Symmetry principles in old and new physics
Author:
Eugene P. Wigner
Journal:
Bull. Amer. Math. Soc. 74 (1968), 793815
MathSciNet review:
1566474
Fulltext PDF
References 
Additional Information
 1.
J. Willard Gibbs', Elementary principles in statistical mechanics, a finished work 284 pages long, was written in 1901 and published by Yale University in 1902. It was reprinted by Longmans, Green (New York, 1928). An appraisal, by A. Haas (or, rather, two appraisals, a short and a long one) remain worth reading (in Vol. 2 of A commentary on the writings of J. Willard Gibbs, Yale University Press, 1936). Gibbs' first writing on the concept to which his name is rightly attached, on phase space, is only one page long. It was published in 1884 (Proc. Amer. Assoc. 33, 57) and is reprinted, as page 16, in Vol. 2 of The scientific papers of Willard Gibbs (Longmans Green, London, 1906).
 2.
See, e.g., F. P. Bowden and D. Tabor, Friction and lubrication (Wiley, New York, 1956).
 3.
Il Saggiatore, (Volume VI of Le opere di Galileo Galilei, Firenze, 1896) Section 6: Egei e scritto in lingua matematica. . . .
 4.
J. F. C. Hessel's paper was published, originally, in Gehler's Physikalische Wörterbücher (Leipzig, 1830); it is reprinted in Ostwald's Klassiker der exakten Naturwissenschaften No. 89 (Leipzig, 1897), see p. 91 ff. The history of crystallography was described by P. Groth in his Entwicklungsgeschichte der mineralogischen Wissenschaften (Berlin, 1926). A modern history, very wide ranging, is J. G. Burke's Origins of the science of crystals (University of California Press, 1966).
 5.
R. J. Haüy's most relevant article is that in the Journ. de Physique 20 (1782), 33. For a complete bibliography, see J. G. Burke, 1. c. p. 190. Actually, Burke expresses doubts (see pp. 8384) concerning Haüy's independence from his predecessors, in particular from T. Bergman.
 6.
Actually, N. Steno's, De solido intra solidem naturaliter contento dissertationis prodromus (Florence, 1669) only contains the germs of the ideas of the crystal lattice.
 7.
A. Schönflies, Kristallsysteme und Kristallstruktur (Leipzig, 1891); E. S. Fedorov, Zap. Min. Obsh. (Trans. Min. Soc.) 28 (1891), 1.
 8.
P. Groth, Physikalische Krystallographie (W. Engelman, Leipzig, 1905). A more openly group theoretical attitude is adopted by W. Voigt in his Lehrbuch der Kristallphysik (B. G. Teubner, Leipzig, 1910).
 9.
According to classical, that is nonrelativistic theory, the equations of motion remain invariant if the three positional coordinates x_{} (i = 1, 2, 3) are subjected to a Galilei transformation where the O_{} form a 3×3 orthogonal matrix and the time t is invariant or subject to a change of origin . The vector v gives the velocity of the two coordinate systems with respect to each other; the vector a the displacement of the origin of the second coordinate system with respect to the first one at t=0. The corresponding invariance transformation of relativity theory, the Poincaré transformation, treats the time more on a par with the space coordinates. It introduces, instead of t,the variable x_{0}=ct (where c is the velocity of light). The transformation is, in terms of these where L is a Lorentz transformation, i.e. an element of O(1, 3). The Poincaré transion is called also inhomogeneous Lorentz transformation.
 10.
This is now a commonplace statement. It is due, originally, in the precise form stated, to J. v. Neumann. See his Mathematische Grundlagen der Quantenmechanik (J. Springer, Berlin, 1932). English translation by R. T. Beyer, Princeton University Press, 1955.
 11.
This phase space is the mathematical structure introduced by W. Gibbs and mentioned in Reference [l].
 12.
V.
Bargmann, On unitary ray representations of continuous groups,
Ann. of Math. (2) 59 (1954), 1–46. MR 0058601
(15,397b)
 13.
