Comments on the Dynamics of the Pais-Uhlenbeck Oscillator

TL;DR

This paper analyzes the quantum behavior of the Pais-Uhlenbeck oscillator, highlighting how its spectrum and unitarity depend on frequency differences and interactions, with implications for higher-derivative quantum models.

Contribution

It provides a detailed examination of the spectral properties and unitarity of the Pais-Uhlenbeck oscillator, including the nonstandard realization by Bender and Mannheim.

Findings

Spectrum is dense for different frequencies

Spectrum becomes continuous when frequencies are equal

Unitarity can be preserved or broken depending on conditions

Abstract

We discuss the quantum dynamics of the Pais-Uhlenbeck oscillator. The Lagrangian of this higher-derivative model depends on two frequencies. When the frequencies are different, the free PU oscillator has a pure point spectrum that is dense everywhere. When the frequencies are equal, the spectrum is continuous. It is not bounded from below, running from minus to plus infinity, but this is not disastrous as the Hamiltonian is still Hermitian and the evolution operator is still unitary. Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preserved. We discuss also the nonstandard realization of the PU oscillator suggested by Bender and Mannheim, where the spectrum of the free Hamiltonian is positive definite, but wave functions grow exponentially for large real values of canonical coordinates. The free nonstandard PU oscillator is unitary when the frequencies are different, but unitarity is broken in the equal frequencies limit.