Quantization of three-wave equations

Abstract

The subject of this paper is the revision of discretization and quantization of two similar classical integrable Hamiltonian systems in three-dimensional spacetime: the standard three-wave equations and the ‘modified’ three-wave equations (the Darboux–Manakov–Zakharov system in two distinguished gauges). The quantized systems in discrete spacetime may be understood as the regularized integrable quantum field theories. Integrability of the theories, and in particular the quantum tetrahedron equations for vertex operators, follow from the quantum auxiliary linear problems. The principal object of the lattice field theories is the Heisenberg discrete-time evolution operator constructed with the help of vertex operators.