  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.
