Abstract
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 $4\ifmmode\times\else\texttimes\fi{}4$ matrix can be expressed as a linear combination of these matrices, the presented ansatz allows for a comprehensive and complete treatment of two-dimensional 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 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 one-turn transfer matrix or directly to the coefficient matrix of the linear differential equation.
Highlights
Even though there is continuous interest in this field, the treatment of coupled linear optics in two degrees of freedom has not yet reached the same level of generality, transparency, and conceptual clarity as provided by the Courant-Synder theory for one degree of freedom.This article is about coupled linear optics as required to describe, for instance, the motion of charged particles in accelerators and ion beam optics
We emphasize that a lot of the presented formalism, i.e., practically all equations that do not refer to other matrices than 0, may be applied in arbitrary dimensions, if 0 is extended correspondingly. This holds especially for the concept of the symplex, which we introduce to identify the components of the ‘‘force matrix.’’ We will demonstrate the significance of the real Dirac matrices (RDMs) for the treatment of coupled linear motion, in the context of transfer matrices and eigensystems
We demonstrate how the RDMs can be used in the context of the second moments using Poisson brackets
Summary
Even though there is continuous interest in this field (see, for instance, [1,2,3,4,5,6,7,8,9,10,11]), the treatment of coupled linear optics in two (or more) degrees of freedom has not yet reached the same level of generality, transparency, and conceptual clarity as provided by the Courant-Synder theory for one degree of freedom. In accelerators like cyclotrons or synchrotrons, the beam circulates in the horizontal plane and the electric and magnetic fields are symmetric with respect to this so-called median plane. In this case vertical motion is neither coupled to the horizontal nor to the longitudinal motion. The use of real instead of the complex Dirac matrices has several reasons: First, linear coupled optics is a classical theory and the relevant terms in the Hamilton function are real. The introduction of RDMs into classical mechanics may provide new insights into the relationship of Hamiltonian mechanics, special relativity, and the Dirac equation It has been known for a long time that the mathematical formalism of Twiss parameters and
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