This was shown, first, by the present writer, Ann. of Math. 40 (1939), 149.
 14.
Gelfand, Naimark, and their collaborators are extremely prolific writers. The most important books, from our point of view, are: I. M. Gelfand, R. A. Minlos, and Z. Y. Shapiro, Representations of the rotation and Lorentz groups and their applications, Macmillan, New York, 1963; M. A. Naimark, Linear representations of the Lorentz group, Pergamon Press, London, 1964; I. M. Gelfand and M. A. Neumark, Unitäre Darstellungen der klassischen Gruppen, Akademie Verlag, Berlin. See also the articles of these authors in Vols. 2 and 36 of the Amer. Math. Soc. Translations (Amer. Math. Soc, Providence, R. I., 1956 and 1964). HarishChandra's articles are too numerous for a complete listing and no review of his results is available in book form. His early papers appeared in the Proc. Nat. Acad. Sci. (Vols. 3740, 195154). Among his later writings, we wish to mention his Invariant eigendistributions on a semisimple Lie algebra (Inst. Hautes Études Sci. Publ. Math. No. 27, p. 5) and his article in Ann. of Math. 83 (1966), 74.
 15.
A. Young, Proc. London Math. Soc. 33 (1900), 97; 34 (1902), 361; G. Frobenius, Sitzungsberichte Preuss. Akad. Wiss. 1903, p. 328; I. Schur, ibid. 1908, p. 64. See for a physicist's approach, A. J. Coleman's article in Advances of quantum chemistry (Academic Press, New York, 1966).
 16.
To be published in the Racah Memorial Volume (NorthHolland, Amsterdam, 1968), p. 131.
 17.
R. de L. Kronig, Zeits. f. Physik 31 (1925), 885; 33 (1925), 261; H. N. Russell, Proc. Nat. Acad. Sci. 11 (1925), 314; A. Sommerfeld and H. Hönl, Sitzungsberichte Preuss. Akad. Wiss., 1925, p. 141.
 18.
The Kronecker product as the representative of the union of two physical systems was implicit already in Schrödinger's papers. See his Abhandlungen zur Wellentnechanik, Leipzig, 1927. It was made explicit and precise by von Neumann, [10, Chapter VI, §2].
 19.
L. C. Biedenharn, J. Math. Phys. 4 (1963), 436 and subsequent papers, ending with L. C. Biedenharn, A. Giovanni, J. D. Louck, ibid. 8 (1967), 691. M. Moshinsky, ibid. 7 (1966), 691, J. G. Nagel and M. Moshinsky, ibid 6 (1965), 682 and M. Moshinsky ibid. 7 (1966), 691, M. Kushner and J. Quintanilla, Rev. Mex. de Fisica 16 (1967), 251. For a summary, see L. C. Biedenharn, Racah Memorial Volume [16, p. 173].
 20.
Eugene
P. Wigner, On representations of certain finite groups, Amer.
J. Math. 63 (1941), 57–63. MR 0003417
(2,216a)
 21.
As far as this writer was able to ascertain, the first calculation of these coefficients is given in his Gruppentheorie und ihre Anwendungen etc, (Friedr. Vieweg, Braunschweig, 1931) Chapter XVII. See also the English translation by J. J. Griffin (Academic Press, New York, 1959). The more symmetric expressions for these quantities, the threejsymbols, were introduced in the article which is reprinted in the Quantum theory of angular momentum [20, pp. 89133]. For an extensive tabulation of the numerical values of these coefficients, see, e.g., Rotenberg, Bivins, Metropolis, Wooten, The 3j and 3j symbols, Technology Press, MIT, Cambridge, 1959.
 22.
