Integrability and dynamics of the RajeevRanken model
Abstract
This thesis concerns the dynamics and integrability of the RajeevRanken (RR) model, a mechanical system with 3 degrees of freedom describing screwtype 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 rmatrices 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 2tori, 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 actionangle 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 HamiltonJacobi 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 infinitedimensional unitary representation of the above nilpotent Lie algebra.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.12579
 Bibcode:
 2021arXiv210912579V
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 PhD thesis, defended September 15, 2021, Chennai Mathematical Institute, 109 pages. Based on arXiv:1804.02859, arXiv:1906.03141 and arXiv:2004.05791