MATHEMATICAL STRUCTURE OF IDENTITY

 

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Notes and References

1. Sternberg, S. [1994]: Group Theory and Physics, Cambridge, Cambridge Univiversity Press, p.149 "an elementary particle 'is' an irreducible representation of the group, G, of physics"
2. Discussions of this point are in d'Espagnat, B. [1976]: Conceptual Foundations of Quantum Mechanics, 2nd edition, Reading, Benjamin., p. 14 and in Rosen, J. [1995]: Symmetry in Science, Springer, N.Y., p. 66
3. Cohen-Tannoudji, C., Diu, B. and LaloŽ, F. [1977]: Quantum Mechanics, N.Y., Wiley is a good standard text on the subject.
4.Appendix A of Jones, H. F. [1994]: Groups, Representations and Physics, Bristol, Institute of Physics Press gives a unified overview of quantum mechanics expressed solely in Dirac notation. The concordance between Dirac notation and physical meaning is the theme in Chester, M. [1987]: Primer of Quantum Mechanics, N.Y., Wiley
5.See von Neumann, J. [1955]: Mathematical Foundations of Quantum Mechanics, English trans by R. T. Beyer, Princeton, Princeton University Press, p. 419-420
6. An example is given on p.10 of Chaichian, M. and Hagedorn, R. [1998]: Symmetries in Quantum Mechanics, Bristol, Institute of Physics Press.
7.Considerable attention is devoted to this by Chaichian and Hagedorn [1998] on p.17 and then again in Chapter 3.
8. "... symmmetry operations involve a movement of the body.", p.3 of Burns, G. [1977]: Introduction to Group Theory With Applications, N.Y., Academic Press.
9. See Fonda L. and Ghirardi, G. [1970]: Symmetry Principles in Quantum Physics, N.Y., Dekker., p.39
10.Induced transformation is discussed at length in Section 3.7 of Elliott, J. F. and Dawber, P. G. [1979]: Symmetry in Physics, Vol. 1, N.Y., Oxford University Press.
11.Group theory books with the word symmetry in their titles pay attention to definition. Rosen, J. [1995]: Symmetry in Science, Springer, N.Y. 1995, defines symmetry as "immunity to a possible change", p.157. Chaichian and Hagedorn [1995] give a definition on page 17 but in technical language and too long to reproduce here. Elliott and Dawber [1979] quote the dictionary but do not relate this definition to the mathematics. Weyl, H. [1952]: Symmetry, Princeton, Princeton University Press treats us to poetry and history on the word and then uses it eloquently as if it were defined. He finally accepts the word congruence for its essential property.
12. A pithy but complete summary of the structure of quantum mechanics is given on page 49 and 50 of Weinberg, S. [1995]: The Quantum Theory of Fields, Vol.1, Cambridge, Cambridge University Press.
13. Bell, J. S. [1987]: Speakable and unspeakable in quantum mechanics, Cambridge, Cambridge University Press., p.40
14. McIntosh, H.V. [1971]: 'Symmetry and Degeneracy' in E. M. Loebl (ed), Group Theory and Its Applications, vol. 2, N.Y., Academic Press., p.80-84 gives a detailed early history of this pursuit.
15. This pursuit was not always considered worthy. Wigner, E.P. [1959]: Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, English trans. J.J. Griffin, N.Y., Academic Press, dismissed it on page 120 with, "It will be assumed that accidental degeneracy is a very uncommon situation and that it occurs ... only exceptionally."
16. It is called the non-relativistic Kepler problem on p.381 of Barut and Raczka [1986]. Very fine expositions on this seminal excercise are also given in Jauch and Hill [1940], Joshi [1985], p.177-182 and Jones [1994], p.124-127.



List of References:
Barut A. O. and Raczka, R. [1986]: Theory of Group Representations and Applications, Singapore, World Scientific.
Bell, J. S. [1987]: Speakable and unspeakable in quantum mechanics, Cambridge, Cambridge University Press.
Burns, G. [1977]: Introduction to Group Theory With Applications, N.Y., Academic Press.
Chaichian, M. and Hagedorn, R. [1998]: Symmetries in Quantum Mechanics, Bristol, Institute of Physics Press. Chester, M. [1987]: Primer of Quantum Mechanics, N.Y., Wiley Cohen-Tannoudji, C., Diu, B. and LaloŽ, F. [1977]: Quantum Mechanics, N.Y., Wiley d'Espagnat, B. [1976]: Conceptual Foundations of Quantum Mechanics, 2nd edition, Reading, Benjamin. Elliott, J. F. and Dawber, P. G. [1979]: Symmetry in Physics, Vol. 1, N.Y., Oxford University Press. Fonda L. and Ghirardi, G. [1970]: Symmetry Principles in Quantum Physics, N.Y., Dekker. Jauch J. M. and Hill, E. L. [1940]: 'On the Problem of Degeneracy in Quantum Mechanics', Physical Review 57, pp. 641-645. Jones, H. F. [1994]: Groups, Representations and Physics, Bristol, Institute of Physics Press Joshi, A. W. [1985]: Elements of Group Theory, 3rd edition, New Delhi, Wiley Eastern.
McIntosh, H.V. [1971]: 'Symmetry and Degeneracy' in E. M. Loebl (ed), Group Theory and Its Applications, vol. 2, N.Y., Academic Press.
Rosen, J. [1995]: Symmetry in Science, Springer, N.Y. 1995
Sternberg, S. [1994]: Group Theory and Physics, Cambridge, Cambridge Univiversity Press.
von Neumann, J. [1955]: Mathematical Foundations of Quantum Mechanics, English trans by R. T. Beyer, Princeton, Princeton University Press
Weinberg, S. [1995]: The Quantum Theory of Fields, Vol.1, Cambridge, Cambridge University Press Weyl, H. [1952]: Symmetry, Princeton, Princeton University Press. Wigner, E.P. [1959]: Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, English trans. J.J. Griffin, N.Y., Academic Press.

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© m chester, August 1999, Occidental, CA
chester@physics.ucla.edu