Algebraic quantisation approach to integrable differential-difference equations

Abstract

We develop an algebraic quantisation approach, based on quantisation ideals, and apply it to integrable non-Abelian differential--difference equations. We show that the Toda hierarchy admits a bi-quantum structure whose classical (commutative) limit recovers a well-known Poisson pencil. In addition, we discover a non-standard quantisation that has no commutative counterpart. In both cases we present the quantum systems in the Heisenberg form. The generality of the method is illustrated through a wide range of integrable lattices, including the modified Volterra, Bogoyavlensky, Ablowitz-Ladik, relativistic Toda, Merola-Ragnisco-Tu, Adler-Yamilov, Chen-Lee-Liu, Belov-Chaltikian, and Blaszak-Marciniak systems. For each of them, we construct explicit quantisation ideals and present the first few commuting quantum Hamiltonians.