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

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References

  1. B. B. Mandelbrot, in:Fractals for the Classroom, 1 (eds. H. O. Peiten, H. Jurgens, and D. Saupe), Springer (1992), p. 1.

  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. D. V. Yur'ev,Teor. Mat. Fiz.,92, 172 (1992).

    Google Scholar 

  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. 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. D. Juriev,J. Math. Phys.,33, 2819 (1992);34, No. 4 (1993);Acta Appl. Math. (1993).

    Google Scholar 

  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. A. Alekseev, L. Faddeev, and M. Semenov-Tian-Shansky,Lect. Notes Math.,1510, 148 (1992).

    Google Scholar 

  9. M. A. Semenov-Tyan-Shanskii,Teor. Mat. Fiz.,93, 302 (1992).

    Google Scholar 

  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 8(1) , Contemporary Mathematics 121, Am. Math. Soc., Providence (1991).

    Google Scholar 

  11. S. A. Bychkov and D. V. Yur'ev,Teor. Mat. Fiz.,97, No. 3 (1993).

  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. D. V. Yur'ev,Teor. Mat. Fiz.,37, 74 (1987).

    Google Scholar 

  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).

  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. I. N. Bernshtein, I. M. Gel'fand, and S. I. Gel'fand,Funktsional. Analiz i Ego Prilozhen.,5, 1 (1971).

    Google Scholar 

  17. S. L. Luk'yanov,Funktsional. Analiz i Ego Prilozhen.,22, 1 (1988).

    Google Scholar 

  18. A. Yu. Morozov,Fiz.EElem. Chastits At. Yadra,23, 174 (1992).

    Google Scholar 

  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. 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).

  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. 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. 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. 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. 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).

  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. 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).

  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. G. Moore and N. Seiberg,Commun. Math. Phys.,123, 177 (1989).

    Google Scholar 

  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. 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. 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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Mathematics Division, Institute of Systems Research (Information Technology), Russian Academy of Sciences; Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, Paris. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 2, pp. 220–240, February, 1994.

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Yur'ev, D.V. Quantum projective field theory: Quantum-field analogs of the Euler-Arnol'd equations in projectiveG multiplets. Theor Math Phys 98, 147–161 (1994). https://doi.org/10.1007/BF01015793

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