# Quartic propagators, negative norms and the physical spectrum

## TL;DR

This paper explores the viability of theories with quartic propagators and negative norm states, proposing a lattice simulation approach to determine if such theories can be physically consistent, potentially impacting quantum gravity models.

## Contribution

It introduces a specific lattice-amenable model involving a scalar triplet with quartic derivatives coupled to SU(2) gauge fields, and discusses how strong interactions can modify dispersion relations.

## Key findings

- Path integral methods suggest some quartic theories may be acceptable.
- A lattice simulation example involving scalar triplet and SU(2) gauge fields is proposed.
- Strong interactions can alter dispersion relations, maintaining a healthy effective field theory.

## Abstract

Many arguments against quartic propagators, negative norm states and related effects concern the sicknesses which occur when the spectrum of the free particle Hamiltonian is formed. However, if the theory is more complicated, for example involving confinement such that the particle in question does not appear in the physical spectrum, those considerations do not apply directly. Path integral methods suggest that some of these may be acceptable theories. I provide an example that should be able to be simulated on a lattice which then allows a non-perturbative resolution of this question. In its SU(2) version it involves a scalar triplet with a quartic derivative Lagrangian coupled to the SU(2) gauge field. If this is verified to be a healthy theory, it could open new avenues in model building. I also discuss how strong interactions can dynamically modify the dispersion relation leaving a healthy effective field theory, using conformal gravity coupled to a Yang-Mills theory as an example. Such a theory could possibly form a UV completion for quantum gravity.