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

TL;DR

This paper investigates the quantum Pais-Uhlenbeck oscillator at resonance, revealing how different classically equivalent Hamiltonians can lead to inequivalent quantum theories, and constructs a spectrum-generating algebra at the resonant point.

Contribution

It introduces a spectrum-generating algebra for the resonant Pais-Uhlenbeck oscillator and demonstrates the quantum inequivalence of classically equivalent Hamiltonians.

Findings

Constructed differential intertwiners generating a spectrum algebra

Identified a hidden $su(2)$ Lie algebra at resonance

Showed quantum inequivalence of classically equivalent Hamiltonians

Abstract

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 $su(2)$ 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.