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

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.


INTRODUCTION
Even though there is continuous interest in this field (see, for instance, 1-11 ), 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.
This article is about coupled linear optics as required to describe, for instance, the motion of charged particles in accelerators and ion beam optics. Even though ion beam optics is in principle tree-dimensional, often symmetries can be used to reduce the problem to the treatment of two-dimensional systems. 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. But the dispersion of the bending magnets couples horizontal and longitudinal motion. In other configurations, the transversal degrees of freedom are coupled with each other by solenoid magnets, but not with the longitudinal motion -or the longitudinal degree of freedom is "hidden" as the beam is not bunched but continuous. Therefore we will treat coupling in two dimensions like most authors 1- 10 . In an accompanying paper the problem of transverse-longitudinal coupling by space charge forces in isochronous cyclotrons is treated in linear approximation 12 . This special type of coupling does not allow to apply the method of Teng and Edwards without modifications 12 . Other authors like Qin and Davidson restrict themselves to special forms of the a) Electronic mail: christian.baumgarten@psi.ch Hamilton function so that their treatment lacks generality 3 .
The real Dirac matrices (RDMs) have been known for a long time as the Majorana representation, going back to a paper by Ettore Majorana 13,14 . In this article, we will use the RDMs in a very practical way that has -to the knowledge of the author -not yet attracted much attention. The RDMs form a basis of 4 × 4-matrices with remarkable properties also -and maybe especially -in the context of classical mechanics. The RDMs are essential ingredients for a formulation of a theory of symplectic flow in two dimensions. They enable to survey all possible symplectic transformations in an elegant and straight manner.
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. Second, the RDMs are a complete basis: Any real 4 × 4 matrix can be written as a linear combination of the RDMs. This in fact guarantees generality and completeness of the presented theory of linear coupled motion in two dimensions. And finally -the RDMs are discriminable with respect to all relevant structural properties: Each matrix is either symplectic or antisymplectic, either even or odd, two RDMs either commute or anticommute. The use of the RDMs supports another clear distinction -real or imaginary.
Furthermore the introduction of RDMs into classical mechanics may provide new insights into the relationship of Hamiltonian mechanics, special relativity and the Dirac equation. It is known for a long time that the mathematical formalism of Twiss-parameters and Courant-Snyder invariants can be applied to quantum systems 15 . But no attempt has yet been made to apply the tools of quantum mechanics in classical mechanics.
Nevertheless we emphasize that a lot of the presented formalism, i.e. practically all equations that do not refer to other matrices than γ 0 , may be applied in arbitrary dimensions, if γ 0 is extended correspondingly. This holds especially for the concept of the symplex, that we introduce to identify the components of the "force matrix". We will demonstrate the significance of the RDMs for the treatment of coupled linear motion, in the context of transfer matrices and eigensystems. This leads in a natural way to the two-dimensional generalization of the Courant-Snyder theory. In the second part we demonstrate how the RDMs can be used in the context of the second moments using Poisson brackets. We show that both, the Maxwell-equations and the Lorentz force equations, can be formulated casually in terms of the RDMs. This brings up an analogy which we call the "electromechanical equivalence". Effectively we use the isomorphism of symplectic transformations with Lorentz boosts and rotations in Minkowski space to introduce a physical nomenclature of symplectic transformations in two dimensions. Based on this equivalence and on the distinction of even and odd matrices we finally develope a general algorithm that allows to determine the symplectic decoupling transformation. Furthermore the algorithm enables to compute the eigenvalues and the eigenvectors of stable two-dimensional symplectic systems.