The first publications containing recoupling (now called Racah) coefficients are due to G. Racah. His work was clearly independent of earlier considerations of the present writer (cf. the second article of [20]). They arose in connection with problems of atomic spectra. Cf. Phys. Rev. 61 (1942), 186; 62 (1942), 438; 63 (1943), 367; 76 (1949), 1352. These articles are all reprinted in the Quantum theory of angular momentum [20]. For a numerical table of these coefficients, also called 6jsymbols, see [21].
 23.
T. Regge, Nuovo Cimento 11 (1959), 116; 10 (1958), 544. An extension of these relations was given by R. T. Sharp.
 24.
Quantum theory of angular momentum. A collection of reprints and
original papers, Edited by L. C. Biedenharn and H. van Dam, Academic
Press, New YorkLondon, 1965. MR 0198829
(33 #6983)
 25.
A.
P. Yutsis, I.
B. Levinson, and V.
V. Vanagas, Mathematical apparatus of the theory of angular
momentum, Translated from the Russian by A. Sen and R. N. Sen, Israel
Program for Scientific Translations, Jerusalem, 1962. MR 0193903
(33 #2117)
 26.
A. Chakrabarti, Ann. Inst. Henri Poincaré 1 (1964), 301; K. Kumar, Austral. J. Phys. 19 (1966), 719; J. M. LévyLeblond and M. LévyNahas, J. Math. Phys. 6 (1965), 1372.
 27.
A full listing of all the contributions to the subject would require several pages. Furthermore, some noncompact Lie groups, in particular also the Poincaré group, were the subjects of investigations similar to those described for O_{3}. See, for instance, J. Ginibre, J. Math. Phys. 4 (1963), 720, J. R. Derome and W. T. Sharp, ibid. 6 (1965), 1584; D. R. Tompkins, ibid. 8 (1967), 1502. Articles dealing with other properties of the representations of the Poincaré group and also extensions thereof include J. L. Lomont and M. E. Moses, J. Math. Phys. 5 (1964), 294; 8 (1966), 837; I. Raszillier, Nuovo Cimento 38 (1965), 1928; J. M. LévyLeblond, ibid. 40 (1965), 748; C. George and M. LévyNahas, J. Math. Phys. (1966), 980; J. C. Guillot and J. L. Petit, Helv. Phys. Acta 39 (1966), 281; V. Berzi and V. Goroni, Nuovo Cimento 57 (1967), 207; S. Ström, Ark. Fys. 34 (1967), 215; J. Nilsson and A. Beskow, ibid. 34 (1967), 307; H. Joos and R. Schrader, Comm. Math. Phys. 7 (1968), 21 (the characters of the representations); A. Kihlberg, Nuovo Cimento 53 (1968), 592. Other articles investigate more complex noncompact Lie groups as they underlie the de Sitter space (the groups O(4, 1) and O(3,2), or the situation at very high energy O(4, 2) or U(2,2)). See the very early paper of L. H. Thomas, Ann. of Math. 42 (1941), 113. Corrections to this paper and extensions thereof were given by T. D. Newton, Ann. of Math. 51 (1950), 730 and by J. Dixmier, Bull. Soc. Math. France 89 (1960), 9. Other contributions are due to J. B. Ehrman, Proc. Cambridge Philos. Soc. 53 (1957), 290; several articles of A. Kihlberg and S. Ström, including some mentioned before but in particular Ark. Fys. 31 (1966), 491; W. Rühl, Nuovo Cimento 44 (1966), 572; A. Chakrabarti, J. Math. Phys. 7 (1966), 949; A. J. Macfarlane, L. O'Raifeartaigh and P. S. Rao, ibid. 8 (1967), 536; O. Nachtman, Acta Physica Austriaca 25 (1967), 118; H. Bacry, Comm. Math. Phys. 5 (1967), 97. The Trieste Institute of the International Atomic Energy Agency is intensely working on the subject and has issued many reports thereon. The authors include R. Delbourgo, A. Salam and J. Strathdee; R. L. Anderson, R. Raczka, M. A. Rashid and P. Winternitz; D. T. Stoyanovand I. T. Todorov. A rather complete review of the more important articles from the mathematical point of view, up to 1965, was presented by G. A. Pozzi, Nuovo Cimento Suppl. 4 (1966), 37. See also H. Baumgärtel, Wiss. Z. Humboldt Univ., Berlin, Math.Naturw. Reihe 13 (1964), 881.
