Coping with the Pais-Uhlenbeck oscillator's ghosts in a canonical approach

TL;DR

This paper introduces a complex canonical transformation that converts the Pais-Uhlenbeck oscillator into two harmonic oscillators, ensuring a stable, unitary quantum model free of ghosts and negative norm states.

Contribution

The paper presents a novel complex canonical transformation method that resolves ghost issues in higher-order oscillators, extending to field theories.

Findings

The transformed model has energy bounded from below.

The approach ensures unitarity and positive definite inner product.

Negative norm states are eliminated in the transformed framework.

Abstract

A {\em complex} canonical transformation is found that takes the fourth order derivative Pais-Uhlenbeck oscillator into two independent harmonic oscillators thus showing that this model has energy bounded from below, unitary time-evolution and no negative norm states, or ghosts. Such transformation yields a positive definite inner product consistent with reality conditions in the Hilbert space. The method is illustrated by eliminating the negative norm states in a complex oscillator. Extensions to other higher order mechanical models and field theory are discussed.