I. REAL DIRAC MATRICES IN COUPLED LINEAR OPTICS
The position and direction of a charged particle within a beam is usually described by its coordinates and angles relative to the reference trajectory. In case of two transversal coordinates, the position of the particle is described by x, x ′ , y, y ′ , where x is the horizontal and y the vertical ("axial") coordinate. The dashes represent the derivative with respect to the path length s along the reference orbit. In case of transverse-longitudinal coupling, a common choice of the coordinates is x, x ′ , l, δ where l is the longitudinal coordinate with respect to the bunch center and δ = p−p0 p0 is the relative momentum deviation, where p is the momentum of a specific ion and p 0 is the average momentum.
Since the formalism is related to classical Hamiltonian mechanics, we prefer to write q i and p i for the dynamical variables and we collect them in a vector ψ: ψ = (q 1 , p 1 , q 2 , p 2 ) T . (1) We could as well change the ordering of the variables and use for example ψ = (q 1 , q 2 , p 1 , p 2 ) T .
This may (of course) have no influence on the physical situation and its description, but it leads to a different ordering of the RDMs. Four elements can be ordered in 4! = 24 different ways, but since we do not distinguish the indices, there are 4!/2 = 12 different permutations of the elements in ψ and corresponding systems of RDMs as listed in Tab. III. We identify the Dirac matrix γ 0 with the symplectic unit matrix, which is usually denoted by I, J or S. In case of Eq. 1, the "time direction" γ 0 is in case of Eq. 2, the form is γ 0 is a skew-symmetric matrix that squares to the negative unit matrix γ 2 0 = −1, while in the usual definition of the Dirac matrices one has γ 2 0 = 1. The other three basic matrices γ 1 . . . γ 3 are defined by the requirement that they must anticommute with γ 0 and with each other and that they square to the opposite sign of γ 2 0 . The signature of the metric tensor g µν , hence, is (−1, +1, +1, +1) -instead of (+1, −1, −1, −1) 16, 17 : The other matrices that we use to form the symplectic basis are The remaining matrices are defined by  e   TABLE I. Properties of the real γ-matrices. The type encoding is V for vectors, B for bi-vector, A for axial vector, P for pseudoscalar and S for scalar and refers to the symplectic transformation properties. The subscripts t (s) indicate time-(space-) like components of a four-vector, respectively. The column labeled γa gives the sign s that fulfills the following equation: γa γx γa = s γx. The column labeled "s/a" indicates, whether γx is a symplex or an antisymplex, respectively. Accordingsly, the column labeled "S/A" tells, if γx is symplectic or antisymplectic, respectively. The column labeled "e/o" indicates, whether the corresponding γ-matrix is even or odd. Even matrices are non-zero only in the block-diagonal components, odd matrices are zero in the block-diagonal components. Finally, the column labeledγ0 denotes, if a basis exists such that γx appears as the "time direction", i.e. if a basis exists in which γx plays the role of γ0. This is only the case for antisymmetric matrices, which are either both a symplex and symplectic or none of it.
Note that the definition deviates from the conventions in particle physics, where the product γ 0 γ 1 γ 2 γ 3 is usually labeled γ 5 . The matrices are explicitly given in App. B. If A is an arbitrary real-valued 4 × 4 matrix, then A can be written as a linear combination of RDMs where the RDM-coefficients a k are given by the scalar product and Tr(X) is the trace of X. The RDMs form a group and are the basis of a vector space. The associated algebra is the real Clifford algebra Cl(3, 1) 16,17 .

A. The Hamiltonian
Linear coupled optics is characterized by a Hamiltonian function of the classical harmonic oscillator form where the superscript "T" denotes a transposed vector or matrix. A is a (generally time dependent) symmetric matrix. The Hamilton equations of motion (EQOM) have the familiar formq or, in vector notation,ψ where the dot represents the time derivative. It is wellknown that the Jacobian Matrix M of a canonical transformation is symplectic, i.e. that 18 A matrix M is antisymplectic, if Eq. 15 includes also The product of symplectic matrices is symplectic. But note: Also the product of two anti-symplectic matrices is symplectic (see also Tab. I).

B. The Force Matrix and the Definition of a Symplex
From Eq. 12 and Eq. 14 one derives the following linear EQOM:ψ where F is the (generally time-dependent) force matrix. The antisymmetry of γ 0 and the symmetry of A yield: In the following we call a matrix F that fulfills EQ. 19 a symplex (not "simplex"). Symplices are sometimes called "infinitesimally symplectic" or "Hamiltonian" 19 , but the author prefers a unique and short name. A matrix F a that holds is called an antisymplex. γ 0 itself is a symplex as it is antisymmetric and squares to the negative unit matrix. By definition the basic matrices γ 1 . . . γ 3 are also symplices.
The most important property of symplices is the superposition principle. According to this principle the sum of two symplices is a symplex: The superposition principle includes scalability: A symplex multiplied by a scalar is still a symplex. Given that the product of two symplices F 1 and F 2 is also a symplex, then one finds: The product of two symplices is a symplex, if (and only if) the symplices anticommute. Since the four basic matrices γ 0 . . . γ 3 anticommute with each other, the six possible bi-vectors γ ν γ µ are symplices.
Since a symmetric n × n-matrix A is described by ν parameters with the force matrix F = γ 0 A must contain the same number of independent components. In case of n = 4 we expect ν = 10 force components. These are the four basic matrices, and the mentioned six bi-vectors γ ν γ µ = γ 4 . . . γ 9 . Hence, a general force matrix in two dimensions has the form

C. Symmetric Products and Projectors
The symmetric product of a matrix F 1 , which is either a symplex or an antisymplex, and a symplex F 2 is again a symplex: Since all γ-matrices are either symplices or antisymplices, any expression of the form 15 k=0 a k γ k F γ k with arbitrary coefficients a k is a symplex, if F is a symplex. Tab. I shows the result of γ a γ x γ a for a = [0, 10,14].
The result of the operation is a projection. For a = 14 for instance one has that is, γ 14 separates the "vector"-components from the "bi-vector" components.

