On the Hamiltonian formulation, integrability and algebraic structures of the Rajeev-Ranken model

TL;DR

This paper analyzes the Hamiltonian and algebraic structures of the integrable Rajeev-Ranken model, a reduced classical wave model related to the SU(2) principal chiral model, revealing its integrability and connections to other models.

Contribution

It provides a detailed algebraic and Hamiltonian analysis of the Rajeev-Ranken model, establishing its integrability and uncovering its relation to the Neumann model.

Findings

Identification of Darboux coordinates, Lax pairs, and classical r-matrices.

Proof of Liouville integrability through conserved quantities in involution.

Discovery of a new Hamiltonian formulation of the Neumann model.

Abstract

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is 'pseudodual' to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serve as a toy-model for non-perturbative properties of theories with a Landau pole. Unlike the 'Euclidean' current algebra of the PCM, its pseudodual is based on a nilpotent current algebra. Recently, Rajeev and Ranken obtained a mechanical reduction by restricting the nilpotent scalar field theory to a class of constant energy-density classical waves expressible in terms of elliptic functions, whose quantization survives the passage to the strong-coupling limit. We study the Hamiltonian and Lagrangian formulations of this model and its classical integrability from an algebraic perspective, identifying Darboux coordinates, Lax pairs, classical r-matrices and a degenerate Poisson pencil. We identify Casimirs as well as a complete set of conserved quantities in involution and the canonical transformations they generate. They are related to Noether charges of the field theory and are shown to be generically independent, implying Liouville integrability. The singular submanifolds where this independence fails are identified and shown to be related to the static and circular submanifolds of the phase space. We also find an interesting relation between this model and the Neumann model allowing us to discover a new Hamiltonian formulation of the latter.