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


SIGMA 4 (2008), 045, 21 pages      arXiv:0712.1718      http://dx.doi.org/10.3842/SIGMA.2008.045
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Rapidities and Observable 3-Velocities in the Flat Finslerian Event Space with Entirely Broken 3D Isotropy

George Yu. Bogoslovsky
Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119991 Moscow, Russia

Received December 09, 2007, in final form May 08, 2008; Published online May 26, 2008

Abstract
We study the geometric phase transitions that accompany the dynamic rearrangement of vacuum under spontaneous violation of initial gauge symmetry. The rearrangement may give rise to condensates of three types, namely the scalar, axially symmetric, and entirely anisotropic condensates. The flat space-time keeps being the Minkowski space in the only case of scalar condensate. The anisotropic condensate having arisen, the respective anisotropy occurs also in space-time. In this case the space-time filled with axially symmetric condensate proves to be a flat relativistically invariant Finslerian space with partially broken 3D isotropy, while the space-time filled with entirely anisotropic condensate proves to be a flat relativistically invariant Finslerian space with entirely broken 3D isotropy. The two Finslerian space types are described briefly in the extended introduction to the work, while the original part of the latter is devoted to determining observable 3-velocities in the entirely anisotropic Finslerian event space. The main difficulties that are overcome in solving that problem arose from the nonstandard form of the light cone equation and from the necessity of correct introducing of a norm in the linear vector space of rapidities.

Key words: Lorentz, Poincaré, and gauge invariances; spontaneous symmetry breaking; dynamic rearrangement of vacuum; Finslerian space-time.

