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Classical and Semi-Classical Solutions of the Yang-Mills Theory

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Particles and Fields

Abstract

This Review summarizes what is known at present about classical solutions to Yang-Mills theory both in Euclidean and Minkowski space. The quantal meaning of these solutions is also discussed. Solutions in Euclidean space expose multiple vacua and tunnelling of the quantum theory. Those in Minkowski space provide a semi-classical spectrum for a conformal generator.

This work is supported in part through funds provided by ERDA under Contract EY-76-C-02-3069.*000

Material based on lectures presented at: Workshop on Theoretical Problems in Quantum Chromodynamics Crete, Greece, 20–30 June, 1977 Workshop on Solitons Salerno, Italy, 27 June – 23 July, 1977

Banff Summer Institute on Particles and Fields Banff, Canada, 26 August – 3 September, 1977

Mystery of the Soliton Warsaw, Poland, 26–30 September, 1977

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References

  1. For previous reviews see R. Rajaraman, Phys. Rep. 21C, 227 (1975); R. Jackiw, Acta Physica Pol. B6, 919 (1975); S. Coleman, Erice Lectures (1975); J.-L. Gervais and A. Neveu, Phys. Rep. 23C, 237 (1976); R. Jackiw, Rev. Mod. Phys. 49, 681 (1977); S. Coleman, Erice Lectures (1977).

    Article  ADS  Google Scholar 

  2. S. Coleman, Erice Lectures (1975); H. Pagels, Phys. Lett. 68B, 466 (1977); S. Coleman, Commun. Math. Phys. 55, 113 (1977).

    Google Scholar 

  3. A. Belavin, A. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. 59B, 85 (1975).

    MathSciNet  ADS  Google Scholar 

  4. G.’ t Hooft, these Proceedings.

    Google Scholar 

  5. R. Jackiw and C. Rebbi, Phys. Rev. D14, 517 (1976).

    MathSciNet  ADS  Google Scholar 

  6. S. Adler, Phys. Rev. D6, 3445 (1972); S. Adler, Phys. Rev. D8, 2400 (1973).

    ADS  Google Scholar 

  7. F. Ore, Phys. Rev. D15, 470 (1977).

    MathSciNet  ADS  Google Scholar 

  8. F. Ore, Phys. Rev. D16, 1041 (1977); Phys. Rev. D (in press).

    MathSciNet  ADS  Google Scholar 

  9. S. Chadha, A. D’Adda, P. DiVecchia and F. Nicodemi, Phys. Lett. 67B, 103 (1977).

    MathSciNet  ADS  Google Scholar 

  10. G. ’t Hooft, Phys. Rev. D14, 3432 (1976).

    ADS  Google Scholar 

  11. F. Wilczek in Quark Confinement and Field Theory, D. Stump and D. Weingarten, eds., (Wiley, New York, 1977); F. Corrigan and D. Fairlie, Phys. Lett. 67B, 69 (1977).

    Google Scholar 

  12. R. Jackiw, C. Nohl and C. Rebbi, Phys. Rev. D15, 1642 (1977).

    ADS  Google Scholar 

  13. A less general version of this result was obtained by G. ’t Hooft, Coral Gables proceedings, 1977.

    Google Scholar 

  14. The first construction of multi-pseudoparticle configurations with arbitrary Pontryagin index was achieved, with a method different from the one described here, by E. Witten, Phys. Rev. Lett. 38, 121 (1976). His pseudoparticles are distributed in an 0(3) symmetric configuration.

    Article  ADS  Google Scholar 

  15. R. Jackiw and C. Rebbi, Phys. Lett. 67B, 189 (1977).

    MathSciNet  ADS  Google Scholar 

  16. R. Jackiw and C. Rebbi, Phys. Rev. D16, 1052 (1977).

    MathSciNet  ADS  Google Scholar 

  17. R. Ward, Phys. Lett 61A, 81 (1977); C. N. Yang, Phys. Rev. Lett. 38, 1377 (1977); M. Atiyah and R. Ward, Commun. Math. Phys. 55, 117 (1977).

    ADS  Google Scholar 

  18. B. Grossman, Phys. Lett. 61A, 86 (1977).

    MathSciNet  ADS  Google Scholar 

  19. S. Adler in Lectures on Elementary Particles and Quantum Field Theory, Vol, 1, S. Deser, M. Grisaru and H. Pendleton, eds, (MIT Press, Canbridge, 1970); S. Treiman, R. Jackiw, D. Gross, Lectures on Current Algebra and Its Applications, (Princeton University Press, Princeton, 1972).

