RSS feeds
РусскийEnglish
Issue 11, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2013

 / 

May

←  →

Physics of our days


New life of complete integrability


St. Petersburg Branch of the V.A. Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg, Russian Federation

We briefly review new trends in how the notion of complete integrability of dynamical systems and their quantum counterpart developed over the last four decades. We describe a new technique for working with integrable models and outline its main applications.

Fulltext pdf (832 KB)
Fulltext is also available at DOI: 10.3367/UFNe.0183.201305b.0487
PACS: 02.30.Ik, 45.20.Jj, 75.10.Pq (all)
DOI: 10.3367/UFNe.0183.201305b.0487
URL: https://ufn.ru/en/articles/2013/5/b/
000322890700002
2-s2.0-84882953609
2013PhyU...56..465F
Citation: Faddeev L D "New life of complete integrability" Phys. Usp. 56 465–472 (2013)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 21st, February 2013, 27th, February 2013

Оригинал: Фаддеев Л Д «Новая жизнь полной интегрируемости» УФН 183 487–495 (2013); DOI: 10.3367/UFNr.0183.201305b.0487

References (61) ↓ Cited by (19) Similar articles (6)

  1. Arnol’d V I Matematicheskie Metody Klassicheskoi Mekhaniki 3-e izd. (M.: Nauka, 1989); Arnold V I Mathematical Methods Of Classical Mechanics (New York: Springer, 1997)
  2. Gardner G S, Greene J M, Kruskal M D, Miura R M "Method for solving the Korteweg-deVries equation" Phys. Rev. Lett. 19 1095 (1967)
  3. Korteweg D J, de Vries G "On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves" Phil. Mag. 5 39 422 (1895)
  4. Faddeev L D "Obratnaya zadacha kvantovoi teorii rasseyaniya" Usp. Mat. Nauk 14 (4) 57 (1959)
  5. Faddeev L D "O svyazi S-matritsy i potentsiala dlya odnomernogo operatora Shredingera" Dokl. Akad. Nauk SSSR 121 63 (1958); Faddeev L D "On the relation between the S-matrix and potential for the one-dimensional Schrödinger operator" Sov. Phys. Dokl. 3 747 (1959)
  6. Lax P D "Integrals of nonlinear equations of evolution and solitary waves" Commun. Pure Appl. Math. 21 467 (1968)
  7. Zakharov V E, Shabat A B "Tochnaya teoriya samofokusirovki i odnomernoi samomodulyatsii volny v nelineinoi srede" Zh. Eksp. Teor. Fiz. 61 118 (1971); Zakharov V E, Shabat A B "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media" Sov. Phys. JETP 34 62 (1972)
  8. Zakharov V E, Faddeev L D "Uravnenie Kortevega—De Frisa — vpolne integriruemaya gamil’tonova sistema" Funkts. Analiz Ego Pril. 5 (4) 18 (1971); Zakharov V E, Faddeev L D "Korteweg—de Vries equation: A completely integrable Hamiltonian system" Funct. Anal. Appl. 5 280 (1971)
  9. Okun’ L B "V.A. Fok i kalibrovochnaya simmetriya" Usp. Fiz. Nauk 180 871 (2010); Okun L B "V A Fock and gauge symmetry" Phys. Usp. 53 835 (2010)
  10. Fock V Z. Phys. 39 226 (1926)
  11. Fok V A "Ob invariantnoi forme volnovykh uravnenii i uravnenii dvizheniya zaryazhennoi tochechnoi massy" Usp. Fiz. Nauk 180 874 (2010); Fock V A "On the invariant form of the wave and motion equations for a charged point mass" Phys. Usp. 53 839 (2010)
  12. Takhtadzhyan L A, Faddeev L D "Sushchestvenno-nelineinaya odnomernaya model’ klassicheskoi teorii polya" Teor. Mat. Fiz. 21 160 (1974); Takhtadzhyan L A, Faddeev L D "Essentially nonlinear one-dimensional model of classical field theory" Theor. Math. Phys. 21 1046 (1974)
  13. Dashen R F, Hasslacher B, Neveu A "Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods" Phys. Rev. D 10 4114 (1974)
  14. Faddeev L D "Adrony iz leptonov?" Pis’ma ZhETF 21 141 (1975); Faddeev L D "Hadrons from leptons?" JETP Lett. 21 64 (1975)
