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The Courant-Snyder theory for two-dimensional coupled linear optics ispresented, based on the systematic use of the real representation of the Diracmatrices. Since any real $4\times 4$-matrix can be expressed as a linearcombination of these matrices, the presented Ansatz allows for a comprehensiveand complete treatment of two-dim. linear coupling. A survey of symplectictransformations in two dimensions is presented. A subset of thesetransformations is shown to be identical to rotations and Lorentz boosts inMinkowski space-time. The transformation properties of the classical statevector are formulated and found to be analog to those of a Dirac spinor. Theequations of motion for a relativistic charged particle - the Lorentz forceequations - are shown to be isomorph to envelope equations of two-dimensionallinear coupled optics. A universal and straightforward method to decoupletwo-dimensional harmonical oscillators with constant coefficients by symplectictransformations is presented, which is based on this isomorphism. The methodyields the eigenvalues (i.e. tunes) and eigenvectors and can be applied to aone-turn transfer matrix or directly to the coefficient matrix of the lineardifferential equation.

About the Use of Real Dirac Matrices in 2-dimensional Coupled Linear Optics