×

Classical and quantum algebras of non-local charges in \(\sigma\) models. (English) Zbl 0536.58013

In field theories the notion of complete integrability involves the existence of an infinite number of commuting local conserved charges. Some models possess also nonlocal conserved charges which do not commute among themselves. Thus nonabelian dyamical symmetry algebras arise. The paper is devoted to the investigation of such algebras both in classical and quantum cases in the framework of nonlinear \(\sigma\)-models. For a large class of classical integrable models the Poisson bracket \(\{\) \(T(\lambda)\otimes T(\mu)\}\) of monodromy matrices is ultralocal and can be expressed using r-matrix. The classical Yang-Baxter equation on the r- matrix guarantees usual properties of the Poisson bracket. In the case of classical chiral \(\sigma\)-models it turns out that there exists no definition of the Poisson bracket satisfying the antisymmetry property and the Jacobi identity. The problems found in classical theory are absent in the quantum case. In the case of the quantum 0(N) nonlinear \(\sigma\)-model the conserved quantum monodromy operator is determined which generates the quantum nonlocal charges. The expression for [\(T(\lambda)\otimes T(\mu)]\) is derived consistent with all the properties of a Lie algebra. The quantum R-matrix is found. It is shown that the quantum nonlocal charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.
Reviewer: I.Ja.Dorfman

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81T08 Constructive quantum field theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Lüscher, M., Pohlmeyer, K.: Scattering of massless lumps and non-local charges in the two-dimensional classical non-linear ?-model. Nucl. Phys. B137, 46 (1978)
[2] Brézin, E., Itzykson, C., Zinn-Justin, J., Zuber, J.B.: Remarks about the existence of non-local charges in two-dimensional models. Phys. Lett.82 B, 442 (1979)
[3] de Vega, H.J.: Field theories with an infinite number of conservation laws and Bäcklund transformations in two dimensions. Phys. Lett.87 B, 233 (1979)
[4] Eichenherr, H., Forger, M.: On the dual symmetry of the non-linear sigma models. Nucl. Phys. B155, 381 (1979) · Zbl 0482.53040
[5] Zakharov, V.E., Mikhailov, A.V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Sov. Phys. JETP47, 1017 (1978)
[6] Faddeev, L.D.: Les Houches lectures 1982, Saclay preprint T/82/76
[7] Kulish, P.P., Skylanin, E.K.: Nectures Notes in Physics, Vol. 151. Berlin, Heidelberg, New York: Springer 1982
[8] Izergin, A.G., Korepin, V.E.: The inverse scattering method approach to the quantum Shabat-Mikhailov model. Commun. Math. Phys.79, 303 (1981) · Zbl 0441.35053
[9] Polyakov, A.M.: String representations and hidden symmetries for gauge fields. Phys. Lett.82 B, 247 (1979)
[10] de Vega, H.J., Zuber, J.B.: Unpublished
[11] Dolan, L.: Kac-Moody algebra is hidden symmetry of chiral models. Phys. Rev. Lett.47, 1371 (1981)
[12] Ueno, K.: Kyoto University preprint RIMS-374 (1981)
[13] Davies, M.C., Houston, P.J., Leinaas, J.M., Macfarlane, A.J.: Hidden symmetries as canonical transformations for the chiral model. Phys. Lett.119 B, 187 (1982)
[14] Lüscher, M.: Quantum non-local charges and absence of particle production in the two-dimensional nonlinear ?-model. Nucl. Phys. B135, 1 (1978)
[15] Zamolodchikov, Al.: Dubna preprint E2-11485 (1978)
[16] Devchand, C., Fairlie, D.B.: A generating function for hidden symmetries of chiral models. Nucl. Phys. B194, 232 (1982)
[17] Korepin, V.E.: Zapisky Nauchny Seminarov101, 90 (1980)
[18] Brézin, E., Zinn-Justin, J., Le Guillou, J.C.: Renormalization of the nonlinear ? model in 2 + ? dimensions. Phys. Rev. D14, 2615 (1976)
[19] Zamolodchikov, A.B., Zamolodchikov, Al.B.: FactorizedS-Matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys.120, 253 (1979) · Zbl 0946.81070
[20] See, for example, Babelon, O., de Vega, H.J., Viallet, C.-M.: Exact solutions of theZ n+1 {\(\times\)}Z n+1 symmetric generalization of theXXZ model. Nucl. Phys. B200 FS4, 266 (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.