  In this review we discuss the connection between two seemingly disparate
topics, macroscopic gravity on astrophysical scales and Hamiltonians that are
not Hermitian but $PT$ symmetric on microscopic ones. In particular we show
that the quantum-mechanical unitarity problem of the fourth-order derivative
conformal gravity theory is resolved by recognizing that the scalar product
appropriate to the theory is not the Dirac norm associated with a Hermitian
Hamiltonian but is instead the norm associated with a non-Hermitian but
$PT$-symmetric Hamiltonian. Moreover, the fourth-order theory Hamiltonian is
not only not Hermitian, it is not even diagonalizable, being of Jordan-block
form. With $PT$ symmetry we establish that conformal gravity is consistent at
the quantum-mechanical level. In consequence, we can apply the theory to data,
to find that the theory is capable of naturally accounting for the systematics
of the rotation curves of a large and varied sample of 138 spiral galaxies
without any need for dark matter. The success of the fits provides evidence for
the relevance of non-diagonalizable but $PT$-symmetric Hamiltonians to physics.
