A superintegrable quantum field theory (2511.03373v1)

Abstract: G\'erard and Grellier proposed, under the name of the cubic Szeg\H{o} 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.

• The paper demonstrates superintegrability in quantum field theory via the Gerard-Grellier Hamiltonian, with all eigenvalues and conservation laws exhibiting integer spectra.
• It employs resonant interaction terms and a quartic, particle-number-conserving operator framework to enable finite-dimensional diagonalization and recursive eigenstate construction.
• The study bridges classical Lax integrability with quantum corrections, offering explicit energy bounds and analytical control over conserved quantities.

Superintegrability in Quantum Field Theory: The Gerard-Grellier Hamiltonian

Introduction and Context

The paper presents a systematic paper of a quantum field theory defined by the Gerard-Grellier (GG) Hamiltonian, a quartic, particle-number-conserving operator acting on bosonic modes indexed by non-negative integers. The GG Hamiltonian is distinguished by its resonance condition $n+m=k+l$ in the interaction terms, which imparts a highly constrained integrability structure. The classical counterpart, known as the cubic Szegő equation, is Lax-integrable and exhibits explicit solutions, dynamically invariant manifolds, and turbulent energy transfer. The quantum version, however, displays even more remarkable features: all eigenvalues of the Hamiltonian and its associated conservation laws are integers, a property not typical of quantum integrable systems and indicative of superintegrability.

Resonant Hamiltonian Systems and Physical Realizations

The GG Hamiltonian is a special case of a broader class of resonant Hamiltonians arising from weakly nonlinear approximations to Hamiltonian PDEs with equispaced linear spectra. Examples include the nonlinear Schrödinger equation with harmonic trapping and wave dynamics in asymptotically AdS spacetimes. The resonance condition ensures that only certain mode couplings survive in the effective dynamics, leading to a finite-dimensional reduction in each sector of fixed particle number $N$ and momentum $M$. This structure facilitates numerical diagonalization and analytic investigation, independent of the specific coupling coefficients.

Physical realizations of such resonant systems are possible in ultracold atomic gases in harmonic traps and chiral edge states in condensed matter systems, provided the mode couplings can be engineered appropriately. The quantum Hamiltonian emerges naturally as the weak-coupling limit of interacting quantum fields with highly degenerate spectra.

Classical Integrability and Lax Structure

The classical GG Hamiltonian admits a Lax pair formulation, leading to an infinite hierarchy of conservation laws. The dynamics preserves invariant manifolds characterized by meromorphic functions with a fixed number of poles, corresponding to solitonic solutions. The explicit general solution for arbitrary initial data is available via the Lax formalism, and the system supports a rich set of conserved quantities beyond those generated by the standard trace construction.

The Lax pair in mode space is given by operators $\mathcal{L}$ and $\mathcal{M}$ acting on auxiliary vectors, with compatibility conditions ensuring the conservation of traces and more general quantities involving projectors onto special vectors. These structures underpin the integrability and allow for explicit analytic control over the classical dynamics.

Quantum Integrability, Integer Spectra, and Conservation Laws

Upon quantization, the GG Hamiltonian retains the resonance structure but introduces operator ordering ambiguities. Notably, naive quantization of classical conservation laws fails to produce valid quantum conserved operators. Instead, the quantum system exhibits a hierarchy of conservation laws, all with integer spectra, including the Hamiltonian itself and operators such as $H_{\min}$, which commutes with $H$ and is constructed via mode index minimization.

The Hilbert space decomposes into finite-dimensional blocks labeled by $(N, M)$, with basis states corresponding to integer partitions. Diagonalization within these blocks reveals the integer eigenvalue property, and the presence of ladder operators (e.g., $K_+$) enables transitions between eigenstates with integer shifts in energy. The algebraic structure is further enriched by shift operators and recursive constructions of eigenbases.

Explicit Construction of Eigenstates and Energy Bounds

Energy bounds for $H$ and $H_{\min}$ are derived via sum-of-squares decompositions, leading to explicit criteria for the saturation of these bounds. The highest energy state in each $(N, M)$ block is unique and constructed by repeated application of a raising operator $Z$ on the vacuum state. The corresponding eigenvalues are $(N-1)(N+2M)/2$ for $H$ and $M^2$ for $H_{\min}$.

Within the top eigenspace of $H_{\min}$, a complete basis is constructed by shifting highest energy states from lower blocks and populating the zero mode. The Hamiltonian $H$ is tridiagonal in this basis, and its spectrum follows a square integer sequence, a property proved by induction and related to the factorization method in quantum mechanics. The eigenvectors are explicitly given in terms of combinatorial coefficients and correspond to coherent-like states in the semiclassical limit.

Lower eigenspaces of $H_{\min}$ are analyzed similarly, with recursive constructions and tridiagonal Hamiltonian matrices yielding further square integer spectra. Degeneracies arise in these sectors, and the ladder operator $K_+$ plays a central role in connecting eigenstates across different energy levels.

Quantum Lax Pair and Higher Conservation Laws

A quantum Lax pair is constructed, closely mirroring the classical structure but with quantum corrections. The evolution equation $[HI+\mathcal{M},\mathcal{L}]=0$ holds, and an infinite family of conservation laws is generated via operators of the form $(\vec{1},\mathcal{L}^n\vec{1})$. However, normal ordering introduces divergences, necessitating systematic subtraction of lower-order conservation laws to obtain finite operators. Empirically, all such conservation laws exhibit integer spectra.

An alternative construction of conservation laws is provided via recursive commutators, yielding higher-order polynomial operators with explicit quantum corrections. These towers of conservation laws are numerically verified to commute with the Hamiltonian within each block.

Connections to Classical Invariant Manifolds and Other Integrable Systems

The eigenstates constructed correspond to classical invariant manifolds in the semiclassical limit, with mode occupation numbers matching classical observables. The shift operator structure mirrors the embedding of invariant manifolds in the classical theory, and the ladder of eigenstates reflects the hierarchy of classical solutions.

Parallels are drawn with Calogero systems and the quantum Benjamin-Ono equation, both of which exhibit integer spectra and non-standard constructions of conservation laws. However, the GG Hamiltonian remains distinct as a field theory, and direct mappings to these systems are nontrivial due to differences in mode structure and conservation law generation.

Implications and Future Directions

The GG Hamiltonian exemplifies superintegrability in quantum field theory, with integer spectra, explicit energy bounds, and a rich algebraic structure of conservation laws and ladder operators. The analytic control achieved in the top eigenspaces and the recursive construction of eigenstates provide a foundation for further exploration of the full spectrum and eigenfunctions.

Theoretical implications include the possibility of extending the quantum Lax pair formalism, resolving divergences in conservation law construction, and uncovering deeper connections to classical integrable systems and other quantum models. Practically, the structure of the Hilbert space and the explicit eigenstate construction may inform numerical algorithms for diagonalization and the paper of quantum turbulence and energy transfer.

Future developments may involve the application of $r$-matrix theory, reformulation of the Lax pair in terms of finite-dimensional matrices with spectral parameters, and exploration of integrable deformations. The similarities with the quantum Benjamin-Ono equation suggest promising avenues for analytic solution and classification of eigenstates.

Conclusion

The paper of the GG Hamiltonian reveals a quantum field theory with superintegrable properties, integer spectra, and a hierarchy of conservation laws. The interplay between classical integrability, quantum corrections, and algebraic structures provides a fertile ground for further research in quantum integrable systems, with potential applications in mathematical physics, condensed matter, and quantum information. The explicit constructions and analytic results presented lay the groundwork for a comprehensive understanding of superintegrability in quantum field theory and its connections to broader integrable phenomena.