Bethe Ansatz and Classical Hirota Equations

Abstract

A brief non-technical review of the recent study of classical integrable structures in quantum integrable systems is given. It is explained how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's $Q$-operators, etc) with elements of classical non-linear integrable difference equations ($τ$-functions, Baker-Akhiezer functions, etc). The nested Bethe ansatz equations for $A_{k-1}$-type models emerge as discrete time equations of motion for zeros of classical $τ$-functions and Baker-Akhiezer functions. The connection with discrete time Ruijsenaars-Schneider system of particles is discussed.