D. The Transfer Matrix
The solution of Eq. 18 can be written by a symplectic transfer matrix M(t, t 0 ): If the force matrix is constant in time, then The time derivative of EQ. 28 is: so that the following differential equation holds for M: Comparison with EQ. 18 shows that the column vectors of the transfer matrix M are solutions of EQ. 18. If n is the dimension of ψ, the complete transfer matrix can be obtained by integrating Eq. 18 n times, using the n euclidean unit vectors as starting conditions ψ(t 0 ). If the transfer matrix is expressed by a time-dependent matrix Φ according to Eq. 32 can be written as: so that in this case one finds with τ = t − t 0 so that the transfer matrix can be written as If Φ andΦ would anticommute, i.e. if then the square -and any even power of -Φ would be constant and therefore one would find: We define the matrices M s and M c according to so that Eq. 37 would yielḋ In Sec. I H it will become clear that Eq. 37 has to be rejected in the case of stable focused systems. Beam transfer lines and circular accelerators are typically composed of guiding elements that provide a constant force matrix for a certain time (or better: length). Examples are dipole-or quadrupole magnets, drifts and bends. In this case, it is possible to express the transfer matrix as a product of transfer-matrices for the individual elements: The exponential of a symplex is symplectic.

E. The Definition of Coupling
Before starting an investigation on decoupling there should be a clear definition of coupling. A possible (and typical) definition refers to the structure of the force matrix. As we defined two degrees of freedom q 1 and q 2 , the obvious form of decoupling is a block-diagonal force matrix: where A and B are 2 × 2-matrices. There are less obvious forms of decoupling -consider a constant force matrix. The second derivative of the state vector ψ is:ψ If the square of a constant force matrix is (block-) diagonal, then the system can be regarded as decoupled in second order. The solutions can be found separately for each degree of freedom. In this case the coupling fixes the relative phases between the different degrees of freedom -but it does not determine the functional form of the solution.
A force matrix of the Dirac-form is an example: Even though the odd component p 2 γ 2 is not blockdiagonal, the second order differential equation is decoupled. Another interesting example is a constant force matrix X of the form Here one finds that neither X nor X 2 or X 3 are blockdiagonal. Nevertheless the fourth time derivative of ψ is "decoupled". We refer to systems as decoupled in n-th order, if the n-th order EQOM have the form where B is block-diagonal. In case of time-dependent force matrices, the second derivative is Hence we consider systems with time-dependent force matrices to be decoupled to second order, if the expres-sionḞ + F 2 is block-diagonal.

F. Symplectic Symplices
Tab. I lists all RDMs with their main properties. Each RDM ist either a symplex or an anti-symplex, symplectic or anti-symplectic. Four RDMs are both -symplectic and symplex. If F is a symplectic symplex (SYSY), then the combination of EQ. 42 and EQ. 19 yields: A symplex is symplectic, if (and only if) its square equals the negative unit matrix. But note that symplectic matrices are in principle not scalable, i.e. without unit. Symplices are scalable and may therefore have arbitrary units. If the symplex F appears in the form of Eq. 18, then it has the natural unit of a frequency or wavenumber if the dot is interpreted as time or pathlength derivative, respectively. In practical problems of accelerator physics, it will always be possible to find a typical length or time interval that can be used to redefine the dotderivative in such a way that the force matrix is unitless. In consequence this means that force matrices which fulfill F 2 = −Ω 2 1 with a constant Ω are equivalent to SYSYs:ψ In this respect the Dirac-operator (Eq. 45) is a SYSY.
From Eq. 49 and Eq. 29 it is quickly derived that the transfer matrix of a constant SYSY F is given by Eq. 51 is known in the Courant-Synder theory of 1-dim. ion beam optics. The unitless parameter t is called phase advance. If t represents the phase advance for one turn, then it is called tune. In the 1-dim. theory, the matrix F has the form where α, β and γ are the so-called Twiss-Parameters 20 .
According to the concept of the Sec. I E, a constant SYSY is decoupled in second order and Eq. 51 shows that this definition is meaningful. Besides this a SYSY also has equal eigenfrequencies -as will be shown in Sec. I Gand in consequence there is a common phase advance for both degrees of freedom. Let U be a time dependent symplectic transformation, then we obtain for the transformed "spinor"ψ: The time derivative of EQ. 18 is The formal difference between Eq. 53 and Eq. 54 is that U is symplectic and F is a symplex. If U is also a symplex (or if F is symplectic), then these equations through light on the structural equivalence between a (time) derivative and the multiplication with a SYSY. If the dynamics of a system is described by a SYSY, then the (time) derivative is itsself a symplectic transformation. One could say that SYSYs are operators, which are equivalent to derivatives.

G. Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors play an important role, if the system has some kind of self-feedback. Circular accelerators like storage rings are a simple example for a system with self-feedback. Another example are systems with a constant force matrix, so-called "constant focusing channels". If λ is the diagonal matrix containing the eigenvalues of F and E is the matrix of columnwise eigenvectors, then If E can be reversed, then In the simplest case of a constant force matrix one finds that where is the diagonal matrix of the eigenvalues of M.
The trace of the product of an antisymmetric and a symmetric matrix is zero. Hence F has zero trace. As similarity-transformations preserve the trace and the eigenvalues, we find T r(λ) = T r(F) = 0 and hence the sum of all eigenvalues is zero. The eigenvalues are either real or (two) pairs of complex conjugate values. A symplectic transformation is said to be strongly stable, if all eigenvalues of M are distinct and lie on the unit circle in the complex plane 18 . This means that for stable (oscillatory) solutions the eigenvalues of F are two conjugate pairs of imaginary values: Eq. 56 yields: This shows that F is (isomorphic to) a SYSY, if ω 2 1 = ω 2 2 . SYSYs are degenerate. The absolute values of all eigenfrequencies of a SYSY are equal.

H. The Form of the Transfer Matrix
Eq. 36 allows specific functional forms f (τ ) for the matrix elements -depending on the dimensionality and the properties of the force matrix. The force matrix of a strongly stable system has eigenvalues that are grouped in two pairs of imaginary values -the eigenfrequencies. Ring-accelerators always have this property -but not in each section. Ion beam transport systems are usually composed of sections with separate elements like dipole-, quadrupole or sextupole magnets, drifts, solenoids, and so on. These elements are characterized by their individual transfer matrices and are not necessarily "stable". Only the product of the transfer matrices of all elements in a ring-accelerator has to be stable. Eq. 36 is actually computed as a series: In case of pure RDMs, i.e.F = γ k with k ∈ [0 . . . 9], the functional form of the elements of the corresponding transfer matrices are exponentials of τ -if we include (hyperbolic) sine-and cosine forms. In the case of arbitrary symplices, other forms are possible: A square matrix F is called nilpotent, if F q = 0 for some positive integer q > 1. A simple example is the "force" matrix of a drift, which is given by (see Eq. 133 in Sec. II G below): Since the maximal (non-zero) power of a n×n matrix F is n−1, polynomials up to third order as well as products of polynomials and exponentials are also possible solutions of Eq. 61. In case of a constant force or in case of a transfer matrix for a complete turn in a stable circular accelerator, the diagonalized transfer matrix can be computed from Eq. 58 and Eq. 59. We introduce the abbreviations so that the diagonal matrx Λ (Eq. 57 and Eq. 58) can be written as and the one-turn-transfer matrix of strongly stable systems is according to Eq. 57 given by: Eq. 66 is the generalization of the Twiss-matrix for 2dimensional systems 21 . It shows that transfer matrices usually have -in contrast to symplices -a scalar component as well as components which are antisymplices. The second and the third term of Eq. 66 have the same form as a force matrix (though different eigenvalues). The last term vanishes in case of a degenerate system with equal eigenfrequencies. In this case the one-turn transfer matrix has the form of Eq. 51. In order to split the transfer matrix into the components according to Eq. 66, we make use of the method of projections, modified for this purpose, as follows: The inverse of a symplectic (transfer-) matrix M has the same eigenvectors as M: so that using Eq. 17 yields: If we recall Eq. 39, then we can determine the RDMcoefficients of the transfer matrix and the matrices M s and M c according to Eq. 10 and Eq. 11. Let m k be the resulting coefficients, then insertion into Eq. 68 yields: A decoupled force matrix logically implies a decoupled transfer matrix. A comparison of Eq. 56, Eq. 59 and Eq. 69 shows that the force matrix and M s differ only in the eigenvalues. Hence all the information that is required to compute the decoupling transformation can be obtained either from the force matrix or from M s . M c can be ignored in the context of decoupling. Even more than that: The matrix of eigenvectors E of the matrix M s diagonalizes M c , too. We come back to this in Sec. III D and Sec. III F, after the construction of the matrix of eigenvectors E.
From Eq. 69 one quickly derives that M s is a symplex while M c is an antisymplex. As M s and M c share the same eigenvectors and since all diagonal matrices, i.e. γ 3 , γ 4 , γ 12 and the unit matrix commute, also M s and M c commute: From Eq. 15 and Eq. 70 one derives which includes that in agreement with Eq. 63.

