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

TL;DR

This work analyzes the resonant (degenerate) Pais-Uhlenbeck oscillator within a ghostly two-dimensional Hamiltonian framework, revealing a hidden spectrum-generating $\mathfrak{su}(2)$ algebra that organises generalised eigenvectors into finite Jordan chains. It demonstrates that classically equivalent Hamiltonians can quantise to inequivalent quantum theories: one yields a non-diagonalisable spectrum with Jordan blocks, while another provides a fully diagonalisable spectrum with genuine degeneracies. The study also shows that a bi-Hamiltonian approach cannot produce a positive-definite Hamiltonian at resonance, highlighting a sharp qualitative difference from the non-degenerate case. A partial factorisation of the ground state suggests a restricted, effectively one-dimensional ghost-free sector, but does not resolve the overall ghost problem. Together, these results position the resonant PU oscillator as a crucial testbed for algebraic quantisation methods and for understanding ghost issues in higher-time-derivative systems.

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.