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Quantum and classical integrable systems

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Book cover Integrability of Nonlinear Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 495))

Abstract

The key concept discussed in these lectures is the relation between the Hamiltonians of a quantum integrable system and the Casimir elements in the underlying hidden symmetry algebra. (In typical applications the latter is either the universal enveloping algebra of an affine Lie algebra or its q-deformation.) A similar relation also holds in the classical case. We discuss different guises of this very important relation and its implication for the description of the spectrum and the eigenfunctions of the quantum system. Parallels between the classical and the quantum cases are thoroughly discussed.

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References

  • Adler M. (1979). On a trace functional for formal pseudodifferential operators and the symplectic structure for the KdV type equations, Invent. Math. 50, 219–248.

    Article  MATH  ADS  Google Scholar 

  • Baxter R.J. (1982). Exactly Solved Models in Statistical Mechanics. Academic Press, London.

    MATH  Google Scholar 

  • Belavin A.A., Drinfeld V.G. (1983). Triangle equations and simple Lie algebras, Sov. Sci. Rev. 4C, 93–165.

    MathSciNet  Google Scholar 

  • Babujian H.M., Flume R. (1993). Off-shell Bethe ansatz equations for Gaudin magnets and solutions of KZ equations, Preprint Bonn-HE-93-30.

    Google Scholar 

  • Bogolyubov N.M., Izergin A.G., Korepin V.E. (1994). Correlation Functions of Integrable Systems and Quantum Inverse Scattering Method. Cambridge University Press, Cambridge.

    Google Scholar 

  • Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D. (1977). Deformation theory and quantization, I, II. Ann. Phys. 111, 61–151.

    ADS  MathSciNet  Google Scholar 

  • Cartier P. (1994). Some fundamental techniques in the theory of integrable systems, in: Lectures on integrable systems, O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, eds. World Scientific.

    Google Scholar 

  • Ding J., Etingof P. (1994). Center of a quantum affine algebra at the critical level. Math. Res. Lett. 1, 469–480.

    MATH  MathSciNet  Google Scholar 

  • Ding J,. Frenkel I. (1993). Commun. Math. Phys. 156, 277–300.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Dixmier J. (1974). Algèbres enveloppantes. Cahiers Scientifiques, fasc. XXXVII. Gauthier-Villars, Paris.

    Google Scholar 

  • Drinfeld V.G. (1983). Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation. Sov. Math. Dokl., 27, 68–71.

    MathSciNet  Google Scholar 

  • Drinfeld V.G. (1987). Quantum Groups. Proc. ICM-86 (Berkeley), vol. 1, p. 798. AMS.

    MathSciNet  Google Scholar 

  • Drinfeld V.G. (1987). Sov. Math. Dokl., 36, 212–216.

    MathSciNet  Google Scholar 

  • Faddeev L.D. (1980). Quantum Completely Integrable Systems in Quantum Field Theory. Sov. Sci. Rev. 1C, 107–155.

    Google Scholar 

  • Faddeev L.D. (1984). Integrable Models in (1+1)-dimensional Quantum Field Theory. Les Houches 1982. Elsevier Science Publ.

    Google Scholar 

  • Faddeev L.D. (1995). Algebraic aspects of Bethe ansatz, Int. J. Mod. Phys. A10, 1845–1878, e-print archive: hep-th/9404013.

    ADS  MathSciNet  Google Scholar 

  • Faddeev L.D., Reshetikhin N.Yu., Takhtajan L.A. (1989). Quantization of Lie groups and Lie algebras. Leningrad Math. J., 1, 178–207.

    MathSciNet  Google Scholar 

  • Faddeev L.D., Takhtajan L.A. (1979). Quantum Inverse Scattering Method. Russian Math. Surveys, 34:5, 11–68.

    Article  ADS  Google Scholar 

  • Faddeev L.D., Takhtajan L.A. (1987). Hamiltonian Methods in the Theory of Solitons. Springer-Verlag.

