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A basic problem in linear particle optics is to find a symplectictransformation that brings the (symmetric) beam matrix to a special diagonalform, called normal form. The conventional way to do this involves aneigenvalue-decomposition of a matrix related to the beam matrix, and may beapplied to the case of 1, 2 or 3 particle degrees of freedom. For 2 degrees offreedom, a different normalization method involving "real Dirac matrices" hasrecently been proposed. In the present article, the mathematics of real Diracmatrices is presented differently. Another normalization recipe is given, andmore general decoupling problems are solved. A 3D visual representation of thebeam matrix is provided. The corresponding normalization method for 1 degree offreedom involving "real Pauli matrices" is also given.

Symplectic transformations of a beam matrix with real Pauli and Dirac matrices