Quantization of three-wave equations

Abstract

The subject of this paper is the consecutive procedure of discretization and quantization of two similar classical integrable systems in three-dimensional space-time: the standard three-wave equations and less known modified three-wave equations. The quantized systems in discrete space-time 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. Principal object of the lattice field theories is the Heisenberg discrete time evolution operator constructed with the help of vertex operators.

Contribution to J. Phys. A. Special issue "Symmetries and Integrability of Difference Equations (SIDE) VII"

References in corpus (15)

• Classification of integrable equations on quad-graphs. The consistency approach
• Discrete Heisenberg-Weyl Group and Modular Group
• Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality
• Multidimensional Quadrilateral Lattices are Integrable
• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• Transformations of Quadrilateral Lattices
• Functional Tetrahedron Equation
• Lattice and q-difference Darboux-Zakharov-Manakov systems via $\bar{\partial }$-dressing method
• The vertex formulation of the Bazhanov-Baxter Model
• Quantum 2+1 evolution model
• 3D symplectic map
• Strongly Coupled Quantum Discrete Liouville Theory. II: Geometric Interpretation of the Evolution Operator
• Integrability of $q$-oscillator lattice model
• Bazhanov-Stroganov model from 3D approach
• Theta function parameterization and fusion for 3-D integrable Boltzmann weights

Cited by in corpus (9)

• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Non-commutative lattice modified Gel'fand-Dikii systems
• Super-tetrahedra and super-algebras
• The pentagon relation and incidence geometry
• The Coxeter relations and KP map for non-commuting symbols
• Pentagon Relations in Direct Sums and Grassmann Algebras
• Classical integrable field theories in discrete 2+1 dimensional space-time
• Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
• Geometry of quadrilateral nets: second Hamiltonian form