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by S. Deser, M. Grisaru, and H. Pendleton, MIT Press, Cambridge, Mass., 1970; and S. B. Treiman in "Lectures on Current Algebra and Its Applications," Princeton University Press, 1972. Phenomenological lagrangians are discussed by S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys., vol. 41, p. 531, 1969. References to specific points discussed in the text are as follows.
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Sum rules N. Cabibbo and L. A. Radicati, Phys. Lett., vol. 19, p. 697, 1966. S. Adler, Phys. Rev., vol. 143, p. 1144, 1966. J. D. Bjorken, Phys. Rev. Lett., vol. 16, p. 408, 1966. Consequences of PCAC M. L. Goldberger and S. B. Treiman, Phys. Rev., vol. 110, p. 1178, 1958. Low-energy theorems F. E. Low, Phys. Rev., vol. 96, p. 1428, 1954, and vol. 110, p. 974,1958. M. Gell-Mann and M. L. Goldberger, Phys. Rev., vol. 96, p. 1433, 1954. S. D. Drell and A. C. Hearn, Phys. Rev. Lett., vol. 16, p. 908, 1966. S. L. Adler, Phys. Rev., vol. 140, ser. B, p. 736, 1965. W. I. Weisberger, Phys. Rev., vol. 143, p. 1302, 1966. Soft pion theorems S. Adler, Phys. Rev., vol. 139, ser. B, p. 1638, 1965. S. Weinberg, Phys. Rev. Lett., vol. 17, p. 616, 1966. SU(3) x SU(3) breaking M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev., vol. 175, p. 2195,1968.
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The (J model was proposed by M. Gell-Mann and M. Levy, Nuov. Cim., vol. 16, p. 705, 1960, and is reviewed by B. W. Lee in "Chiral Dynamics," Gordon and Breach, 1972. The neutral pion decay was considered by J. Steinberger, Phys. Rev., vol. 76, p. 1180, 1949, and in the context of current algebra by D. G. Sutherland, Nucl. Phys., ser. B, vol. 2, p. 433, and by M. Veltman, Proc. Roy. Soc., ser. A, vol. 301, p. 107, 1967. Anomalies appear in J. Schwinger's work, Phys. Rev., vol. 82, p. 664, 1951, and were analyzed by S. Adler, Phys. Rev., vol. 177, p. 2426, 1969, and by J. S. Bell and R. Jackiw, Nuov. Cim., vol. 60, ser. A, p. 47, 1969. The general structure of anomalies is discussed by W. A. Bardeen, Phys. Rev., vol. 184, p. 1848, 1969. See also the Brandeis 1970 Lectures by S. Adler, edited by S. Deser, M. Grisaru, and H. Pendleton, MIT Press, Cambridge, Mass., 1970, and R. Jackiw in "Lectures on Current Algebra and Its Applications," Princeton University Press, 1972. Massless two-dimensional quantum electrodynamics was solved by J. Schwinger, Phys. Rev., vol. 128, p. 2425, 1962.
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After reviewing their geometrical background we present in detail the quantum theory of Yang-Mills fields. The emphasis is on the quantization and the renormalization procedures, both in the symmetric and spontaneously broken cases. Vacuum degeneracy and classical solutions are briefly evoked. Applications to the unified theory of weak and electromagnetic interactions are studied in the framework of the Weinberg-Salam model.
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12-1 CLASSICAL THEORY
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The generalization of gauge in variance to nonabelian groups was a very appealing and natural idea, first proposed by Yang and Mills in 1954. In spite of the large amount of work they inspired, these nonabelian gauge theories had a rather slow development until the end of the 1960s. The important problems of quantization, renormalization, and mass generation were then brilliantly solved. It is believed that a theory of this sort can provide a unified description of weak and electromagnetic interactions, and possibly also of strong interactions. These models offer all possible intricacies. Their quantization and renormalization are difficult, they exhibit remarkable mechanisms of symmetry breaking, and they have a unique behavior at short distances, while most aspects of their long-range behavior are not yet fully elucidated. It has recently been realized that even the classical theory has a fascinating complexity. It is almost impossible to present such a wealth of information within a few dozens of pages. We shall restrict ourselves to the basic topics and only mention some of the possible developments and problems.
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