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Matrix Elements of the Octet Operator of SU 3
1.M. Gell‐Mann, Cal. Tech. Rept. CTSL‐20, 1961 (unpublished),
1.and Phys. Rev. 125, 1067 (1962).
1.S. Okubo, Progr. Theoret. Phys. (Kyoto) 27, 949;
1.S. Okubo, 28, 24 (1962).
2.By regular tensor operator, we mean the tensor operator which transforms under the group according to the regular representation of the group. For this representation is the octet or the IR (1, 1).
3.See, for example, C. Dullemond, A. J. Macfarlane, and E. C. G. Sudarshan, Phys. Rev. Letters 10, 423 (1961);
3.and E. C. G. Sudarshan, Proc. Athens Conference on Recently Discovered Resonant Particles, Ohio University, Athens, Ohio, 1963.
4.Particle mixtures in a theory with charge‐independent, strong interactions and electromagnetic interaction have been considered by various authors, e.g., S. Okubo, Nuovo Cimento 16, 963 (1960);
4.S. L. Glashow, Phys. Rev. Letters 7, 469 (1961).
4.Use of the mixing to break exact invariance has been studied by J. J. Sakurai, Phys. Rev. Letters 9, 472 (1962);
4.S. Okubo, Phys. Letters 5, 165 (1962);
4.S. L. Glashow, Phys. Rev. Letters 11, 48 (1963).
5.E. U. Condon and G. H. Shortley, Theory of Atomic Spectra (Cambridge University Press, Cambridge, England, 1955), p. 61.
5.See also E. Feenberg and G. E. Pake, Notes on the Quantum Theory of Angular Momentum (Addison‐Wesley Publishing Company, Inc., Cambridge, Massachusetts, 1953), p. 29.
5.The original treatment appears in M. Born and P. Jordan, Elementare Quantenmechanik (Springer‐Verlag, Leipzig, 1930).
6.A. J. Macfarlane and E. C. G. Sudarshan, Proc. Stanford Conference on Nucleon Structure, Stanford, California, 1963 (to be published), and Electromagnetic Properties of Stable Particles and Resonances in the Unitary Symmetry Theory (to be published).
6.See also S. P. Rosen, Phys. Rev. Letters 11, 100 (1963)
6.and C. A. Levinson, H. J. Lipkin and S. Meshkow, Phys. Letters (to be published).
7.It will be convenient to abbreviate commutation rule and irreducible representation to CR and IR.
8.See G. Racah, Princeton lectures, 1951 (unpublished).
9.R. E. Behrends, J. Dreitlein, C. Fronsdal, and B. W. Lee, Rev. Mod. Phys. 34, 1 (1962).
10.A. J. Macfarlane, E. C. G. Sudarshan, and C. Dullemond, Nuovo Cimento 30, 845 (1963).
11.L. C. Biedenharn, Phys. Letters 3, 69 (1962).
12.M. Harvey and J. P. Elliott, Proc. Roy. Soc. London A 272, 557 (1963).
12.K. T. Hecht, “Reduction Coefficients and Fractional Parentage Coefficients,” (University of Michigan preprint, 1963).
13.See, e.g., M. E. Rose, Elementary Theory of Angular Momentum (John Wiley & Sons, Inc., New York, 1957), Eqs. (3.16a) and (3.17a).
14.A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1957), Eq. (5.5.3).
15.Y. Ne’eman, Nucl. Phys. 26, 222 (1961);
15.M. Gell‐Mann, footnote 1.
16.J. Ginibre, J. Math. Phys. 4, 720 (1963).
17.A. R. Edmonds, Proc. Roy. Soc. London A 268, 567 (1962).
18.D. R. Speiser, Proc. Istanbul Summer School, Istanbul, 1962 (to be published).
19.Familiar results can be verified to follow from this.
20.See Eq. (A.8) of the first of the papers by Okubo cited in footnote 1.
21.M. Gell‐Mann, footnote 1.
22.G. Racah, Phys. Rev. 76, 1352 (1949).
23.For a discussion of antiunitary operators see E. P. Wigner, Group Theory (Academic Press Inc., New York, 1959), p. 325,
23.and A. Messiah, Quantum Mechanics (North‐Holland Publishing Company, Amsterdam, 1962), Vol. II, p. 633.
24.Using Eq. (XV. 22) of Messiah (footnote 23), we write the definition of the adjoint of an antilinear operator as where and are any two states. If and with M a linear operator belonging to the set (2.6), we can develop Here we have used
25.See Ref. 10, Sec. 2.
26.L. C. Biedenharn, J. Math. Phys. 4, 436 (1963), Appendix.
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