<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi mathvariant="script">PT</mml:mi> </mml:math> -symmetric quantum mechanics
Carl M. Bender, Daniel Hook
Abstract
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 $\mathcal{PT}$-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of $\mathcal{PT}$-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 $\mathcal{PT}$ symmetry have been applied in the context of many different branches of physics. This review introduces the concepts of $\mathcal{PT}$ symmetry by focusing on elementary one-dimensional $\mathcal{PT}$-symmetric quantum and classical mechanics and relies, in particular, on oscillator models to illustrate and explain the basic properties of $\mathcal{PT}$-symmetric quantum theory.