  We study a model of n coupled scalar fields in Minkowski spacetime where all
masses degenerate, which is considered as a toy model of polycritical gravity
on AdS spacetime. We quantize this model within the Becchi-Rouet-Stora-Tyutin
(BRST) scheme by introducing n Faddeev-Popov (FP) ghost fields. Extending a
BRST quartet generated by two scalars and two FP ghosts to n scalars and n FP
ghosts, there remains a physical subspace with positive norm for odd n, but
there exists only the vacuum for even n. This clearly shows a non-triviality of
odd-higher order derivative scalar field theories. This is helpful to
understand the truncation mechanism which is used to obtain a unitary conformal
field theory dual to linearized polycritical gravity. It turns out that the
truncation mechanism is nothing but a general quartet mechanism appeared when
introducing the FP ghost action.
