Integrability and dynamics of the Rajeev-Ranken model

TL;DR

This thesis thoroughly analyzes the integrability, solutions, and quantization of the Rajeev-Ranken model, a 3D mechanical system linked to a scalar field theory, revealing its classical and quantum properties.

Contribution

It provides a comprehensive study of the model's integrability, solutions, and quantization, including new action-angle variables and an infinite-dimensional representation of its Lie algebra.

Findings

Liouville integrability established with conserved quantities

Solutions expressed in elliptic functions and classified by root analysis

Quantization leads to a generalized Lame equation and a unitary Lie algebra representation

Abstract

This thesis concerns the dynamics and integrability of the Rajeev-Ranken (RR) model, a mechanical system with 3 degrees of freedom describing screw-type nonlinear wave solutions of a scalar field theory dual to the 1+1D SU(2) Principal Chiral Model. This field theory is strongly coupled in the UV and could serve as a toy model to study nonperturbative features of theories with a perturbative Landau pole. We begin with a Lagrangian and a pair of Hamiltonian formulations based on compatible degenerate nilpotent and Euclidean Poisson brackets. Darboux coordinates, Lax pairs and classical r-matrices are found. Casimirs are used to identify the symplectic leaves on which a complete set of independent conserved quantities in involution are found, establishing Liouville integrability. Solutions are expressible in terms of elliptic functions and their stability is analyzed. The model is compared and contrasted with those of Neumann and Kirchhoff. Common level sets of conserved quantities are generically 2-tori, though horn tori, circles and points also arise. On the latter, conserved quantities develop relations and solutions degenerate from elliptic to hyperbolic, circular and constant functions. The common level sets are classified using the nature of roots of a cubic polynomial. We also discover a family of action-angle variables that are valid away from horn tori. On the latter, the dynamics is expressed as a gradient flow. In Darboux coordinates, the model is reinterpreted as an axisymmetric quartic oscillator. It is quantized and variables are separated in the Hamilton-Jacobi and Schrodinger equations. Analytic properties and weak and strong coupling limits of the radial equation are studied. It is shown to reduce to a generalization of the Lame equation. Finally, we use this quantization to find an infinite-dimensional unitary representation of the above nilpotent Lie algebra.