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Spectrum-generating algebra and intertwiners of the resonant Pais–Uhlenbeck oscillator

Andreas Fring, Ian Marquette, Takano Taira

Published Jul 24, 2026• 0 citations
Publisher

Abstract

We study the quantum Pais–Uhlenbeck (PU) oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden (Formula presented) (Formula presented) Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant PU oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.

DOI
10.26181/33456340

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We study the quantum Pais–Uhlenbeck (PU) oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden (Formula presented) (Formula presented) Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant PU oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.
We study the quantum Pais-Uhlenbeck oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden su(2)su(2)su(2) Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant Pais-Uhlenbeck oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.
Hidden u(2,1)\mathfrak{u}(2,1)u(2,1) symmetry and Jordan chains in a resonant ghostly three-dimensional model
We investigate a three-dimensional ghostly Hamiltonian realisation of the fully degenerate resonant sixth-order Pais-Uhlenbeck oscillator. On the classical level, the phase-space flow is non-diagonalisable and decomposes into two complex-conjugate Jordan chains of length three, explaining the appearance of oscillatory solutions with secular terms. Upon quantisation, we construct intertwining operators whose quadratic combinations generate a hidden spectrum-generating u(2,1)\mathfrak{u}(2,1)u(2,1)-algebra. The associated descendant spaces are finite-dimensional invariant subspaces carrying non-trivial Jordan structure. Although these spaces admit a natural decomposition into irreducible modules of a distinguished sl2\mathfrak{sl}_2sl2​-subalgebra, this decomposition does not in general coincide with the Jordan decomposition of the Hamiltonian. We further derive a tri-Hamiltonian formulation from Lie point symmetries of the classical flow and show that the corresponding Hamiltonians are naturally encoded by the same hidden algebra. Nevertheless, unlike in the non-resonant case, no positive-definite linear combination of them generates the same dynamics. Finally, we analyse the common centraliser of the tri-Hamiltonian family in U(u(2,1))U(\mathfrak u(2,1))U(u(2,1)), showing that the natural higher-order candidate QQQ is reducible and yields no independent classical or quantum integral. The model thus provides a resonant higher-derivative system in which hidden u(2,1)\mathfrak{u}(2,1)u(2,1) symmetry, classical and quantum Jordan structures, and multi-Hamiltonian geometry coexist.
In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian HPU{H}_{\mathrm{PU}}HPU​ of the model is PT\mathcal{P}\mathcal{T}PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that HPU{H}_{\mathrm{PU}}HPU​ becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT\mathcal{P}\mathcal{T}PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schr\"odinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.
We discuss the quantum dynamics of the PU oscillator, i.e. the system with the LagrangianWhen Ω 1 = Ω 2 , the free PU oscillator has a pure point spectrum that is dense everywhere.When Ω 1 = Ω 2 , the spectrum is continuous, E ∈ {-∞, ∞}.The spectrum is not bounded from below, but that is not disastrous as the Hamiltonian is Hermitian and the evolution operator is unitary.Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preserved.We discuss also the nonstandard realization of the PU oscillator suggested by Bender and Mannheim, where the spectrum of the free Hamiltonian is positive definite, but wave functions grow exponentially for large real values of canonical coordinates.The free nonstandard PU oscillator is unitary at Ω 1 = Ω 2 , but unitarity is broken in the equal frequencies limit.
We develop a Bohmian analysis of a two-dimensional ghost Hamiltonian and its mapping to the degenerate Pais-Uhlenbeck model. Using Gaussian wavepackets, we derive the corresponding guidance equations, the centre and width evolution, and the quantum potential. We use these quantities to characterise bounded, quasi-semiclassical, spiral, and runaway regimes. The Bohmian trajectories provide a direct dynamical diagnostic of coherence, packet deformation, and quantum-classical separation. We then compare a bi-Hamiltonian pair consisting of the ghost Hamiltonian and a classically equivalent alternative formulation. While the two descriptions produce identical classical trajectories, they lead to different Bohmian trajectories and different quantum potentials evaluated along those trajectories. This demonstrates that classical equivalence need not extend to Bohmian quantum dynamics and identifies a concrete quantum ambiguity in the degenerate higher-derivative system.
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of uncoupled oscillators. This motivates us to propose a generalized Pais-Uhlenbeck Hamiltonian in terms of the usual harmonic oscillator creation and annihilation operators by including terms quadratic and linear in them. Such a Hamiltonian turns out to be essentially non-Hermitian but has an equivalent Hermitian representation which is reducible to a typically position-dependent reduced mass form.