Purely virtual particles versus Lee-Wick ghosts: physical Pauli-Villars fields, finite QED and quantum gravity

TL;DR

This paper compares Lee-Wick models with purely virtual particles, arguing for the latter's advantages in maintaining Hermitian limits and consistent observable properties, and explores methods to partially remove Lee-Wick ghosts for finite QED and quantum gravity.

Contribution

It introduces a novel approach to treat Lee-Wick ghosts by replacing them with superpositions, enabling finite QED and consistent Pauli-Villars fields without infinite masses.

Findings

Purely virtual particles lead to Hermitian classical limits.

Replacing LW ghosts with superpositions can make Pauli-Villars fields observable.

The method faces challenges with general covariance in quantum gravity.

Abstract

We reconsider the Lee-Wick (LW) models and compare their properties to the properties of the models that contain purely virtual particles. We argue against the LW premise that unstable particles can be removed from the sets of incoming and outgoing states in scattering processes. The removal leads to a non-Hermitian classical limit, besides clashing with the observation of the muon. If, on the other hand, all the states are included, the LW models have a Hamiltonian unbounded from below or negative norms. Purely virtual particles, on the contrary, lead to a Hermitian classical limit and are absent from the sets of incoming and outgoing states without implications on the observation of long-lived unstable particles. We give a vademecum to summarize the properties of most options to treat abnormal particles. We study a method to remove the LW ghosts only partially, by saving the physical particles they contain. Specifically, we replace a LW ghost with a certain superposition of a purely virtual particle and an ordinary particle, and drop only the former from the sets of the external states. The trick can be used to make the Pauli-Villars fields consistent and observable, without sending their masses to infinity, or to build a finite QED, by tweaking the original Lee-Wick construction. However, it has issues with general covariance, so it cannot be applied as is to quantum gravity, where a manifestly covariant decomposition requires the introduction of a massive spin-2 multiplet.