 1975, V.16, N 8, 1615-1624.
  14. Bogoslovsky G.Yu., Goenner H.F., On the generalization of the fundamental field equations for locally anisotropic space-time, in Proceedings of XXIV International Workshop "Fundamental Problems of High Energy Physics and Field Theory" (June 27-29, 2001, Protvino, Russia), Editor V.A. Petrov, Protvino, Insitute for High Energy Physics, 2001, 113-125.
    Bogoslovsky G.Yu., Goenner H.F., Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation, Phys. Lett. A, 2004, V.323, 40-47, hep-th/0402172.
  15. Fushchych W.I., Zhdanov R.Z., Symmetry and exact solutions of nonlinear spinor equations, Phys. Rep., 1989, V.172, N 4, 123-174.
    Fushchych W.I., Zhdanov R.Z., Symmetry and exact solutions of nonlinear Dirac equations, Kyiv, Mathematical Ukraina Publisher, 1997 (and references therein).

Previous article   Next article   Contents of Volume 1 (2005)