  15. Takhtadzhyan L A, Faddeev L D Gamil’tonov Podkhod v Teorii Solitonov (M.: Nauka, 1986); Faddeev L D, Takhtajan L A Hamiltonian Methods In The Theory Of Solitons (Berlin: Springer-Verlag, 1987)
  16. Bethe H "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette" Z. Phys. 71 205 (1931)
  17. Sklyanin E K, Takhtadzhyan L A, Faddeev L D "Kvantovyi metod obratnoi zadachi. I" Teor. Mat. Fiz. 40 194 (1979); Sklyanin E K, Takhtadzhyan L A, Faddeev L D "Quantum inverse problem method. I" Theor. Math. Phys. 40 688 (1979)
  18. Takhtadzhyan L A, Faddeev L D "Kvantovyi metod obratnoi zadachi i XYZ model’ Geizenberga" Usp. Mat. Nauk 34 (5) 13 (1979); Takhtadzhan L A, Faddeev L D "The quantum method of the inverse problem and the Heisenberg XYZ model" Russ. Math. Surv. 34 (5) 11 (1979)
  19. Takhtadzhyan L A, Faddeev L D "Spektr i rasseyanie vozbuzhdenii v odnomernom izotropnom magnetike Geizenberga" Zap. Nauchn. Sem. LOMI 109 134 (1981); Translated into English, Translated into English: Faddeev L D, Takhtadzhyan L A "Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model" J. Sov. Math. 24 241 (1984)
  20. Hulthén L "Über das Austauschproblem eines Kristalls" Ark. Mat. Astron. Fysik A 26 (11) 1 (1938)
  21. Faddeev L D, Takhtajan L A "What is the spin of a spin wave?" Phys. Lett. A 85 375 (1981)
  22. Kulish P P, Reshetikhin N Yu, Sklyanin E K "Yang—Baxter equation and representation theory. I" Lett. Math. Phys. 5 393 (1981)
  23. Zamolodchikov A B, Fateev V A "Faktorizovannaya S-matritsa i integriruemaya tsepochka Geizenberga spina 1" Yad. Fiz. 32 581 (1980); Zamolodchikov A B, Fateev V A "A model factorized S-matrix and an integrable spin-1 Heisenberg chain" Sov. J. Nucl. Phys. 32 298 (1980)
  24. Kulish P P, Reshetikhin N Yu "Kvantovaya lineinaya zadacha dlya uravneniya sinus-Gordon i vysshie predstavleniya" Zap. Nauch. Sem. LOMI 101 101 (1981); Kulish P P, Reshetikhin N Yu "Quantum linear problem for the sine-Gordon equation and higher representations" J. Sov. Math. 23 2435 (1983)
  25. Sklyanin E K "Ob odnoi algebre, porozhdaemoi kvadratichnymi sootnosheniyami" Usp. Mat. Nauk 40 (2) 214 (1985)
  26. Drinfel’d V G "Kvantovye gruppy" Zap. Nauchn. Sem. LOMI 155 18 (1986)
  27. Jimbo M "A q-difference analogue of U(g) and the Yang—Baxter equation" Lett. Math. Phys. 10 63 (1985)
  28. Reshetikhin N Yu, Takhtadzhyan L A, Faddeev L D "Kvantovanie grupp Li i algebr Li" Algebra Analiz 1 (1) 178 (1989); Reshetikhin N Yu, Takhtadzhyan L A, Faddeev L D "Quantization of Lie groups and Lie algebras" Leningr. Math. J. 1 (1) 193 (1989)
  29. Faddeev L D, Takhtajan L A "Liouville model on the lattice" Lect. Notes Phys. 246 166 (1986)
  30. Gervais J-L, Neveu A "Non-standard 2D critical statistical models from Liouville theory" Nucl. Phys. B 257 59 (1985)
  31. Faddeev L D "History and perspectives of quantum groups" Milan J. Math. 74 279 (2006)
  32. Reshetikhin N "S-matrices in integrable models of isotropic magnetic chains. I" J. Phys. A Math. Gen. 24 3299 (1991)
  33. Faddeev L D ""How Algebraic Bethe Ansatz works for integrable model" hep-th/9605187
  34. Onsager L "Crystal statistics. I. A two-dimensional model with an order-disorder transition" Phys. Rev. 65 117 (1944)
  35. Lieb E H "Residual entropy of square ice" Phys. Rev. 162 162 (1967)
  36. Baxter R J Exactly Solved Models In Statistical Mechanics (London: Academic Press, 1982)
  37. Stroganov Yu G "A new calculation method for partition functions in some lattice models" Phys. Lett. A 74 116 (1979)
