6533b833fe1ef96bd129b674

RESEARCH PRODUCT

Statistical mechanics of the NLS models and their avatars

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description

“In Vishnuland what avatar? Or who in Moscow (Leningrad) towards the czar [1]”. The different manifestations (avatars) of the Nonlinear Schrodinger equation (NLS models) are described including both classical and quantum integrable cases. For reasons explained the sinh-Gordon and sine-Gordon models, which can be interpreted as covariant manifestations of the ‘repulsive’ and ‘attractive’ NLS models, respectively, are chosen as generic models for the statistical mechanics. It is shown in the text how the quantum and classical free energies can be calculated by a method of functional integration which uses the classical action-angle variables on the real line with decaying boundary conditions, even though we define the finite density thermodynamic limit by imposing periodic boundary conditions. Both this method and an equivalent method of ‘generalised Bethe ansatz’ exploit the classical complete integrability of the models, and in quantum form are manifestly fermi-bose equivalent. The state of current knowledge of quantum and classical integrability is reviewed. Algebraic manifestations of the NLS models as supersymmetric NLS models are described. The fermi-bose equivalence of the quantum models has a natural algebraic origin.