Abstract
We investigate the algebras of the non-local charges and their generating functionals (the monodromy matrices) in classical and quantum non-linear σ models. In the case of the classical chiral σ models it turns out that there exists no definition of the Poisson bracket of two monodromy matrices satisfying antisymmetry and the Jacobi identity. Thus, the classical non-local charges do not generate a Lie algebra. In the case of the quantum O(N) non-linear σ model, we explicitly determine the conserved quantum monodromy operator from a factorization principle together withP,T, and O(N) invariance. We give closed expressions for its matrix elements between asymptotic states in terms of the known two-particleS-matrix. The quantumR-matrix of the model is found. The quantum non-local charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.
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Communicated by K. Osterwalder
Laboratoire associé au CNRS No. LA 280
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de Vega, H.J., Eichenherr, H. & Maillet, J.M. Classical and quantum algebras of non-local charges in σ models. Commun.Math. Phys. 92, 507–524 (1984). https://doi.org/10.1007/BF01215281
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DOI: https://doi.org/10.1007/BF01215281