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References

  1. Kostelecký A., Samuel S., Spontaneous breaking of Lorentz symmetry in string theory, Phys. Rev. D 39 (1989), 683-685.
  2. Colladay D., Kostelecký A., CPT violation and the standard model, Phys. Rev. D 55 (1997), 6760-6774, hep-ph/9703464.
    Colladay D., Kostelecký A., Lorentz-violating extension of the standard model, Phys. Rev. D 58 (1998), 116002, 23 pages, hep-ph/9809521.
  3. Kostelecký A. (Editor), CPT and Lorentz symmetry III, Singapore, World Scientific, 2005.
    Kostelecký A. (Editor), CPT and Lorentz symmetry IV, Singapore, World Scientific, 2008.
  4. Bogoslovsky G.Yu., A special relativistic theory of the locally anisotropic space-time, Nuovo Cimento B 40 (1977), 99-134, Erratum, Nuovo Cimento B 43 (1978), 377-378.
  5. Tavakol R., van den Bergh N., Finsler spaces and the underlying geometry of space-time, Phys. Lett. A 112 (1985), 23-25.
  6. Tavakol R., van den Bergh N., Viability criteria for the theories of gravity and Finsler spaces, Gen. Relativity Gravitation 18 (1986), 849-859.
  7. Bogoslovsky G.Yu., Theory of locally anisotropic space-time, Moscow, Moscow Univ. Press, 1992.
  8. Bogoslovsky G.Yu., Finsler model of space-time, Phys. Part. Nucl. 24 (1993), 354-379.
    Bogoslovsky G.Yu., A viable model of locally anisotropic space-time and the Finslerian generalization of the relativity theory, Fortschr. Phys. 42 (1994), 143-193.
  9. Bogoslovsky G.Yu., Goenner H.F., Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation, Phys. Lett. A 323 (2004), 40-47, hep-th/0402172.
  10. Bogoslovsky G.Yu., Subgroups of the group of generalized Lorentz transformations and their geometric invariants, SIGMA 1 (2005), 017, 9 pages, math-ph/0511077.
    Bogoslovsky G.Yu., Lorentz symmetry violation without violation of relativistic symmetry, Phys. Lett. A 350 (2006), 5-10, hep-th/0511151.
  11. Kostelecký A., Gravity, Lorentz violation, and the Standard Model, Phys. Rev. D 69 (2004), 105009, 20 pages, hep-th/0312310.
  12. Bailey Q.G., Kostelecký A., Signals for Lorentz violation in post-Newtonian gravity, Phys. Rev. D 74 (2006), 045001, 46 pages, gr-qc/0603030.
  13. Girelly F., Liberati S., Sindoni L., Planck-scale modified dispersion relations and Finsler geometry, Phys. Rev. D 75 (2007), 064015, 9 pages, gr-qc/0611024.
  14. Ghosh S., Pal P., Deformed special relativity and deformed symmetries in a canonical framework, Phys. Rev. D 75 (2007), 105021, 11 pages, hep-th/0702159.
  15. Bogoslovsky G.Yu., Some physical displays of the space anisotropy relevant to the feasibility of its being detected at a laboratory, arXiv:0706.2621.
  16. Gibbons G.W., Gomis Joaquim, Pope C.N., General very special relativity is Finsler geometry, Phys. Rev. D 76 (2007), 081701(R), 5 pages, arXiv:0707.2174.
    Cohen A.G., Glashow S.L., Very special relativity, Phys. Rev. Lett. 97 (2006), 021601, 3 pages, hep-ph/0601236.
  17. Mavromatos N.E., Lorentz invariance violation from string theory, arXiv:0708.2250.
  18. Sindoni L., The Higgs mechanism in Finsler spacetimes, arXiv:0712.3518.
  19. Bogoslovsky G.Yu., Goenner H.F., On the possibility of phase transitions in the geometric structure of space-time, Phys. Lett. A 244 (1998), 222-228, gr-qc/9804082.
    Bogoslovsky G.Yu., Goenner H.F., Finslerian spaces possessing local relativistic symmetry, Gen. Relativity Gravitation 31 (1999), 1565-1603, gr-qc/9904081.
  20. Bogoslovsky G.Yu., On a special relativistic theory of anisotropic space-time, Dokl. Akad. Nauk SSSR 213 (1973), 1055-1058.
  21. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental groups of physics. II. The similitude group, J. Math. Phys. 16 (1975), 1615-1624.
  22. Winternitz P., Fris I., Invariant expansions of relativistic amplitudes and subgroups of the proper Lorentz group, Yadern. Fiz. 1 (1965), 889-901.
  23. Bogoslovsky G.Yu., The proper time, spatial distances and clock synchronization in the locally anisotropic space-time, JINR Communication E2-82-779, Dubna, JINR, 1982.
  24. Bogoslovsky G.Yu., The relativistic inert mass tensor, Vestn. Mosk. Univ. Ser. Fiz. Astron. 24 (1983), no. 1, 70-71.
  25. Bogoslovsky G.Yu., On the local anisotropy of space-time, inertia and force fields, Nuovo Cimento B 77 (1983), 181-190.
  26. Bogoslovsky G.Yu., A generalized Klein-Gordon equation and Mach's principle, Vestn. Mosk. Univ. Ser. Fiz. Astron. 24 (1983), no. 3, 59-61.
  27. 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, http://dbserv.ihep.su/~pubs/tconf01/c2-5.htm.
  28. Berwald L., Projective Krümmung allgemeiner affiner Räume und Finslersche Räume skalarer Krümmung, Ann. Math. 48 (1947), 755-781 (und die Literaturhinweise darin).
  29. Moór A., Ergänzung, Acta. Math. 91 (1954), 187-188.
  30. Weyl H., Gravitation und Elektrizität, Sitzber. preuss Akad. Wiss., Physik-math. Kl., 1918, 465-480.
    Weyl H., Eine neue Erweiterung der Relativitätstheorie, Ann. Phys. 59 (1919), 101-133.
  31. Bogoslovsky G.Yu., From the Weyl theory to a theory of locally anisotropic space-time, Classical Quantum Gravity 9 (1992), 569-575.
  32. Bogoslovsky G.Yu., 4-momentum of a particle and the mass shell equation in the entirely anisotropic space-time, in Space-Time Structure (Algebra and Geometry), Editors D.G. Pavlov, Gh. Atanasiu and V. Balan, Moscow, Lilia Print, 2007, 156-173.
  33. Arbuzov B.A., Infrared non-perturbative QCD running coupling from Bogolubov approach, Phys. Lett. B 656 (2007), 67-73, hep-ph/0703237.

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