CPT Symmetry Without Hermiticity

In the literature the $CPT$ theorem has only been established for Hamiltonians that are Hermitian. Here we extend the $CPT$ theorem to quantum field theories with non-Hermitian Hamiltonians. Our derivation is a quite minimal one as it requires only the time independent evolution of scalar products and invariance under complex Lorentz transformations. The first of these requirements does not force the Hamiltonian to be Hermitian. Rather, it forces its eigenvalues to either be real or to appear in complex conjugate pairs, forces the eigenvectors of such conjugate pairs to be conjugates of each other, and forces the Hamiltonian to admit of an antilinear symmetry. The latter requirement then forces this antilinear symmetry to be $CPT$, with Hermiticity of a Hamiltonian thus only being a sufficient condition for $CPT$ symmetry and not a necessary one. $CPT$ symmetry thus has primacy over Hermiticity, and it rather than Hermiticity should be taken as a guiding principle for constructing quantum theories. With conformal gravity being a non-Hermitian theory, our approach allows us to construct a positive, ghost-free norm for the theory, to thereby establish the unitarity of conformal gravity. Since our approach allows for complex energies and decays, our work justifies the use of the $CPT$ theorem in establishing the equality of the lifetimes of unstable particles that are charge conjugates of each other. In the quantum-mechanical limit where charge conjugation is separately conserved, the key results of the $PT$ symmetry program of Bender and collaborators are recovered.