Use of real Dirac matrices in twodimensional coupled linear optics
Abstract
The CourantSnyder theory for twodimensional coupled linear optics is presented, based on the systematic use of the real representation of the Dirac matrices. Since any real 4×4 matrix can be expressed as a linear combination of these matrices, the presented ansatz allows for a comprehensive and complete treatment of twodimensional 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 spacetime. 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 twodimensional linear coupled optics. A universal and straightforward method to decouple twodimensional harmonic 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 oneturn transfer matrix or directly to the coefficient matrix of the linear differential equation.
 Publication:

Physical Review Accelerators and Beams
 Pub Date:
 November 2011
 DOI:
 10.1103/PhysRevSTAB.14.114002
 arXiv:
 arXiv:1109.2001
 Bibcode:
 2011PhRvS..14k4002B
 Keywords:

 47.10.Df;
 41.75.i;
 05.45.Xt;
 03.30.+p;
 Hamiltonian formulations;
 Chargedparticle beams;
 Synchronization;
 coupled oscillators;
 Special relativity;
 Physics  Accelerator Physics;
 Mathematical Physics
 EPrint:
 25 pages, no figures