  Second-order-derivative plus fourth-order-derivative gravity is the
ultraviolet completion of second-order-derivative quantum Einstein gravity.
While it achieves renormalizability through states of negative Dirac norm, the
unitarity violation that this would entail can be postponed to Planck energies.
As we show in this paper the theory has a different problem, one that occurs at
all energy scales, namely that the Dirac norm of the vacuum of the theory is
not finite. To establish this we present a procedure for determining the norm
of the vacuum in any quantum field theory. With the Dirac norm of the vacuum of
the second-order-derivative plus fourth-order-derivative theory not being
finite, the Feynman rules that are used to establish renormalizability are not
valid, as is the assumption that the theory can be used as an effective theory
at energies well below the Planck scale. This lack of finiteness is also
manifested in the fact that the Minkowski path integral for the theory is
divergent. Because the vacuum Dirac norm is not finite, the Hamiltonian of the
theory is not Hermitian. However, it turns out to be $PT$ symmetric. And when
one continues the theory into the complex plane and uses the $PT$ symmetry
inner product, viz. the overlap of the left-eigenstate of the Hamiltonian with
its right-eigenstate, one then finds that for the vacuum this norm is both
finite and positive, the Feynman rules now are valid, the Minkowski path
integral now is well behaved, and the theory now can serve as a low energy
effective theory. Consequently, the theory can now be offered as a fully
consistent, unitary and renormalizable theory of quantum gravity.
