Theoretical and Mathematical Physics

, Volume 98, Issue 2, pp 147–161 | Cite as

Quantum projective field theory: Quantum-field analogs of the Euler-Arnol'd equations in projectiveG multiplets

  • D. V. Yur'ev
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Keywords

Field Theory Projective Field 

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References

  1. 1.
    B. B. Mandelbrot, in:Fractals for the Classroom, 1 (eds. H. O. Peiten, H. Jurgens, and D. Saupe), Springer (1992), p. 1.Google Scholar
  2. 2.
    R. Penrose,The Emperor's New Mind, Oxford University Press, Oxford (1989); Ya. A. Smorodinskii,Usp. Fiz. Nauk,161, 203 (1991); T. L. Saati,Lett. Appl. Math.,1, 79 (1988).Google Scholar
  3. 3.
    D. V. Yur'ev,Teor. Mat. Fiz.,92, 172 (1992).Google Scholar
  4. 4.
    D. Juriev,Lett. Math. Phys.,21, 113 (1991);22, 141 (1991); D. V. Yur'ev,Teor. Mat. Fiz.,86, 338 (1991).Google Scholar
  5. 5.
    S. A. Bychkov and D. V. Yur'ev,Usp. Mat. Nauk,46, 167 (1991); S. A. Bychkov,Usp. Mat. Nauk,47, 187 (1992); D. Juriev,J. Math. Phys.,33, 1153 (1992); S. A. Bychkov, S. V. Plotnikov, and D. V. Yur'ev,Usp. Mat. Nauk,47, 153 (1992); D. Juriev,J. Math. Phys.,33, 3112 (1992); D. V. Yur'ev,Teor. Mat. Fiz.,93, 32 (1992).Google Scholar
  6. 6.
    D. Juriev,J. Math. Phys.,33, 2819 (1992);34, No. 4 (1993);Acta Appl. Math. (1993).Google Scholar
  7. 7.
    D. V. Yur'ev,Algebra i Analiz,2, 209 (1990);3, 197 (1991); D. Juriev,J. Math. Phys.,33, 492 (1992);Commun. Math. Phys.,138, 569 (1991);146, 427 (1992);J. Funct. Anal.,101, 1 (1991);J. Math. Phys.,32, 2034 (1991); D. V. Yur'ev,Usp. Mat. Nauk,46, No. 4, 115.Google Scholar
  8. 8.
    A. Alekseev, L. Faddeev, and M. Semenov-Tian-Shansky,Lect. Notes Math.,1510, 148 (1992).Google Scholar
  9. 9.
    M. A. Semenov-Tyan-Shanskii,Teor. Mat. Fiz.,93, 302 (1992).Google Scholar
  10. 10.
    A. Z. Patashinskii and V. L. Pokrovskii,Fluctuation Theory of Phase Transitions [in Russian], Nauka, Moscow (1982); I. Frenkel, J. Lepowsky, and A. Meurman,Unified String Theories, World Scientific, Singapore (1986), p. 533;Vertex Operators in Mathematics and Physics, Springer (1985), p. 231;Mathematical Aspects of String Theory, World Scientific, Singapore (1987), p. 150; R. E. Borcherds,Proc. Nat. Acad. Sci. USA,83, 3068 (1986); I Frenkel, J. Lepowsky, and A. Meurman,Vertex Operator Algebras and the Monster, Academic Press, New York (1988); P. Goddard,Infinite-Dimensional Lie Algebras and Groups, World Scientific, Singapore (1988); I. Frenkel', Dzh. Lepovskii, and A. Merman,Funktsional. Analiz i Ego Prilozhen.,25, 36 (1991); A. J. Feingold, I. B. Frenkel, and J. F. X. Ries,Spinor Construction of Vertex Operator Algebras, Triality andE(1)8, Contemporary Mathematics 121, Am. Math. Soc., Providence (1991).Google Scholar
  11. 11.
    S. A. Bychkov and D. V. Yur'ev,Teor. Mat. Fiz.,97, No. 3 (1993).Google Scholar
  12. 12.
    S. A. Alekseev, L. Faddeev, and S. Shatashvili,J. Geom. Phys.,5, 391 (1988) (reprinted inGeometry and Physics. Essays in Honor of 1. M. Gel'fand (eds. S. Gindikin and I. M. Singer), Elsevier Sci. Publ. and Pitagora Editrice (1991), p. 391); A. Alekseev and S. Shatashvili,Nucl. Phys. B,323, 719 (1989);Commun. Math. Phys.,128, 197 (1990).Google Scholar
