Equivalence of a Complex PT -Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly

In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian PT -symmetric wrong-sign quartic Hamiltonian H = 1 p 2 − gx 4 has the same spectrum as the conventional Hermitian Hamiltonian ˜ H = 1 p 2 + 4gx 4 − √ 2g x. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian PT -symmetric Hamiltonian. This anomaly in the Hermitian form of a PT -symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into H. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to −φ 4 quantum field theory in higher-dimensional space-time are discussed.