 28.
Y.
Ne′eman, Derivation of strong interactions from a gauge
invariance, Nuclear Phys. 26 (1961), 222–229.
MR
0129821 (23 #B2856)
 29.
L. O'Raifeartaigh, Phys. Rev. 139B (1965), 1952. A mathematically more precise formulation of the O'Raifeartaigh theorem was given by R. Jost, Helv. Phys. Acta 39 (1966), 369. For an extension of the theorem, see I. Segal, J. Functional Anal. 1 (1967), 1; A. Galindo, J, Math. Phys. 8 (1967), 768.
 30.
S. Okubo, Progr. Theor. Phys. 27 (1962), 949; M. GellMann, Phys. Rev. 125 (1962), 1067.
 31.
The interested reader will find an illuminating introduction to, and a collection of, articles dealing with the problems of the extension of the Poincaré group in F. J. Dyson's Symmetry groups in nuclear and particle physics, Benjamin, New York, 1966. A more mathematically oriented review of the "higher symmetries" is given in M. Gourdin's book, Unitary symmetries and their applications to high energy physics, NorthHolland, Amsterdam, 1967. See also A. O. Barut, Proc. Seminar on High Energy Physics and Elementary Particles, Trieste, 1965 (International Atomic Energy Agency, Vienna, 1965).
 1.
 J. Willard Gibbs', Elementary principles in statistical mechanics, a finished work 284 pages long, was written in 1901 and published by Yale University in 1902. It was reprinted by Longmans, Green (New York, 1928). An appraisal, by A. Haas (or, rather, two appraisals, a short and a long one) remain worth reading (in Vol. 2 of A commentary on the writings of J. Willard Gibbs, Yale University Press, 1936). Gibbs' first writing on the concept to which his name is rightly attached, on phase space, is only one page long. It was published in 1884 (Proc. Amer. Assoc. 33, 57) and is reprinted, as page 16, in Vol. 2 of The scientific papers of Willard Gibbs (Longmans Green, London, 1906).
 2.
 See, e.g., F. P. Bowden and D. Tabor, Friction and lubrication (Wiley, New York, 1956).
 3.
 Il Saggiatore, (Volume VI of Le opere di Galileo Galilei, Firenze, 1896) Section 6: Egei e scritto in lingua matematica. . . .
 4.
 J. F. C. Hessel's paper was published, originally, in Gehler's Physikalische Wörterbücher (Leipzig, 1830); it is reprinted in Ostwald's Klassiker der exakten Naturwissenschaften No. 89 (Leipzig, 1897), see p. 91 ff. The history of crystallography was described by P. Groth in his Entwicklungsgeschichte der mineralogischen Wissenschaften (Berlin, 1926). A modern history, very wide ranging, is J. G. Burke's Origins of the science of crystals (University of California Press, 1966).
 5.
 R. J. Haüy's most relevant article is that in the Journ. de Physique 20 (1782), 33. For a complete bibliography, see J. G. Burke, 1. c. p. 190. Actually, Burke expresses doubts (see pp. 8384) concerning Haüy's independence from his predecessors, in particular from T. Bergman.
 6.
 Actually, N. Steno's, De solido intra solidem naturaliter contento dissertationis prodromus (Florence, 1669) only contains the germs of the ideas of the crystal lattice.
 7.
 A. Schönflies, Kristallsysteme und Kristallstruktur (Leipzig, 1891); E. S. Fedorov, Zap. Min. Obsh. (Trans. Min. Soc.) 28 (1891), 1.
 8.