    Google Scholar 

  20. An example of the Dirac equation in a potential with no definite duality properties has been analyzed by L. Dolan, Harvard University preprint. No zero-eigenvalue modes are found.

    Google Scholar 

  21. S. Coleman (unpublished) suggested that the index theorem may be derived from the axial-vector anomaly. For further discussion see L. Brown, R. Carlitz and C. Lee, Phys. Rev. D16, 417 (1977); as well as Jackiw and Rebbi, Ref. 16.

    ADS  Google Scholar 

  22. R. Jackiw and C. Rebbi, Phys. Rev. D13, 3398 (1976).

    MathSciNet  ADS  Google Scholar 

  23. For further discussion of the index theorem, signature defect, etc., see J. Kiskis, Phys. Rev. D15, 2329 (1977); M. Ansourian, Phys. Lett. (in press); B. Schroer and K. Nielsen, Nordita preprint. For a discussion from the mathematical point of view see M. Atiyah, V. Patodi, and I. Singer, Math. Proc. Camb. Phil. Soc. 77. 43 (1975); 78, 405 (1975); 79, 71 (1976).

    Google Scholar 

  24. This application of the index theory is due to Brown, Carlitz and Lee, Ref. 21. It is also possible to apply index theory directly to the small deformation equation, without reference to fermions; see A. Schwarz, Phys. Lett. 67B, 172 (1977); M. Atiyah, N. Hitchin and I. Singer, Proc. Nat. Acad. Sci. USA 74, 2662 (1977).

    ADS  Google Scholar 

  25. D. Freedman and D. Gross, unpublished; Chadha, D’Adda, DiVecchia and Nicodemi, Ref. 9; B. Zumino, Phys. Lett. 69B, 369 (1977).

    ADS  Google Scholar 

  26. M. Lüscher, DESY preprint; B. Schechter, Phys. Rev. D (in press).

    Google Scholar 

  27. V. deAlfaro, S. Fubini and G. Furlan, Phys. Lett. 65B, 163 (1976); see also M. Cervero, L. Jacobs and C. Nohl, Phys. Lett. B (in press); W. Bernreuther, MIT preprint.

    MathSciNet  ADS  Google Scholar 

  28. For a review see R. Jackiw, Springer Tracts in Modern Physics, Vol. 62, G. Höhler, ed., (Springer-Verlag, Berlin, 1972).

    Google Scholar 

  29. S. Fubini, A. Hanson and R. Jackiw, Phys. Rev. D7, 1732 (1973).

    ADS  Google Scholar 

  30. S. Fubini, Nuovo Cim. 34A, 521 (1976).

    Article  ADS  Google Scholar 

  31. V. deAlfaro, S. Fubini, G. Furlan, Nuovo Cim. 34A, 569 (1976).

    Article  ADS  Google Scholar 

  32. C. Rebbi, Phys. Rev. D, in press.

    Google Scholar 

  33. V. Maslov, Teor. Mat. Fiz. 2, 30 (1970) [Theor. Math. Phys. 2, 21 (1970)]; M. Gutzwiller, J. Math. Phys. 12, 343 (1971).

    Google Scholar 

  34. R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D10, 4114 1974); 10, 4130 (1974); V. Korepin and L. Faddeev, Teor. Mat. Fiz. 25, 147 (1975) [Theor. Math. Phys. 25, 1039 (1976)].

    ADS  Google Scholar 

  35. C. Nohl, Ann. Phys. 96, 234 (1976); A. Klein and F. Krejs, Phys. Rev. D13, 3282 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. S. Coleman and R. Jackiw, Ann. Phys. 67, 552 (1971).

    Article  MathSciNet  ADS  Google Scholar 

  37. C. Lovelace, Nucl. Phys. B99, 109 (1975).

    Article  MathSciNet  ADS  Google Scholar 

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© 1978 Plenum Press, New York

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Jackiw, R., Nohl, C., Rebbi, C. (1978). Classical and Semi-Classical Solutions of the Yang-Mills Theory. In: Boal, D.H., Kamal, A.N. (eds) Particles and Fields. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-4000-3_8

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  • DOI: https://doi.org/10.1007/978-1-4613-4000-3_8

  • Publisher Name: Springer, Boston, MA

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