In recent years a great deal of work has been done on understanding
quantum theories which have a non-Hermitian Hamiltonian, but which
nevertheless possess a real energy spectrum, due in many cases to an
underlying PT symmetry. In some circumstances, however, this symmetry
may be broken, in which case some or all of the eigenvalues coalesce
into complex conjugate pairs.
These ideas have now found practical application in the field of
classical optics, where the paraxial equation
of propagation is formally identical to the Schroedinger equation,
with the role of the quantum potential being
played by the variations in the refractive index.
Wave guides and optical lattices with the carefully coordinated
regions of loss and gain implied by PT symmetry have many remarkable
properties, some of which may prove useful in optical devices needed
for optical computers.
We review both the original concepts of PT symmetry in quantum
mechanics and their application in classical optics.