 P. Groth, Physikalische Krystallographie (W. Engelman, Leipzig, 1905). A more openly group theoretical attitude is adopted by W. Voigt in his Lehrbuch der Kristallphysik (B. G. Teubner, Leipzig, 1910).
 9.
 According to classical, that is nonrelativistic theory, the equations of motion remain invariant if the three positional coordinates x_{} (i = 1, 2, 3) are subjected to a Galilei transformation where the O_{} form a 3×3 orthogonal matrix and the time t is invariant or subject to a change of origin . The vector v gives the velocity of the two coordinate systems with respect to each other; the vector a the displacement of the origin of the second coordinate system with respect to the first one at t=0. The corresponding invariance transformation of relativity theory, the Poincaré transformation, treats the time more on a par with the space coordinates. It introduces, instead of t,the variable x_{0}=ct (where c is the velocity of light). The transformation is, in terms of these where L is a Lorentz transformation, i.e. an element of O(1, 3). The Poincaré transion is called also inhomogeneous Lorentz transformation.
 10.
 This is now a commonplace statement. It is due, originally, in the precise form stated, to J. v. Neumann. See his Mathematische Grundlagen der Quantenmechanik (J. Springer, Berlin, 1932). English translation by R. T. Beyer, Princeton University Press, 1955.
 11.
 This phase space is the mathematical structure introduced by W. Gibbs and mentioned in Reference [l].
 12.
 V. Bargmann, Ann. of Math. 59 (1954), 1. MR 58601
 13.
 This was shown, first, by the present writer, Ann. of Math. 40 (1939), 149.
 14.
 Gelfand, Naimark, and their collaborators are extremely prolific writers. The most important books, from our point of view, are: I. M. Gelfand, R. A. Minlos, and Z. Y. Shapiro, Representations of the rotation and Lorentz groups and their applications, Macmillan, New York, 1963; M. A. Naimark, Linear representations of the Lorentz group, Pergamon Press, London, 1964; I. M. Gelfand and M. A. Neumark, Unitäre Darstellungen der klassischen Gruppen, Akademie Verlag, Berlin. See also the articles of these authors in Vols. 2 and 36 of the Amer. Math. Soc. Translations (Amer. Math. Soc, Providence, R. I., 1956 and 1964). HarishChandra's articles are too numerous for a complete listing and no review of his results is available in book form. His early papers appeared in the Proc. Nat. Acad. Sci. (Vols. 3740, 195154). Among his later writings, we wish to mention his Invariant eigendistributions on a semisimple Lie algebra (Inst. Hautes Études Sci. Publ. Math. No. 27, p. 5) and his article in Ann. of Math. 83 (1966), 74.
 15.
 A. Young, Proc. London Math. Soc. 33 (1900), 97; 34 (1902), 361; G. Frobenius, Sitzungsberichte Preuss. Akad. Wiss. 1903, p. 328; I. Schur, ibid. 1908, p. 64. See for a physicist's approach, A. J. Coleman's article in Advances of quantum chemistry (Academic Press, New York, 1966).
 16.
 To be published in the Racah Memorial Volume (NorthHolland, Amsterdam, 1968), p. 131.
 17.
 R. de L. Kronig, Zeits. f. Physik 31 (1925), 885; 33 (1925), 261; H. N. Russell, Proc. Nat. Acad. Sci. 11 (1925), 314; A. Sommerfeld and H. Hönl, Sitzungsberichte Preuss. Akad. Wiss., 1925, p. 141.
 18.
 The Kronecker product as the representative of the union of two physical systems was implicit already in Schrödinger's papers. See his Abhandlungen zur Wellentnechanik, Leipzig, 1927. It was made explicit and precise by von Neumann, [10, Chapter VI, §2].
 19.
