  Optics has proved a fertile ground for the experimental simulation of quantum
mechanics. Most recently, optical realizations of $\mathcal{PT}$-symmetric
quantum mechanics have been shown, both theoretically and experimentally,
opening the door to international efforts aiming at the design of practical
optical devices exploiting this symmetry. Here, we focus on the optical
$\mathcal{PT}$-symmetric dimer, a two-waveguide coupler were the materials show
symmetric effective gain and loss, and provide a review of the linear and
nonlinear optical realizations from a symmetry based point of view. We go
beyond a simple review of the literature and show that the dimer is just the
smallest of a class of planar $N$-waveguide couplers that are the optical
realization of Lorentz group in 2+1 dimensions. Furthermore, we provide a
formulation to describe light propagation through waveguide couplers described
by non-Hermitian mode coupling matrices based on a non-Hermitian generalization
of Ehrenfest theorem.