    Google Scholar 

  • Faddeev L.D., Volkov A.Yu. (1993). Abelian current algebra and the Virasoro algebra on the lattice. Phys. Lett. B 315, 311–318.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Feigin B, Frenkel E. (1992). Affine Kac-Moody algebras at the critical level and Gelfand-Dikij algebras, Int. J. Mod. Phys. A 7, Suppl. 1A, 197–215.

    Article  MATH  ADS  Google Scholar 

  • Feigin B., Frenkel E., and Reshetikhin N. (1994). Gaudin model, Bethe ansatz and correlation functions at the critical level. Commun. Math. Phys. 166, 27–62.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Frenkel E. (1995). Affine algebras, Langlands duality and Bethe ansatz. Proc. International Congr. of Math. Physics, Paris 1994. International Press, pp. 606–642.

    Google Scholar 

  • Frenkel I.B., Reshetikhin N.Yu. (1992). Quantum affine algebras and holonomic difference equations. Commun. Math. Phys. 146, 1–60.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M. (1967). Method for solving the KdV equation, Phys. Rev. Lett. 19, 1095–1097.

    Article  MATH  ADS  Google Scholar 

  • Gaudin M. (1983). La Fonction d'Onde de Bethe. Masson, Paris.

    MATH  Google Scholar 

  • Goodman R., and Wallach N.R. (1982). Classical and quantum-mechanical systems of Toda lattice type. I. Commun. Math. Phys. 83, 355–386.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Goodman R., and Wallach N.R. (1984). Classical and quantum-mechanical systems of Toda lattice type. II. Commun. Math. Phys. 94, 177–217.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Harnad J., Winternitz P. (1993). Classical and quantum integrable systems in \(\bar gl(2)^{ + * }\) and separation of variables. Preprint CRM 1921, Montreal.

    Google Scholar 

  • Jimbo M. (1986). A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 62–69.

    Google Scholar 

  • Jurčo B. (1989). Classical Yang-Baxter equation and quantum integrable systems. J. Math. Phys. 30, 1289–1293.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kazhdan D., Kostant B., Sternberg S. (1978). Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. 31, 481–508.

    Article  MATH  MathSciNet  Google Scholar 

  • Kirillov A.A. (1976). Elements of representation theory. Springer Verlag. Berlin a.o.

    Google Scholar 

  • Kirillov A.N., Reshetikhin N.Yu. (1990). q-Weyl group and a multiplicative formula for the universal R-matrix, Commun. Math. Phys. 134, 421–431.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Kostant B. (1978). On Whittaker vectors and representation theory, Inventiones Math. 48, 101–184.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Kostant B. (1979). Quantization and Representation Theory, in: Representation Theory of Lie Groups, Proc. SRC/LMS Res. Symp., Oxford 1977. London Math. Soc. Lecture Notes Series 34, 287–316.

    Google Scholar 

  • Kulish P.P., Sklyanin E.K. (1981). Quantum Spectral Transform Method. In: Integrable Quantum Field Theories. Lecture Notes in Physics 151, 61–119, Springer.

    Article  ADS  MathSciNet  Google Scholar 

  • Kuznetsov V. (1994). Equivalence of two graphical calculi. J. Phys. A25, 6005–6026.

    ADS  Google Scholar 

  • Lax P.D. (1968). Integrals of nonlinear equations and solitary waves. Commun. Pure Appl. Math. 21, 467–490.

    Article  MATH  MathSciNet  Google Scholar 

  • Olshanetsky M.A., Perelomov A.M. (1994). Integrable systems and finite-dimensional Lie algebras, in: Encyclopaedia of Math. Sciences, vol. 16. Springer-Verlag, Berlin a.o.

    Google Scholar 

  • Reshetikhin N.Yu., Semenov-Tian-Shansky M.A. (1990). Central extensions of quantum current groups. Lett. Math. Phys. 19, 133–142.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Reshetikhin N.Yu., Semenov-Tian-Shansky M.A. (1991). Quantum R-matrices and factorization problems, in: Geometry and Physics, essays in honour of I.M. Gelfand, S. Gindikin and I.M. Singer, eds. North Holland, Amsterdam-London-New York, pp. 533–550.

