An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator

TL;DR

This paper introduces a complex canonical transformation approach to the Pais-Uhlenbeck oscillator, effectively resolving ghost issues and unbounded energy problems at both classical and quantum levels by transforming it into two standard oscillators.

Contribution

It presents a novel complexification and transformation method that eliminates negative norm states and unbounded energy in the Pais-Uhlenbeck oscillator.

Findings

Negative norm states are eliminated at the quantum level.

The approach transforms the PU oscillator into two standard oscillators.

Quantum propagators are explicitly calculated.

Abstract

Our purpose in this paper is to analyze the Pais-Uhlenbeck (PU) oscillator using complex canonical transformations. We show that starting from a Lagrangian approach we obtain a transformation that makes the extended PU oscillator, with unequal frequencies, to be equivalent to two standard second order oscillators which have the original number of degrees of freedom. Such extension is provided by adding a total time derivative to the PU Lagrangian together with a complexification of the original variables further subjected to reality conditions in order to maintain the required number of degrees of freedom. The analysis is accomplished at both the classical and quantum levels. Remarkably, at the quantum level the negative norm states are eliminated, as well as the problems of unbounded below energy and non-unitary time evolution. We illustrate the idea of our approach by eliminating the negative norm states in a complex oscillator. Next, we extend the procedure to the Pais-Uhlenbeck oscillator. The corresponding quantum propagators are calculated using Schwinger's quantum action principle. We also discuss the equal frequency case at the classical level.