  The Courant-Snyder theory for two-dimensional coupled linear optics is
presented, based on the systematic use of the real representation of the Dirac
matrices. Since any real $4\times 4$-matrix can be expressed as a linear
combination of these matrices, the presented Ansatz allows for a comprehensive
and complete treatment of two-dim. linear coupling. A survey of symplectic
transformations in two dimensions is presented. A subset of these
transformations is shown to be identical to rotations and Lorentz boosts in
Minkowski space-time. The transformation properties of the classical state
vector are formulated and found to be analog to those of a Dirac spinor. The
equations of motion for a relativistic charged particle - the Lorentz force
equations - are shown to be isomorph to envelope equations of two-dimensional
linear coupled optics. A universal and straightforward method to decouple
two-dimensional harmonical oscillators with constant coefficients by symplectic
transformations is presented, which is based on this isomorphism. The method
yields the eigenvalues (i.e. tunes) and eigenvectors and can be applied to a
one-turn transfer matrix or directly to the coefficient matrix of the linear
differential equation.
