Comment on “PT-Symmetric versus Hermitian Formulations of Quantum Mechanics”

We explain why the main conclusion of Bender et al, J. Phys. A 39, 1657 (2006), regarding the practical superiority of the non-Hermitian description of PT -symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable. PACS number: 03.65.-w Recently Bender, Chen, and Milton [1] have employed the path integral method to examine the perturbative calculation of the ground-state energy and a one-point Green’s function for the PT -symmetric cubic anharmonic oscillator. They performed this calculation in both the PT -symmetric (pseudo-Hermitian) and the Hermitian descriptions of this model and concluded that the use of the latter description leads to practical difficulties that are “severe and virtually insurmountable · · · ”, and that such difficulties do not arise in the former description. As explained in [2], the level of the difficulty of a calculation in PT -symmetric quantum mechanics depends on the quantity that one chooses to calculate. If one wishes to calculate the expectation value of a canonical pair of basic observables of the theory, both the descriptions/representations involve dealing with practical problems with the same degree of difficulty. The reason why the difficulties with the PT -symmetric representation do not surface in the interesting calculations reported in [1] is that the authors only calculate the ground-state energy and the one-point Green’s function for the operator x. The difficulties with the calculation of groundand excited-state energies in the Hermitian representation is already clear from the complicated expression for the Hermitian Hamiltonian