The odd-order Pais-Uhlenbeck oscillator

TL;DR

This paper analyzes the Hamiltonian structure of the odd-order Pais-Uhlenbeck oscillator, revealing issues with unbounded energy and proposing an alternative formulation to address the ghost problem in quantum theory.

Contribution

It introduces a new canonical formulation for the odd-order Pais-Uhlenbeck oscillator to circumvent the unbounded Hamiltonian and ghost issues.

Findings

The system's Noether integral is unbounded from below.

An alternative canonical formulation is constructed.

The new formulation aims to resolve the ghost problem.

Abstract

We consider a Hamiltonian formulation of the (2n+1)-order generalization of the Pais-Uhlenbeck oscillator with distinct frequencies of oscillation. This system is invariant under time translations. However, the corresponding Noether integral of motion is unbounded from below and can be presented as a direct sum of 2n one-dimensional harmonic oscillators with an alternating sign. If this integral of motion plays a role of a Hamiltonian, a quantum theory of the Pais-Uhlenbeck oscillator faces a ghost problem. We construct an alternative canonical formulation for the system under study to avoid this nasty feature.