Comment on "Dirac Quantization of Pais-Uhlenbeck Fourth Order Oscillator"

TL;DR

This paper critically examines the equal-frequency limit of the Pais-Uhlenbeck oscillator, clarifying the subtlety in the limiting process and showing that the resulting theory remains unsatisfactory.

Contribution

It provides a detailed analysis of the equal-frequency limit, clarifying the proper procedure and discussing the implications for the space of states in different metric scenarios.

Findings

The equal-frequency limit is subtle and involves complex state space structures.

The limiting theory is generally unsatisfactory regardless of the limiting procedure.

Proper limit definition does not resolve issues with the theory's physical viability.

Abstract

The structure of Pais-Uhlenbeck oscillator in the equal-frequency limit has been recently studied by Mannheim and Davidson [Phys.Rev. A71 (2005), 042110]. It appears that taking this limit, as presented in the above paper, is quite subtle and the resulting structure of space of states - involved. In order to clarify the situation we present here the proper way of taking the equal-frequency limit, first under the assumption that the scalar product in the space of states is positive defined. We discuss also the case of indefinite metric space of states. We show that, irrespective of the way the limit is defined, the limiting theory can be hardly viewed as satisfactory.