    Google Scholar 

  • Reshetikhin N., Varchenko A. (1993). Quasiclassical asymptotics of the KZ equations. Preprint.

    Google Scholar 

  • Reyman A.G. (1996). Poisson structures related to quantum groups, in: International School of Physics “E. Fermi”, Course 127, Varenna, 1994. Quantum Groups and their Applications in Physics, L. Castellani and J. Wess, eds. Amsterdam.

    Google Scholar 

  • Reyman A.G., Semenov-Tian-Shansky M.A. (1994). Group-theoretical methods in the theory of finite-dimensional integrable systems, in: Encyclopaedia of Math. Sciences, vol. 16. Springer-Verlag, Berlin a.o.

    Google Scholar 

  • Semenov-Tian-Shansky M.A. (1983). What is the classical r-matrix. Funct. Anal. Appl. 17, 259–272.

    Article  Google Scholar 

  • Semenov-Tian-Shansky M.A. (1994a) Quantization of open Toda lattices, in: Encyclopaedia of Math. Sciences, vol. 16. Springer-Verlag, Berlin a.o.

    Google Scholar 

  • Semenov-Tian-Shansky M.A. (1994b). Lectures on r-matrices, Poisson Lie groups and integrable systems, in: Lectures on integrable systems, O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, eds. World Scientific.

    Google Scholar 

  • Semenov-Tian-Shansky M.A. (1994c). Poisson Lie groups, quantum duality principle and the quantum double. Contemporary Math. 175, 219–248.

    MathSciNet  Google Scholar 

  • Sklyanin E.K. (1979) On complete integrability of the Landau-Lifshitz equation. Preprint LOMI E-3-79, Leningrad.

    Google Scholar 

  • Sklyanin E.K. (1992). Quantum Inverse Scattering Method. Selected Topics, in: Quantum Groups and Quantum Integrable Systems, Ge M.L., ed. World Scientific, Singapore.

    Google Scholar 

  • Sklyanin E.K. (1995). Separation of variables. New trends, in: QFT, Integrable Models and beyond, T. Inami and R. Sasaki, eds. Progr. Theor. Phys. Suppl. 118, 35–60.

    Article  ADS  MathSciNet  Google Scholar 

  • Sklyanin E.K., Takhtajan L.A., Faddeev L.D. (1979). The quantum inverse scattering method. I. Theor. Math. Phys. 101, 194–220.

    Google Scholar 

  • Smirnov F.A. (1992). Form Factors in Completely Integrable Models of Quantum Field Theory. Adv. Series in Math. Phys. 14. World Scientific, Singapore.

    Google Scholar 

  • Tarasov V.O. (1985). Irreducible monodromy matrices for the R-matrix of the XXZ-model and lattice local quantum Hamiltonians. Theor. Math. Phys. 63, 440–454.

    Article  MathSciNet  Google Scholar 

  • Tarasov V.O., Varchenko A. (1994). Jackson integral representations for solutions of the quantized KZ equation. St. Petersburg Math. J. 6, no 2.

    Google Scholar 

  • Weinstein A. (1994). Deformation quantization. Séminaire Bourbaki, exposé no 789.

    Google Scholar 

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Authors and Affiliations

  1. Physique Mathématique, Université de Bourgogne, Dijon, France

    M. A. Semenov-Tian-Shansky

  2. Steklov Mathematical Institute, St. Petersburg, Russia

    M. A. Semenov-Tian-Shansky

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  1. M. A. Semenov-Tian-Shansky
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Y. Kosmann-Schwarzbach B. Grammaticos K. M. Tamizhmani

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© 1997 Springer-Verlag

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Semenov-Tian-Shansky, M.A. (1997). Quantum and classical integrable systems. In: Kosmann-Schwarzbach, Y., Grammaticos, B., Tamizhmani, K.M. (eds) Integrability of Nonlinear Systems. Lecture Notes in Physics, vol 495. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0113700

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  • DOI: https://doi.org/10.1007/BFb0113700

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  • Print ISBN: 978-3-540-63353-2

  • Online ISBN: 978-3-540-69521-9

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