Imaginary-Scaling versus Indefinite-Metric Quantization of the Pais-Uhlenbeck Oscillator

TL;DR

This paper proposes a new quantization method for the Pais-Uhlenbeck Oscillator that maintains unitarity and redefines ghost states without altering the energy spectrum, offering an alternative to indefinite-metric schemes.

Contribution

It introduces an imaginary-scaling quantization approach that constructs a positive-definite Hilbert space and modifies creation operators to handle ghost states.

Findings

The method preserves the energy spectrum of the oscillator.

It achieves a consistent, unitary quantization without indefinite metrics.

Ghost states are incorporated without negative norm issues.

Abstract

Using the Pais-Uhlenbeck Oscillator as a toy model, we outline a consistent alternative to the indefinite-metric quantization scheme that does not violate unitarity. We describe the basic mathematical structure of this method by giving an explicit construction of the Hilbert space of state vectors and the corresponding creation and annihilation operators. The latter satisfy the usual bosonic commutation relation and differ from those of the indefinite-metric theories by a sign in the definition of the creation operator. This change of sign achieves a definitization of the indefinite-metric that gives life to the ghost states without changing their contribution to the energy spectrum.