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

Abstract

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1dimensional 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 strongcoupling 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 rmatrices 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.

Invariant tori, action-angle variables, and phase space structure of the Rajeev-Ranken model

“…Interestingly, the topology of M E cm can change with energy: this will be discussed in §3.4. In addition to the Hamiltonian and Casimirs c and m , the helicity hk 2 = Tr SL is a fourth (generically independent) conserved quantity (see §5.7 of [2]). Thus each trajectory must lie on one of the level surfaces M Eh cm of h that foliate M E cm .…”

“…The Rajeev-Ranken model [1,2] is a Hamiltonian system with three degrees of freedom. It arises as a reduction of a 1 + 1 -dimensional scalar field theory [3,4] dual to the SU(2) principal chiral model (PCM) [5].…”

“…These equations were shown to admit a Hamiltonian formulation based on a quadratic Hamiltonian and a step-2 nilpotent Lie algebra mimicking those of the scalar field theory, with solutions expressible in terms of elliptic functions. As we describe in Appendix A, the equations (1) may also be viewed as the Euler equations for a centrally extended Euclidean e(3) algebra and the quadratic Hamiltonian 2H = L 2 + (S − K/λ) 2 .…”

“…In [2], we studied the classical integrability of the Rajeev-Ranken model. We showed that it admits a family of degenerate but compatible Poisson structures, identified their Casimirs and found Lax pairs and r -matrices for the model.…”

“…The Rajeev-Ranken model is related to two other interesting dynamical systems: (a) In [2], a formal relation between its equations and those of the Neumann model [6,7] was obtained. Though not an equivalence (as the corresponding dynamical variables live in different spaces), it was exploited to find a new Hamiltonian formulation for the Neumann model.…”