QUANTUM AND CLASSICAL INTEGRABILITY - NEW APPROACHES IN STATISTICAL-MECHANICS

The present status of the statistical mechanics (SM), quantum and classical, of integrable models is reviewed by reporting new results for their partition functions Z obtained for anyon type models in one space and one time (1 + 1) dimensions. The methods of functional integration developed already are extended further. Bose-Fermi equivalence and anyon descriptions are natural parts of the quantum theory and the anyon phase is quantised. The classical integrability is exploited throughout and both classical and quantum integrability theory are reviewed this way, and related to underlying algebraic structures-notably the Hopf algebras ("quantum groups"). A new "q-boson" lattice gas is solved in this connection. Further results to be realised within 10 years concerning integrability, integrable systems coupled to large dimensional chaos provided by a heat bath, and their application in physics e.g. to high-T(c) superconductivity, are suggested.