Overview
This research introduces a definition for the superintegrability of Hamiltonian systems operating within a stratified symplectic space. The investigation specifically concentrates on spin Calogero-Moser-Sutherland (sCMS) systems. A primary finding indicates that the sCMS systems corresponding to the $SU(3)$ Lie group exhibit superintegrability.
Research Context
The study operates within the theoretical framework of Hamiltonian systems, which are fundamental in classical mechanics and quantum mechanics. The specific environment for these systems is defined as a 'stratified symplectic space.' This type of space represents a generalization of standard symplectic manifolds, accommodating structures that may possess singularities or a stratification into simpler components. Within this context, the research defines the property of 'superintegrability'.
The sCMS systems are a class of exactly solvable many-body systems that have connections to various areas of physics and mathematics, including integrable systems, quantum field theory, and representation theory. The phase space for these sCMS systems is explicitly characterized as a stratified symplectic space. This particular stratified symplectic space is derived through a process known as Hamiltonian reduction, applied to the cotangent bundle over a compact Lie group. This construction method provides the specific setting for analyzing the superintegrability of sCMS systems.
Approach
The research methodology involved two key steps:
- Defining the concept: The initial step was to establish a formal definition for superintegrability when a Hamiltonian system is situated on a stratified symplectic space.
- Application and demonstration: Following the conceptual definition, the study applied this new notion to a specific class of systems: spin Calogero-Moser-Sutherland (sCMS) systems. The sCMS systems' phase space is identified as a stratified symplectic space resulting from the Hamiltonian reduction of the cotangent bundle over a compact Lie group. The research then proceeded to demonstrate that, for the specific case of $SU(3)$, these sCMS systems are superintegrable.
Findings
The core finding of this research is the demonstration of superintegrability for spin Calogero-Moser-Sutherland (sCMS) systems when the Lie group is $SU(3)$. This demonstration follows from the definition of superintegrability applied to Hamiltonian systems residing on a stratified symplectic space. The sCMS systems' phase space, critical to this analysis, is characterized as a stratified symplectic space obtained via the Hamiltonian reduction of the cotangent bundle over a compact Lie group.