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

Abstract

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.