Giving up the ghost

TL;DR

This paper demonstrates that the Pais-Uhlenbeck higher-derivative quantum model is free of ghost states when correctly quantized as a PT-symmetric system, challenging previous beliefs about its physical acceptability.

Contribution

It shows that the Pais-Uhlenbeck model is a consistent quantum theory without ghosts when properly treated as PT symmetric, correcting prior misconceptions.

Findings

The model has a real, bounded energy spectrum.

All quantum states have positive norm.

Ghost states are an artifact of incorrect Hermiticity assumptions.

Abstract

The Pais-Uhlenbeck model is a quantum theory described by a higher-derivative field equation. It has been believed for many years that this model possesses ghost states (quantum states of negative norm) and therefore that this model is a physically unacceptable quantum theory. The existence of such ghost states was believed to be attributable to the field equation having more than two derivatives. This paper shows that the Pais-Uhlenbeck model does not possess any ghost states at all and that it is a perfectly acceptable quantum theory. The supposed ghost states in this model arise if the Hamiltonian of the model is (incorrectly) treated as being Dirac Hermitian (invariant under combined matrix transposition and complex conjugation). However, the Hamiltonian is {\it not} Dirac Hermitian, but rather it is PT symmetric. When it is quantized correctly according to the rules of PT quantum mechanics, the energy spectrum is real and bounded below and all of the quantum states have positive norm.