The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including all Known Relativity Symmetries, Classical and Quantum

  • Otto C. W. KongEmail author
  • Jason Payne


In this article, we utilize the insights gleaned from our recent formulation of space(-time), as well as dynamical picture of quantum mechanics and its classical approximation, from the relativity symmetry perspective in order to push further into the realm of the proposed fundamental relativity symmetry SO(2,4). The latter has its origin arising from the perspectives of Planck scale deformations of relativity symmetries. We explicitly trace how the diverse actors in this story change through various contraction limits, paying careful attention to the relevant physical units, in order to place all known relativity theories – quantum and classical – within a single framework. More specifically, we explore both of the possible contractions of SO(2,4) and its coset spaces in order to determine how best to recover the lower-level theories. These include both new models and all familiar theories, as well as quantum and classical dynamics with and without Einsteinian special relativity. Along the way, we also find connections with covariant quantum mechanics. The emphasis of this article rests on the ability of this language to not only encompass all known physical theories, but to also provide a path for extensions. It will serve as the basic background for more detailed formulations of the dynamical theories at each level, as well as the exact connections amongst them.


Relativity symmetry Quantum relativity Lie algebra contractions Deformed relativity Covariant relativistic quantum and classical dynamics 



The authors are partially supported by research grants number 105-2112-M-008-017 and 106-2112-M-008-008 of the MOST of Taiwan.


  1. 1.
    Chew, C.S., Kong, O.C.W., Payne, J.: A quantum space behind simple quantum mechanics advances in high energy physics 2017, special issue on planck-scale deformations of relativistic symmetries, Article ID 4395918, 1–9 (2017)Google Scholar
  2. 2.
    Chew, C.S., Kong, O.C.W., Payne, J.: Observables and dynamics, quantum to classical, from a relativity symmetry perspective, NCU-HEP-k070 (2018) submitted; moreover, see the references thereinGoogle Scholar
  3. 3.
    Inönü, E., Wigner, E.P.: On the Contraction of Groups and their Representations. Proc. Nat. Acad. Sci. (US) 39, 510–524 (1953). See also and E. Saletan, Contraction of Lie Groups, J. Math. Phys. 2 (1961) 1–21ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gilmore, R.: Lie groups, Lie algebras, and some of their applications, Dover (2005)Google Scholar
  5. 5.
    Kong, O.C.W., Payne, J.: The Newtonian limit of special relativity: A relativity symmetry contraction perspective, NCU-HEP-k067 (2018)Google Scholar
  6. 6.
    Kong, O.C.W.: A deformed relativity with the quantum \( \hslash \). Phys. Lett. B 665, 58–61 (2008). see also arXiv:0705.0845 [gr-qc] for an earlier version with some different background discussionsADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Amelino-Camelia, G.: Testable scenario for relativity with minimum length. Phys. Lett. B 510, 255–263 (2001)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Amelino-Camelia, G.: Relativity in spacetimes with short-distance structure governed by an observer-independent (Plankian) length scale. Int. J. Mod. Phys. D 11, 35–59 (2002)Google Scholar
  9. 9.
    Magueijo, J., Smolin, L.: Lorentz invariance with an invariant energy scale. Phys. Rev. Lett. 88(190403), 1904031–1904034 (2002)Google Scholar
  10. 10.
    Magueijo, J., Smolin, L.: Generalized Lorentz invariance with an invariant energy scale. Phys. Rev, D 67, 044017 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kowalski-Glikman, J., Smolin, L.: Triply special relativity. Phys. Rev. D 70, 065020 (2004)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Snyder, H.S.: Quantized space-Time. Phys. Rev. 71, 8–41 (1947)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yang, C.N.: On quantized space-time. Phys. Rev. 72, 874 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mendes, R.V.: Deformations, stable theories and fundamental constants. J. Phys. A 27, 8091–8104 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chryssomalakos, C., Okon, E.: Generalized quantum relativistic kinematics: A stability point of view. Int. J. Mod. Phys. D 13, 1817–1850 (2004). ibid (2004) 2003–2034ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cho, D.N., Kong, O.C.W.: Relativity symmetries and Lie algebra contractions. Ann. Phys. 351, 275–289 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Connes, A.: Noncommutative geometry. Academic Press (1994)Google Scholar
  18. 18.
    Das, A., Kong, O.C.W.: Physics of quantum relativity through a linear realization. Phys. Rev. D 73, 124029 (2006)ADSCrossRefGoogle Scholar
  19. 19.
    Kong, O.C.W., Lee, H.-Y.: Poincaré-Snyder relativity with quantization, NCU-HEP-k036 (2009)Google Scholar
  20. 20.
    Kong, O.C.W., Lee, H.-Y.: Classical and quantum mechanics with poincaré-snyder relativity, NCU-HEP-k037 (2010)Google Scholar
  21. 21.
    Kyprianidis, A.: Scalar time parametrization of relativistic quantum mechanics: The covariant Schrödinger formalism. Phys. Rep. 155, 1–27 (1987). and references thereinADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Fanchi, J.R.: Parametrized relativistic quantum theory, Kluwer Academic Publishers (1993)Google Scholar
  23. 23.
    Aldaya, V., de Azcárraga, J.A.: Quantization as a consequence of the symmetry group: An approach to geometric quantization. J. Math. Phys. 23, 1297–1305 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Aldaya, V., de Azcárraga, J.A.: Symmetries of the pre-Klein-Gordon bundle: a Lagrangian analysis of quantum relativistic symmetry. J. Phys. A: Math. Gen. 18, 2639–2646 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Aldaya, V., de Azcárraga, J.A.: Group manifold analysis of the structure of relativistic quantum dynamics. Ann. Phys. 165, 484–504 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    de Azcárraga, J.A., Izquierdo, J.M.: Lie groups, Lie algebras, cohomology and some applications in physics. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  27. 27.
    Trump, M.A., Schieve, W.C.: Classical relativistic many-body dynamics, Kluwer Academic Publishers (1999)Google Scholar
  28. 28.
    Bacry, H., Levy-Leblond, J.M.: Possible Kinematics. J. Math. Phys. 9, 1605–1614 (1968)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Feynman, R.P.: Mathematical formulation of the quantum theory of electromagnetic interaction. Phys. Rev. 80, 440–457 (1950)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hostler, L.: Quantum field theory of particles of indefinite mass. I. J. Math, Phys. 21, 2461–2467 (1980)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Stückelberg, E.C.G.: La Mecanique du point materiel en theorie de relativite et en theorie des quanta. Helv. Phys. Acta 15, 23–37 (1942)ADSzbMATHGoogle Scholar
  32. 32.
    Feynman, R.P.: A relativistic cut-off for classical electrodynamics. Phys. Rev. 74, 1430–1438 (1948)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physics and Center for High Energy and High Field PhysicsNational Central UniversityChung-liTaiwan

Personalised recommendations