Ghost-Free Quantisation of Higher Time-Derivative Theories via Non-Unitary Similarity Transformations

TL;DR

This paper introduces a novel non-unitary similarity transformation method to quantize higher time-derivative theories, effectively removing ghosts and ensuring a physically consistent, bounded spectrum with normalisable eigenstates.

Contribution

It presents a new approach inspired by PT-symmetric quantum mechanics to transform ghostly sectors into physically viable ones in higher time-derivative theories.

Findings

Successfully applied to a model related to the Pais-Uhlenbeck oscillator.

Achieved a spectrum bounded from below with normalisable eigenstates.

Provided a framework for ghost-free quantisation of higher derivative theories.

Abstract

We address the long-standing ``ghost problem" in higher time-derivative theories (HTDTs), where quantisation typically yields sectors with either unbounded spectra or non-normalisable eigenstates; both rendering the theory unphysical. We propose a novel method that preserves the bounded nature of the spectrum in one particular sector while restoring normalisability by employing a non-unitary similarity transformation. Inspired by techniques from pseudo/quasi-Hermitian PT-symmetric quantum mechanics, we construct a non-unitary map between two Hermitian Hamiltonians, converting ghostly sectors into physically viable ones. We demonstrate the feasibility of this approach using a concrete HTDT model, related to the Pais-Uhlenbeck oscillator, and show that the transformed system admits normalisable eigenstates and a spectrum bounded from below. This framework offers a consistent re-interpretation of HTDTs and extends the toolbox for constructing ghost-free quantum models.