Abstract
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
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References
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).
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)
The Nobel Prize Lectures 1964, Communic. Pure and Appl. Math. 13, 1 (1960), etc.
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 ).
In conventional units, then, X E M-1Q is the position operator.
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 ).
I.e., that it be invariant under space translations and rotations.
This follows from the simplicity assumption that the development transformations form a one-parameter group.
P. Roman and J. P. Leveille, Jour. Math. Phys. 15, 2053 (1974).
The constant ℓ. has the dimension of length.
J. J. Aghassi, P. Roman, R. M. Santilli, Phys. Rev. D1, 2753 (1970).
The more relevant papers are: J. J. Aghassi, P. Roman, R. M. Santilli, Jour. Math. Phys. 11, 2297 (1970);
J. J. Aghassi, P. Roman, R. M. Santilli, Nuovo Cimento 5A, 551 (1971);
R. M. Santilli, Particles and Nuclei 1, 81 (1970);
P. L. Huddleston, M. Lorente, P. Roman, Found. of Physics 5, 75 (1975).
L. Castell, Nuovo Cim. 46A, 1 (1966) and ibid. 49A, 285 (1967).
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.
H. Bacry and J.-M. Lévy-Leblond, Jour. Math. Phys. 9, 1605 (1968).
P. Roman and J. Haavisto, Jour. Math. Phys. 17, 1664 (1976).
P. Roman and J. Haavisto, Internt’l Jour. of Theor. Phys. 16, 915 (1977).
Typical papers in this framework are: F. A. Bais and R. J. Russell, Phys. Rev. D11, 2642 (1975);
Y. M. Cho and P. G. 0. Freund, Phys. Rev. D12, 1588 (1975);
E. Cremer and J. Sherk, Nucl. Phys. B118 61 (1977); and from another viewpoint: P. Nath and R. Arnowitt, Phys. Rev. D15, 1033 (1977).
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.)]
Y. Nambu, Progr. Theor. Phys. Supp. 1, (No’s. 37–38), 368 (1966).
Possible external Poincaré labels are suppressed.
Because of ray-equivalence, setting C1 = 0 is not a restriction of generality.
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.
R. P. Feynman, M. Kíslinger and F. Ravndal, Phys. Rev. D3, 2706 (1971).
Y. S. Kim and M. E. Noz, Phys. Rev. D15, 335 (1977).
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© 1980 Plenum Press, New York
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Roman, P. (1980). Relativistic Dynamical Groups in Quantum Theory and Some Possible Applications. In: Gruber, B., Millman, R.S. (eds) Symmetries in Science. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-3833-8_22
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