Classical and quantum ghosts

TL;DR

This paper reviews the concept of ghost fields in theories, highlighting their classical and quantum instabilities, and discusses conditions under which low-energy ghosts can exist without conflicting with observations.

Contribution

It provides a comprehensive, self-contained review of ghost fields, their instabilities, and the constraints on their existence in effective theories, including implications for Lorentz-invariance and locality.

Findings

Ghost fields cause classical instabilities when interacting with standard fields.

Quantum instabilities of ghosts are more severe and restrict their presence to low-energy effective theories.

Compatibility with observations requires breaking Lorentz-invariance or locality, or having a low cut-off.

Abstract

The aim of these notes is to provide a self-contained review of why it is generically a problem when a solution of a theory possesses ghost fields among the perturbation modes. We define what a ghost field is and we show that its presence is associated to a classical instability whenever the ghost field interacts with standard fields. We then show that the instability is more severe at quantum level, and that perturbative ghosts can exist only in low energy effective theories. However, if we don't consider very ad-hoc choices, compatibility with observational constraints implies that low energy effective ghosts can exist only at the price of giving up Lorentz-invariance or locality above the cut-off, in which case the cut-off has to be much lower that the energy scales we currently probe in particle colliders. We also comment on the possible role of extra degrees of freedom which break Lorentz-invariance spontaneously.