arXiv:2606.02290

Hidden $\mathfrak{u}(2,1)$ symmetry and Jordan chains in a resonant ghostly three-dimensional model

Andreas Fring, Ian Marquette

quant-phhep-thmath-phSubmitted Jun 1, 2026• Announced Jun 2, 2026

Abstract

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 $\mathfrak{u}(2,1)$-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 $\mathfrak{sl}_2$-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 $U(\mathfrak u(2,1))$, showing that the natural higher-order candidate $Q$ is reducible and yields no independent classical or quantum integral. The model thus provides a resonant higher-derivative system in which hidden $\mathfrak{u}(2,1)$ symmetry, classical and quantum Jordan structures, and multi-Hamiltonian geometry coexist.

Comments: 21 pages

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arXiv:2606.02290

Hidden $\mathfrak{u}(2,1)$ symmetry and Jordan chains in a resonant ghostly three-dimensional modelHidden $\mathfrak{u}(2,1)$ symmetry and Jordan chains in a resonant ghostly three-dimensional model

Andreas Fring, Ian Marquette

quant-phhep-thmath-phSubmitted Jun 1, 2026• Announced Jun 2, 2026

arXiv HTML PDF

Abstract

Upon quantisation, we construct intertwining operators whose quadratic combinations generate a hidden spectrum-generating

\mathfrak{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

\mathfrak{sl}_2

-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(\mathfrak u(2,1))

, showing that the natural higher-order candidate

Q

is reducible and yields no independent classical or quantum integral.

The model thus provides a resonant higher-derivative system in which hidden

\mathfrak{u}(2,1)

symmetry, classical and quantum Jordan structures, and multi-Hamiltonian geometry coexist.

Comments: 21 pages

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quant-pharXiv:2607.04052Jul 7, 20260 citations

Hidden

\mathfrak{u}(2,1)

symmetry and Jordan chains in a resonant ghostly three-dimensional model

Upon quantisation, we construct intertwining operators whose quadratic combinations generate a hidden spectrum-generating

\mathfrak{u}(2,1)

\mathfrak{sl}_2

-subalgebra, this decomposition does not in general coincide with the Jordan decomposition of the Hamiltonian.

U(\mathfrak u(2,1))

, showing that the natural higher-order candidate

Q

is reducible and yields no independent classical or quantum integral.

The model thus provides a resonant higher-derivative system in which hidden

\mathfrak{u}(2,1)

symmetry, classical and quantum Jordan structures, and multi-Hamiltonian geometry coexist.

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.

I_4^{\rm o}

and

I_4^{\rm e}

, of odd and even momentum parity. A single octic relation leaves the polynomial invariant ring with transcendence degree four. Solving the complete first-integral equation shows that every rational first integral is a rational function of

H_1,H_2,H_3,I_4^{\rm o}

, so that the rational invariant field likewise has generic functional rank four. A branch-free transcendental invariant raises the generic smooth functional rank to five on the invariant open set

Δ>0

. The system is therefore minimally superintegrable in the polynomial and rational classes and maximally superintegrable on this invariant domain when transcendental integrals are admitted. Weyl quantisation yields exact higher-order differential symmetries in all three Poisson realisations. The branch-free fifth classical invariant also admits a densely defined nonlocal quantum counterpart, providing a concrete quantum realisation of the transcendental completion.

Non-standard quantum algebras and finite dimensional

\mathcal{PT}

-symmetric systems

In this work,

\mathcal{PT}

-symmetric Hamiltonians defined on quantum

sl(2, \mathbb R)

algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard

U_{z}(sl(2, \mathbb R))

Hopf algebra deformation of

sl(2, \mathbb R)

. By making use of a particular boson representation of the generators of

U_{z}(sl(2, \mathbb R))

, both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the

\mathcal{PT}

-transformation. In terms of these operators, we construct several finite dimensional

\mathcal{PT}

-symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact

\mathcal{PT}

-symmetry and the broken

\mathcal{PT}

-symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter

z

has to be identified with the detuning parameter of the system.

C_{μν}{}^λ

and

B_{μν}{}^λ

. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map,

\mathcal D(X,Y)Z=XZY-\frac12\{YX,Z\},

whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations

(B+C)

and

(B-C)

generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.