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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.

Astrophysical evidence for the non‐Hermitian but PT ‐symmetric Hamiltonian of conformal gravity