PT-symmetry in quantum mechanics and its applications in classical optics

One of the key tenets of standard quantum mechanics is Hermiticity, which, among other things, guarantees the reality of energy eigenvalues. However, there exists a whole class of Hamiltonians which are not Hermitian but nonetheless possess a completely real spectrum. These Hamiltonians, of which the paradigm is $H=p^2+ix^3$, are PT symmetric, whereby $x$ goes to $-x$ and $i$ to $-i$. I will review the status of such Hamiltonians, which have been the subject of intensive study over the last few years. An unexpected development was the realization that ideas developed in the context of quantum mechanics could be applied to classical optics. There is a standard approximation in optics, the paraxial approximation, where the equation for propagation has the form of an analogue Schroedinger equation, with the longitudinal distance $z$ playing the role of time and the refractive index taking the role of the potential. PT symmetry implies a medium with both gain and loss balanced in a particular way. The advantage is that real eigenvalues correspond to propagation without exponential growth or decay. Artificial PT-symmetric media have many unusual and potentially useful properties.