A superintegrable quantum field theory

TL;DR

This work analyzes the quantum Gérard-Grellier Hamiltonian, a quartic resonant field theory on a circle, uncovering a remarkably rich, superintegrable structure with strictly integer spectra across the Hamiltonian and its conserved hierarchies. By exploiting block structure in $(N,M)$, introducing the minimal Hamiltonian $H_{\min}$, and constructing explicit eigenvector families, the authors reveal a hierarchy of square-integer energy levels and a ladder framework generated by operators like $K_+$ that connect sectors. They develop a quantum Lax-pair formalism and identify conserved towers that mirror classical invariant manifolds, while drawing fruitful parallels to Calogero and Benjamin-Ono systems to illuminate the spectral structure. These findings point to deep algebraic organization and potential deformations that preserve integrability, with implications for exact solvability in quantum field theories beyond standard quantum integrability.

Abstract

Gérard and Grellier proposed, under the name of the cubic Szegő equation, a remarkable classical field theory on a circle with a quartic Hamiltonian. The Lax integrability structure that emerges from their definition is so constraining that it allows for writing down an explicit general solution for prescribed initial data, and at the same time, the dynamics is highly nontrivial and involves turbulent energy transfer to arbitrarily short wavelengths. The quantum version of the same Hamiltonian is even more striking: not only the Hamiltonian itself, but also its associated conserved hierarchies display purely integer spectra, indicating a structure beyond ordinary quantum integrability. Here, we initiate a systematic study of this quantum system by presenting a mixture of analytic results and empirical observations on the structure of its eigenvalues and eigenvectors, conservation laws, ladder operators, etc.