  With fourth-order derivative theories leading to propagators of the generic
ghost-like $1/(k^2-M_1^2)-1/(k^2-M_2^2)$ form, it would appear that such
theories have negative norm ghost states and are not unitary. However on
constructing the associated quantum Hilbert space for the free theory that
would produce such a propagator, Bender and Mannheim found that the Hamiltonian
of the free theory is not Hermitian but is instead $PT$ symmetric, and that
there are in fact no negative norm ghost states, with all Hilbert space norms
being both positive and preserved in time. Even though perturbative radiative
corrections cannot change the signature of a Hilbert space inner product,
nonetheless it is not immediately apparent how such a ghost-like propagator
would not then lead to negative probability contributions in loop diagrams.
Here we obtain the relevant Feynman rules and show that all states obtained in
cutting intermediate lines in loop diagrams have positive norm. Also we show
that due to the specific way that unitarity (conservation of probability) is
implemented in the theory, negative signatured discontinuities across cuts in
loop diagrams are cancelled by a novel and unanticipated contribution of the
states in which tree approximation (no loop) graphs are calculated, an effect
that is foreign to standard Hermitian theories. Perturbatively, the
fourth-order derivative theory is thus viable. The theory associated with the
pure massless $1/k^4$ propagator is equally shown to be perturbatively viable.