  13. 13.
    D. V. Yur'ev,Teor. Mat. Fiz.,37, 74 (1987).Google Scholar
  14. 14.
    S. L. Adler and R. F. Dashen,Current Algebras and Applications to Particle Physics, New York (1968); J. Bernstein,Elementary Particles and Their Currents, San Francisco (1968).Google Scholar
  15. 15.
    V. G. Kac,Infinite-Dimensional Lie Algebras, Cambridge University Press, Cambridge (1985); G. Segal and G. Wilson, “Loop groups and equations of KdV type”,Publ. Math. IHES,61, 5; A. Pressley and G. Segal,Loop Groups, Oxford University Press, Oxford (1988).Google Scholar
  16. 16.
    I. N. Bernshtein, I. M. Gel'fand, and S. I. Gel'fand,Funktsional. Analiz i Ego Prilozhen.,5, 1 (1971).Google Scholar
  17. 17.
    S. L. Luk'yanov,Funktsional. Analiz i Ego Prilozhen.,22, 1 (1988).Google Scholar
  18. 18.
    A. Yu. Morozov,Fiz.EElem. Chastits At. Yadra,23, 174 (1992).Google Scholar
  19. 19.
    P. Goddard and D. Olive, in:Workshop on Unified String Theories, World Scientific, Singapore (1986), p. 214;Interm. J. Mod. Phys. A,1, 303 (1986); Yu. A. Neretin, in:Modern Problems of Mathematics. Fundamental Directions, Vol. 22 [in Russian], VINITI, Moscow (1988), p. 163; I. Bakas,Commun. Math. Phys.,123, 627 (1989).Google Scholar
  20. 20.
    L. C. Biedenharn and J. D. Louck,Angular Momentum in Quantum Mechanics. Theory and Applications, Encycl. Math. Appl., Vol. 8, Addison Wesley (1981);The Racah-Wigner Algebra in Quantum Theory, Encycl. Math. Appl., Vol. 9, Addison Wesley (1981); A. Barut and R. Raczka,Theory of Group Representations and Applications, Warsaw (1977).Google Scholar
  21. 21.
    I. M. Gelfand and A. A. Kirillov,Publ. Math. IHES,92, 5 (1966); I. M. Gel'fand and A. A. Kirillov,Dokl. Akad. Nauk SSSR,180, 775 (1968); A. A. Kirillov,Elements of the Theory of Representations [in Russian], Nauka, Moscow (1972), p. 169; J. Dixmier, in:Universal Enveloping Algebras [Russian translation], Mir, Moscow (1978), p. 139.Google Scholar
  22. 22.
    V. I. Arnol'd,Ann. Inst. Fourier,16, 319; V. I. Arnol'd,Mathematical Methods of Classical Mechanics ( (Graduate Texts in Mathematics, Vol. 60), Springer, New York (1978).Google Scholar
  23. 23.
    V. I. Arnol'd, V. V. Kozlov, and A. I. Neishtadt, in:Modern Problems of Mathematics. Fundamental Directions, Vol. 4 [in Russian], VINITI, Moscow (1985), p. 5; V. I. Arnol'd and A. B. Givental', in:Modern Problems of Mathematics. Fundamental Directions, Vol. 4 [in Russian], VINITI, Moscow (1985), p. 7; B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, in:Modern Geometry [in Russian], Nauka, Moscow (1987); A. T. Fomenko,Symplectic Geometry. Methods and Applications [in Russian], Moscow State University, Moscow (1988); B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, in:Modern Problems of Mathematics. Fundamental Directions, Vol. 4 [in Russian], VINITI, Moscow (1985), p. 179; M. A. Ol'shanetskii, A. M. Perelomov, A. G. Reiman, and M. A. Semenov-Tyan-Shanskii, in:Modern Problems of Mathematics. Fundamental Directions, Vol. 16 [in Russian], VINITI, Moscow (1987), p. 86; V. V. Trofimov and A. T. Fomenko, in:Modern Problems of Mathematics. Fundamental Directions, Vol. 16 [in Russian], VINITI, Moscow (1987), p. 227.Google Scholar
