On the structure of the canonical tensor operators in the unitary groups. I. An extension of the pattern calculus rules and the canonical splitting in U(3)
1.G. E. Baird and L. C. Biedenharn, J. Math. Phys. (N.Y.) 5, 1730 (1964).
2.See Refs. 1, 3, and 4 for a detailed explanation of the notations of this paper.
3.J. D. Louck, Am. J. Phys. 38, 3 (1970).
4.J. D. Louck and L. C. Biedenharn, J. Math. Phys. (N.Y.) 2, 2368 (1970).
5.L. C. Biedenharn, in Spectroscopic and Group Theoretical Methods in Physics (Racah Memorial Volume), edited by F. Block et al. (North‐Holland, Amsterdam, 1968), p. 59.
6.J. D. Louck and L. C. Biedenharn, J. Math. Phys. (N.Y.) 12, 173 (1971).
7.For convenience of expression, we do not always invert operator patterns. Note that in the notation for a projective operator the upper pattern is inverted, but the lower one is not. One must be very careful, however, not to confuse operator patterns with Gel’fand patterns despite their one‐to‐one correspondence.
8.E. Artin, Geometric Algebra (Interscience, New York, 1964).
9.The importance of the null space has been emphasized in Refs. 3 and 4, where a more complete discussion will be found.
10.The concept of indecomposability is discussed in some detail in Ref. 4.
11.L. C. Biedenharn, A. Giovannini, and J. D. Louck, J. Math. Phys. (N.Y.) 8, 691 (1967).
12.G. E. Baird and L. C. Biedenharn, J. Math. Phys. (N.Y.) 6, 1847 (1965).
13.J. A. Castilho Alcarás, L. C. Biedenharn, K. T. Hecht, and G. Neely, Ann. Phys. (N.Y.) 60, 85 (1970).
14.S. J. Ališauskas, A.‐A. A. Jucys, and A. P. Jucys, J. Math. Phys. (N.Y.) 13, 1349 (1972).
15.E. Chacón, M. Ciftan, and L. C. Biedenharn, J. Math. Phys. (N.Y.) 13, 577 (1972).
16.L. C. Biedenharn and J. D. Louck, Commun. Math. Phys. 8, 89 (1968).
17.For the convenience of the reader, this résumé repeats some previously published material (Secs. 2A, 2B, 2D) in order to summarize in a reasonably self‐contained way the three basic techniques on which the present work depends.
18.The shift is read off from the lower operator pattern in the same way in which is read off from the upper operator pattern [cf. Eqs. (1.3)]. However, the nth component of the shift of the lower pattern is not included in the definition of Let us also note that an extremal operator pattern is one whose Δ pattern is a permutation of the irrep labels (see Ref. 16).
19.Papers which relate most directly to our presentation include: J. Schwinger, Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. van Dam (Academic, New York, 1965), p. 229;
19.V. Bargmann, Rev. Mod. Phys. 34, 829 (1962);
19.T. A. Brody, M. Moshinsky, and I. Renero, J. Math. Phys. 6, 1540 (1965) (and references therein).
20.I. M. Gel’fand and M. I. Graev, Izv. Akad. Nauk SSSR Ser. Mat. 29, 1329 (1965).
21.E. P. Wigner, Group Theory and Its Applications to the Quantum Mechanics of Atomic Spectra, translated by J. J. Griffin (Academic, New York, 1959), p. 190.
22.It is proved in II that is the operator pattern (4.4a) if while it is the operator pattern (4.4b) if
23.I. M. Gel’fand, R. A. Minlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon, Oxford, England, and Macmillan, New York, 1963), p. 362.
24.See Footnote 27 of Ref. 4 for a detailed description of our phase convention.
25.This fact was brought to our attention long ago by Dr. Mark Bolsterli of the Los Alamos Scientific Laboratory.
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