 L. C. Biedenharn, J. Math. Phys. 4 (1963), 436 and subsequent papers, ending with L. C. Biedenharn, A. Giovanni, J. D. Louck, ibid. 8 (1967), 691. M. Moshinsky, ibid. 7 (1966), 691, J. G. Nagel and M. Moshinsky, ibid 6 (1965), 682 and M. Moshinsky ibid. 7 (1966), 691, M. Kushner and J. Quintanilla, Rev. Mex. de Fisica 16 (1967), 251. For a summary, see L. C. Biedenharn, Racah Memorial Volume [16, p. 173].
 20.
 E. P. Wigner, Amer. J. Math. 63 (1941), 57. This and a more detailed paper are reprinted in Quantum theory of angular momentum, L. C. Biedenharn and H. Van Dam, Editors, Academic Press, New York, 1965. This volume contains reprints of several articles which played an important role in the formation of the ideas of physicists on representation theory, in particular also articles of G. Racah. G. W. Mackey, Amer. J. Math. 75 (1953), 387; Pacific J. Math. 8 (1958), 503. See also his Mathematical foundations of quantum mechanics Benjamin, New York, 1963. MR 3417
 21.
 As far as this writer was able to ascertain, the first calculation of these coefficients is given in his Gruppentheorie und ihre Anwendungen etc, (Friedr. Vieweg, Braunschweig, 1931) Chapter XVII. See also the English translation by J. J. Griffin (Academic Press, New York, 1959). The more symmetric expressions for these quantities, the threejsymbols, were introduced in the article which is reprinted in the Quantum theory of angular momentum [20, pp. 89133]. For an extensive tabulation of the numerical values of these coefficients, see, e.g., Rotenberg, Bivins, Metropolis, Wooten, The 3j and 3j symbols, Technology Press, MIT, Cambridge, 1959.
 22.
 The first publications containing recoupling (now called Racah) coefficients are due to G. Racah. His work was clearly independent of earlier considerations of the present writer (cf. the second article of [20]). They arose in connection with problems of atomic spectra. Cf. Phys. Rev. 61 (1942), 186; 62 (1942), 438; 63 (1943), 367; 76 (1949), 1352. These articles are all reprinted in the Quantum theory of angular momentum [20]. For a numerical table of these coefficients, also called 6jsymbols, see [21].
 23.
 T. Regge, Nuovo Cimento 11 (1959), 116; 10 (1958), 544. An extension of these relations was given by R. T. Sharp.
 24.
 L. C. Biedenharn, J. M. Blatt, M. E. Rose, Revs. Mod. Phys. 24 (1952), 249; A. R. Edmonds, Angular momentum in quantum mechanics, Princeton University Press, 1957; G. Ponzano, Nuovo Cimento 35 (1965), 1231; 36 (1965), 385; G. Ponzano and T. Regge, Racah Memorial Volume [16]. MR 198829
 25.
 A. Jucys, J. Levinsonas, V. Vanagas, Mathematical apparatus of the theory of angular momentum, English translation published by the Israel Program of Scientific Translations, Jerusalem, 1962. The original, in Russian, was published in Vilnius (Vilna) in 1960. See also W. T. Sharp's Racah algebra and the contraction of groups, Atomic Energy of Canada Ltd. report 1098 (1960). MR 193903
 26.
 A. Chakrabarti, Ann. Inst. Henri Poincaré 1 (1964), 301; K. Kumar, Austral. J. Phys. 19 (1966), 719; J. M. LévyLeblond and M. LévyNahas, J. Math. Phys. 6 (1965), 1372.
 27.
