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We extend the $CPT$ theorem to quantum field theories with non-HermitianHamiltonians and unstable states. Our derivation is a quite minimal one as itrequires only the time independent evolution of scalar products, invarianceunder complex Lorentz transformations, and a non-standard but nonethelessperfectly legitimate interpretation of charge conjugation as an anti-linearoperator. The first of these requirements does not force the Hamiltonian to beHermitian. Rather, it forces its eigenvalues to either be real or to appear incomplex conjugate pairs, forces the eigenvectors of such conjugate pairs to beconjugates of each other, and forces the Hamiltonian to admit of an anti-linearsymmetry. The latter two requirements then force this anti-linear symmetry tobe $CPT$, while forcing the Hamiltonian to be real rather than Hermitian. Ourwork justifies the use of the $CPT$ theorem in establishing the equality of thelifetimes of unstable particles that are charge conjugates of each other. Weshow that the Euclidean time path integrals of a $CPT$ symmetric theory mustalways 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 linearcomponent of the charge conjugation operator.

Extension of the $CPT$ Theorem to non-Hermitian Hamiltonians and Unstable States