Exact PT-Symmetry Is Equivalent to Hermiticity

We show that a quantum system possessing an exact antilinear symmetry, in particular PT -symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT -symmetry to a Hermitian Hamiltonian. We apply our general results to PT symmetry in finite-dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT -symmetric field theories are equivalent to certain nonlocal Hermitian field theories. The interest in PT -symmetric quantum mechanics [1] has its origin in the idea that since the CPT theorem follows from the axioms of local quantum field theory, one might obtain a more general field theory by replacing the axiom of the Hermiticity of the Hamiltonian by the requirement of CPT -symmetry. The simplest nonrelativistic example of such theories is the PT -symmetric quantum mechanics. During the past five years there have appeared dozens of publications exploring the properties of the PT -symmetric Hamiltonians. Among these is a series of articles [2]-[9] by the present author that attempt to demonstrate that PT -symmetry can be understood most conveniently using the theory of pseudo-Hermitian operators (See also [10].) The recent articles of Bender, Meisimger, and Wang [11, 12], however, show that the mystery associated with PT -symmetry has surprisingly survived the comprehensive treatment offered by pseudo-Hermiticity. The aim of the present article is to provide a conclusive proof that the exact PT -symmetry is equivalent to Hermiticity. In particular, we offer a complete ∗E-mail address: [email protected]