The Marsden-Weinstein reduction structure of integrable dynamical systems and a generalized exactly solvable quantum superradiance model

TL;DR

This paper introduces a geometric approach to integrable systems using Marsden-Weinstein reduction and presents a new exactly solvable quantum superradiance model involving fermionic media and electromagnetic interactions.

Contribution

It develops a novel geometric framework for integrable systems and proposes a new exactly solvable quantum superradiance model with detailed spectral and Hamiltonian analysis.

Findings

Established a connection between Marsden-Weinstein reduction and integrable systems

Derived the spectral problem and R-matrix structure for the new model

Discussed quantum excitations, solitons, and thermodynamical stability

Abstract

An approach to describing nonlinear Lax type integrable dynamical systems of modern mathematical and theoretical physics, based on the Marsden-Weinstein reduction method on canonically symplectic manifolds \ with group symmetry, is proposed. Its natural relationship with the well known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated R-matrix approach is analyzed. A new generalized exactly solvable spatially one-dimensional quantum superradiance model, describing a charged fermionic medium interacting with external electromagnetic field, is suggested. The Lax type operator spectral problem is presented, the related $R$-structure is calculated. The Hamilton operator renormalization procedure subject to a physically stable vacuum is described, the quantum excitations and quantum solitons, related with the thermodynamical equilibrity of the model, are discussed.