  24. 24.
    A. N. Leznov and M. V. Savel'ev,Group Methods of Integration of Nonlinear Dynamical Systems [in Russian], Nauka, Moscow (1985); A. M. Perelomov,Integrable Systems of Classical Mechanics and Lie Algebras [in Russian], Nauka, Moscow (1990).Google Scholar
  25. 25.
    I. N. Bernshtein, I. M. Gel'fand, and S. I. Gel'fand, in:Proceedings of the Seminar of I. G. Petrovskii, Vol. 2 [in Russian] (1976). p. 2; I. M. Gel'fand and A. V. Zelevinskii,Funktsional. Analiz i Ego Prilozhen.,18, 14 (1984).Google Scholar
  26. 26.
    I. M. Gel'fand,Dokl. Akad. Nauk SSSR,288, 14 (1986); I. M. Gel'fand and S. I. Gel'fand,Dokl. Akad. Nauk SSSR,288, 279 (1986); I. M. Gel'fand and A. V. Zelevinskii,Funktsional. Analiz i Ego Prilozhen.,20, 17 (1986); I. M. Gel'fand, M. I. Graev, and A. V. Zelevinskii,Funktsional. Analiz i Ego Prilozhen.,21, 23 (1987); I. M. Gel'fand, A. V. Zelevinskii, and V. V. Serganov,Dokl. Akad. Nauk SSSR,304, 1044 (1989); I. M. Gel'fand, A. V. Zelevinskii, and M. M. Kapranov,Funktsional. Analiz i Ego Prilozhen.,23, 12 (1989); I. M. Gel'fand, M. I. Graev, and V. S. Reshakh,Usp. Mat. Nauk,47, 3 (1991).Google Scholar
  27. 27.
    G. Segal, Preprint No. 85 MPI (1987); E. Witten,Commun. Math. Phys.,113, 529 (1988); L. Alvarez-Gaume, C. Gomez, G. Moore, and C. Vafa,Nucl. Phys. B,303, 455 (1988); K. Gawedzki,Sem. Bourbaki, Ex. 704 (1988).Google Scholar
  28. 28.
    Yu. A. Neretin,Funktsional. Analiz i Ego Prilozhen.,21, 82 (1987);Mat. Sb.,180, 635 (1989); G. I. Ol'shanskii,Funktsional. Analiz i Ego Prilozhen.,22, 23 (1988); G. I. Olshanskii,Adv. Sov. Math.,2, 1, 67 (1991); Yu. A. Neretin,Adv. Sov. Math.,2, 103 (1991).Google Scholar
  29. 29.
    G. Moore and N. Seiberg,Commun. Math. Phys.,123, 177 (1989).Google Scholar
  30. 30.
    I. M. Gel'fand and S. G. Gindikin,Funktsional. Analiz i Ego Prilozhen.,11, 19 (1977); É. B. Vinberg,Funktsional. Analiz i Ego Prilozhen.,14, 1 (1980); G. I. Ol'shanskii,Funktsional. Analiz i Ego Prilozhen.,15, 53 (1981).Google Scholar
  31. 31.
    I. M. Gel'fand, R. A. Minlos, and Z. Ya. Shapiro,Representations of the Rotation and Lorentz Groups and Their Applications. Pergamon Press, Oxford (1963); I. M. Gel'fand, M. I. Graev, and N. Ya. Vilenkin,Integral Geometry and Representation Theory, Academic Press, New York (1966); N. Y. Vilenkin,Special Functions and the Theory of Group Representations, AMS Translations of Math. Monogr., Vol. 22, Providence, R.I. (1968); L. A. Shelepin,Tr. Fiz. Inst. Akad. Nauk,70, 3 (1973); A. U. Klimyk,Matrix Elements and Clebsch-Gordan Coefficients of Group Representations, Naukova Dumka, Kiev (1979).Google Scholar
  32. 32.
    A. A. Kirillov,Elements of the Theory of Representations [in Russian], Nauka, Moscow (1972), p. 212; A. Barut and R. Raczka,Theory of Group Representations and Applications, Warsaw (1977).Google Scholar

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© Plenum Publishing Corporation 1994

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  • D. V. Yur'ev

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