Relativistic Dynamical Groups in Quantum Theory and Some Possible Applications

  • P. Roman


There are many ways to measure the impact of a genius on the future — yet all these assessments are ephemeral. For most physicists, the fame of Einstein rests on his deep understanding of the fundamental coherence between space, time, and matter. Others admire his breadth of command, encompassing, besides relativity, so much of statistical physics and early quantum theory. But for many of us, yet another aspect of Einstein’s monumental work demands our admiration and grateful respect: namely, his clear understanding and insistent emphasis of underlying symmetries that, in a very real sense, govern the laws of Nature. It is important to emphasize that Einstein used the guiding principle of symmetry in its broadest sense: not only as the manifestation of an invariance under some permutation of elements, but also in the deeper connotation of an algebraic systemthat lends unity to the parts and that permits the “divination” of laws even when very little of the details pertaining to the phenomena is known to start with.1


Central Extension Dynamical Group Event Space Internal Symmetry Galilei Group 
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  1. 1.
    Einstein’s pioneering role in implanting the idea of symmetry as a crucial tool of research has been emphasized by E. P. Wigner, Proc. Amer. Phil. Soc. 93, No. 7 (1949).Google Scholar
  2. See also Wigner’s many further philosophical elaborations on the topic of symmetry, for example Proc. Internat. School of Phys. E. Fermi, 29, 40 (1964)Google Scholar
  3. The Nobel Prize Lectures 1964, Communic. Pure and Appl. Math. 13, 1 (1960), etc.Google Scholar
  4. 2.
    A comprehensive review of the Galilei group was given by J.-M. Lévi-Leblond, in “Group Theory and its Applications”, Vol. II. ed. by E. M. Loebl ( Acad. Press, NY, 1971 ).Google Scholar
  5. 3.
    In conventional units, then, X E M-1Q is the position operator.Google Scholar
  6. 4.
    P. Roman and J. P. Leveille, Jour. Math. Phys. 15, 1760 (1974). See also P. Roman’s contribution in “Quantum Theory & the Structures of Time & Space”, p. 85. ( Carl Hanser, Munchen, 1975 ).Google Scholar
  7. 5.
    I.e., that it be invariant under space translations and rotations.Google Scholar
  8. 6.
    This follows from the simplicity assumption that the development transformations form a one-parameter group.Google Scholar
  9. 7.
    P. Roman and J. P. Leveille, Jour. Math. Phys. 15, 2053 (1974).MathSciNetADSCrossRefGoogle Scholar
  10. 8.
    The constant ℓ. has the dimension of length.Google Scholar
  11. 9.
    J. J. Aghassi, P. Roman, R. M. Santilli, Phys. Rev. D1, 2753 (1970).ADSGoogle Scholar
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    The more relevant papers are: J. J. Aghassi, P. Roman, R. M. Santilli, Jour. Math. Phys. 11, 2297 (1970);MathSciNetADSCrossRefGoogle Scholar
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  16. 11.
    L. Castell, Nuovo Cim. 46A, 1 (1966) and ibid. 49A, 285 (1967).Google Scholar
  17. 12.
    I am indebted to Prof. P. Winternitz (Montreal) who some years ago called my attention to ref. 13 which I once read but later forgot.Google Scholar
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    H. Bacry and J.-M. Lévy-Leblond, Jour. Math. Phys. 9, 1605 (1968).ADSMATHCrossRefGoogle Scholar
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    P. Roman and J. Haavisto, Jour. Math. Phys. 17, 1664 (1976).MathSciNetADSCrossRefGoogle Scholar
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    P. Roman and J. Haavisto, Internt’l Jour. of Theor. Phys. 16, 915 (1977).MathSciNetGoogle Scholar
  21. 16.
    Typical papers in this framework are: F. A. Bais and R. J. Russell, Phys. Rev. D11, 2642 (1975);MathSciNetGoogle Scholar
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  24. 17.
    P. Roman and J. Haavisto, “A Relativistic Quantum Dynamical Group for Hadrons”, Preprint BU-PNS-16; to be published. [See also J. Haavisto: “Quantum-Dynamical Symmetry Groups in Curved Spaces”, Ph.D. Thesis, Boston University, 1978. ( Caveat: This thesis contains several minor errors.)]Google Scholar
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    Y. Nambu, Progr. Theor. Phys. Supp. 1, (No’s. 37–38), 368 (1966).ADSCrossRefGoogle Scholar
  26. 19.
    Possible external Poincaré labels are suppressed.Google Scholar
  27. 20.
    Because of ray-equivalence, setting C1 = 0 is not a restriction of generality.Google Scholar
  28. 21.
    It is easy to see that, in Class I reps, m2 will be indeed positive provided we take the constant ß ≷ -n0 depending on whether the constant γ ≷0. Note that, according to the results of ref. 7, ℓ = -h-1 < 0.Google Scholar
  29. 22.
    R. P. Feynman, M. Kíslinger and F. Ravndal, Phys. Rev. D3, 2706 (1971).ADSCrossRefGoogle Scholar
  30. 23.
    Y. S. Kim and M. E. Noz, Phys. Rev. D15, 335 (1977).ADSGoogle Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • P. Roman
    • 1
  1. 1.SUNY at PlattsburghPlattsburghUSA

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