𝒫𝒯-symmetric quantum mechanics

It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a 𝒫𝒯-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of 𝒫𝒯-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of 𝒫𝒯 symmetry have been applied in the context of many different branches of physics. This review introduces the concepts of 𝒫𝒯 symmetry by focusing on elementary one-dimensional 𝒫𝒯-symmetric quantum and classical mechanics and relies, in particular, on oscillator models to illustrate and explain the basic properties of 𝒫𝒯-symmetric quantum theory.