 A full listing of all the contributions to the subject would require several pages. Furthermore, some noncompact Lie groups, in particular also the Poincaré group, were the subjects of investigations similar to those described for O_{3}. See, for instance, J. Ginibre, J. Math. Phys. 4 (1963), 720, J. R. Derome and W. T. Sharp, ibid. 6 (1965), 1584; D. R. Tompkins, ibid. 8 (1967), 1502. Articles dealing with other properties of the representations of the Poincaré group and also extensions thereof include J. L. Lomont and M. E. Moses, J. Math. Phys. 5 (1964), 294; 8 (1966), 837; I. Raszillier, Nuovo Cimento 38 (1965), 1928; J. M. LévyLeblond, ibid. 40 (1965), 748; C. George and M. LévyNahas, J. Math. Phys. (1966), 980; J. C. Guillot and J. L. Petit, Helv. Phys. Acta 39 (1966), 281; V. Berzi and V. Goroni, Nuovo Cimento 57 (1967), 207; S. Ström, Ark. Fys. 34 (1967), 215; J. Nilsson and A. Beskow, ibid. 34 (1967), 307; H. Joos and R. Schrader, Comm. Math. Phys. 7 (1968), 21 (the characters of the representations); A. Kihlberg, Nuovo Cimento 53 (1968), 592. Other articles investigate more complex noncompact Lie groups as they underlie the de Sitter space (the groups O(4, 1) and O(3,2), or the situation at very high energy O(4, 2) or U(2,2)). See the very early paper of L. H. Thomas, Ann. of Math. 42 (1941), 113. Corrections to this paper and extensions thereof were given by T. D. Newton, Ann. of Math. 51 (1950), 730 and by J. Dixmier, Bull. Soc. Math. France 89 (1960), 9. Other contributions are due to J. B. Ehrman, Proc. Cambridge Philos. Soc. 53 (1957), 290; several articles of A. Kihlberg and S. Ström, including some mentioned before but in particular Ark. Fys. 31 (1966), 491; W. Rühl, Nuovo Cimento 44 (1966), 572; A. Chakrabarti, J. Math. Phys. 7 (1966), 949; A. J. Macfarlane, L. O'Raifeartaigh and P. S. Rao, ibid. 8 (1967), 536; O. Nachtman, Acta Physica Austriaca 25 (1967), 118; H. Bacry, Comm. Math. Phys. 5 (1967), 97. The Trieste Institute of the International Atomic Energy Agency is intensely working on the subject and has issued many reports thereon. The authors include R. Delbourgo, A. Salam and J. Strathdee; R. L. Anderson, R. Raczka, M. A. Rashid and P. Winternitz; D. T. Stoyanovand I. T. Todorov. A rather complete review of the more important articles from the mathematical point of view, up to 1965, was presented by G. A. Pozzi, Nuovo Cimento Suppl. 4 (1966), 37. See also H. Baumgärtel, Wiss. Z. Humboldt Univ., Berlin, Math.Naturw. Reihe 13 (1964), 881.
 28.
 Y. Ne'eman, Nuclear Physics 26 (1961), 222; M. GellMann, Phys. Rev. 125 (1962), 1067. See also their reprint collection and review, The eightfold way, Benjamin, New York, 1964. MR 129821
 29.
 L. O'Raifeartaigh, Phys. Rev. 139B (1965), 1952. A mathematically more precise formulation of the O'Raifeartaigh theorem was given by R. Jost, Helv. Phys. Acta 39 (1966), 369. For an extension of the theorem, see I. Segal, J. Functional Anal. 1 (1967), 1; A. Galindo, J, Math. Phys. 8 (1967), 768.
 30.
 S. Okubo, Progr. Theor. Phys. 27 (1962), 949; M. GellMann, Phys. Rev. 125 (1962), 1067.
 31.
 The interested reader will find an illuminating introduction to, and a collection of, articles dealing with the problems of the extension of the Poincaré group in F. J. Dyson's Symmetry groups in nuclear and particle physics, Benjamin, New York, 1966. A more mathematically oriented review of the "higher symmetries" is given in M. Gourdin's book, Unitary symmetries and their applications to high energy physics, NorthHolland, Amsterdam, 1967. See also A. O. Barut, Proc. Seminar on High Energy Physics and Elementary Particles, Trieste, 1965 (International Atomic Energy Agency, Vienna, 1965).
Additional Information
DOI:
http://dx.doi.org/10.1090/S000299041968120476
PII:
S 00029904(1968)120476
