  We extend the $CPT$ theorem to quantum field theories with non-Hermitian
Hamiltonians and unstable states. Our derivation is a quite minimal one as it
requires only the time independent evolution of scalar products, invariance
under complex Lorentz transformations, and a non-standard but nonetheless
perfectly legitimate interpretation of charge conjugation as an anti-linear
operator. 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 anti-linear
symmetry. The latter two requirements then force this anti-linear symmetry to
be $CPT$, while forcing the Hamiltonian to be real rather than Hermitian. 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. We
show that the Euclidean time path integrals of a $CPT$ symmetric theory must
always be real. In the quantum-mechanical limit the key results of the $PT$
symmetry program of Bender and collaborators are recovered, with the
$C$-operator of the $PT$ symmetry program being identified with the linear
component of the charge conjugation operator.
