Is PT -symmetric quantum mechanics just quantum mechanics in a non-orthogonal basis?

One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian, as this guarantees that the eigenvalues are real. Recently there has been an interest in asking if H = H is a necessary condition, and has lead to the development of PT -symmetric quantum mechanics. This note shows that any finite physically acceptable non-Hermitian Hamiltonian is equivalent to doing ordinary quantum mechanics in a non-orthogonal basis. In particular, this means that there is no experimental distinction between PT -symmetric quantum mechanics and ordinary quantum mechanics for finite systems. In particular, the claim that PT -symmetric quantum mechanics allows for faster evolution than Hermitian quantum mechanics is shown to be a problem of physical interpretation. Introduction One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian. This is a sufficient condition for real eigenvalues and unitary time evolution. Recently the question of whether or not this condition is necessary has been raised, and a class of Hamiltonians called unbroken PT -symmetric Hamiltonians are shown to have real eigenvalues, where P and T are the parity and time-reversal operators respectively [1, 2]. To make time evolution unitary the inner product is determined by the Hamiltonian. Because of its novelty and some confusion in the foundations of the field, the first section of this paper will outline the major results of PT -symmetric quantum mechanics (PT -QM). ∗email: [email protected]