A Superintegrable Quantum Field Theory

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.