  38. Bazhanov V V, Stroganov Yu G "Chiral Potts model as a descendant of the six-vertex model" J. Stat. Phys. 59 799 (1990)
  39. Yang C N "Some exact results for the many-body problem in one dimension with repulsive delta-function interaction" Phys. Rev. Lett. 19 1312 (1967)
  40. Brezin E, Zinn-Justin J "Un problème à N corps soluble" C.R. Acad. Sci. B 263 670 (1968)
  41. Baxter R J "One-dimensional anisotropic Heisenberg chain" Ann. Physics 70 323 (1972)
  42. Zamolodchikov A B, Zamolodchikov A B "Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models" Ann. Physics 120 253 (1979)
  43. Belavin A A, Polyakov A M, Zamolodchikov A B "Infinite conformal symmetry in two-dimensional quantum field theory" Nucl. Phys. B 241 333 (1984)
  44. Bazhanov V V, Lukyanov S L, Zamolodchikov A B "Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe Ansatz" Commun. Math. Phys. 177 381 (1996); Bazhanov V V, Lukyanov S L, Zamolodchikov A B hep-th/9412229
  45. Yang C N, Yang C P "Thermodynamics of one-dimensional system of bosons with repulsive delta-function interaction" J. Math. Phys. 10 1115 (1969)
  46. Zamolodchikov Al B "Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state potts and Lee—Yang models" Nucl. Phys. B 342 695 (1990)
  47. Destri C, de Vega H J ""New approach to thermal Bethe Ansatz" hep-th/9203064
  48. Lipatov L N "High energy asymptotics of multi-colour QCD and two-dimensional conformal field theories" Phys. Lett. B 309 394 (1993)
  49. Faddeev L D, Korchemsky G P "High energy QCD as a completely integrable model" Phys. Lett. B 342 311 (1995); Faddeev L D, Korchemsky G P hep-th/9404173
  50. "Lunch Talk on Bethe Ansatz, Jan. 22, 1987" Archives and Special Collections, California Institute of Technology, Section V, Box 52, Folder 11
  51. Faddeev L D, Popov V N "Feynman diagrams for the Yang—Mills field" Phys. Lett. B 25 29 (1967)
  52. Archives and Special Collections, California Institute of Technology, Section VIII, Box 78, Folder 16
  53. Gorsky A, Krichever I, Marshakov A, Mironov A, Morozov A "Integrability and Seiberg—Witten exact solution" Phys. Lett. B 355 466 (1995); Gorsky A, Krichever I, Marshakov A, Mironov A, Morozov A hep-th/9505035
  54. Dubrovin B A, Krichever I M, Novikov S P Dinamicheskie Sistemy—4 (Itogi nauki i tekhniki, Ser. Sovrem. probl. mat. Fundam. napravleniya) Vol. 4 (M.: VINITI, 1985) p. 179; Dubrovin B A, Krichever I M, Novikov S P Encyclopedia Of Mathematical Sciences Vol. 4 (Berlin: Springer, 1990) p. 173
  55. Seiberg N, Witten E "Electric—magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang—Mills theory" Nucl. Phys. B 426 19 (1994); Seiberg N, Witten E "Erratum" Nucl. Phys. B 430 485 (1994); Seiberg N, Witten E hep-th/9407087
  56. Nekrasov N A, Shatashvili S L "Supersymmetric vacua and Bethe ansatz" Nucl. Phys. B Proc. Suppl. 192-193 91 (2009); Nekrasov N A, Shatashvili S L arXiv:0901.4744
  57. Nekrasov N, Shatashvili S "Quantum integrability and supersymmetric vacua" Prog. Theor. Phys. Suppl. 177 105 (2009)
  58. Minahan J A, Zarembo K "The Bethe-ansatz for N=4 super Yang—Mills" JHEP (03) (2003) p. 013 hep-th/0212208
  59. Beisert N et al. "Review of AdS/CFT integrability: An overview" Lett. Math. Phys. 99 3 (2012); Beisert N et al. arXiv:1012.3982
  60. Dorey P, Minahan J, Tseytlin A "Quantum integrable models and gauge-string duality" J. Phys. A Math. Theor. 44 120301 (2011)
  61. Morozov A "Integrability in non-perturbative QFT" arXiv:1303.2578
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions