  We show that some higher derivative theories have a BRST symmetry. This
symmetry is due to the higher derivative structure and is not associated to any
gauge invariance. If physical states are defined as those in the BRST
cohomology then the only physical state is the vacuum. All negative norm
states, characteristic of higher derivative theories, are removed from the
physical sector. As a consequence, unitarity is recovered but the S-matrix is
trivial. We show that a class of higher derivative quantum gravity theories
have this BRST symmetry so that they are consistent as quantum field theories.
Furthermore, this BRST symmetry may be present in both relativistic and
non-relativistic systems.