I. Second Moments and the Envelope Equations
Besides the position of a beam relative to the design orbit, the most important properties of an ion beam are collected in the matrix of second moments. We assume in the following that the beam is centered, i.e. that the first moments are all identically zero. If the state vectors ψ i (τ ) with i = 1 . . . n represent a family of n ions, where τ is the pathlength along the reference orbit, then the matrix of second moments σ is given by Another possible parameterization is given by a density function ρ(ψ, τ ) = ρ(q 1 , p 1 , q 2 , p 2 , τ ) which should be normalized such that . . . ρ(q 1 , p 1 , q 2 , p 2 , τ ) dq 1 dp 1 dq 2 dp 2 = 1 . (74) In this case one writes σ = . . . ρ ψ ψ T dq 1 dp 1 dq 2 dp 2 .
But independent of the specific practical method of computation, we assume that the matrix of second moments σ = ψ ψ T is well-defined and has a non-vanishing determinant. From Eq. 18 one derives: and obtain from Eq. 76 by multiplication from the right with γ 0 :Ṡ In ion beam physics Eq. 76 is called an envelope equation, as the second moments define the envelope of an ion beam. S is a symplex -as any symmetric matrix multiplied by γ 0 : Assuming a constant force matrix F, Eq. 78 tells us that the second moments are constant, if S and F commute. Using the eigenvector-decomposition Eq. 56 of F gives Since λ is diagonal andS commutes with λ, alsõ must be diagonal (see Tab. IV), so that the matrix S has the form: The force matrix and the S-matrix share the same eigenvectors -but will in general have different eigenvalues.

J. Matching
Matching is a concept of circular accelerators, i.e. systems with self-feedback, where eigenvalues and -vectors are well defined. A beam is matched, if the phase space occupied by the ions in the beam fits to the "acceptance" of the machine. The practical consequence of mismatching is an oscillation or "pumping" of the phase space distribution which typically leads to an increase of the beam emittance by filamentation 20 .
Wolski formulated the state of matching for the general case of periodic motion 11 . In this case, the restriction is no more that F has to be constant, but that F must be periodic: F(τ + C) = F(τ ) for a given period C and any τ . Given the transfer matrix over one turn (or period) is M, then the beam is matched, if Using Eq. 17, one quickly finds in analogy to Eq. 80: Following the same arguments as for Eq. 80 one finds that the matrix of second moments is matched, if the transfer matrix over one period and the S-matrix share the same system of eigenvectors. The general form of the matrix S is again given by Eq. 82 where D has a form analogue to Eq. 59 and is given by 11 : where the ε i are the emittances of the two degrees of freedom.

K. Expectation Values and Scalar Product
The "expectation values" ψ γ x ψ are related to the S-matrix according to: The RDM-coefficients of the S-matrix are -apart form the sign -the expectation values of the RDMs (see Eq. 11).

II. POISSON BRACKETS OF SECOND MOMENTS AND THE ELECTROMECHANICAL EQUIVALENCE
The total time derivative of a function f (p, q, t) is given by the Poisson-brackets with the Hamiltonian function 22 : We define the functions f k by the "expectation values" according to f k (p, q) do not explicitely depend on time. Evidently the f k vanish for all non-symmetric matrices γ 0 γ k so that there should be exactly n (n + 1)/2 = 10 non-vanishing functions f k . The gradient ∇ p,q = ∇ ψ yields so that Since all γ-matrices are either symplices or antisymplices, it is obvious, that i.e. only symplices can have a non-vanishing expectation value. For all symplices γ k with k ∈ [0 . . . 9] the Poisson brackets result: On the other hand we find from EQ 18: Combining EQ. 92 and 93 yields: According to EQ. 24, the force matrix F can be written as so that EQ. 93 can be written aṡ The matrix G kl is composed of the expectation values f i and the indices of f i are given by the upper left 10 × 10 part of the commutator table (Tab. IV). Note that all commutators of symplices are either zero or again symplices.
Eq. 97 can be reordered such that it has the form of a linear transformation of a 10-dimensional vector f i : The matrix G kl is antisymmetric. The reordering results in a quite similar matrix B ij . Now we introduce a normalization that assures a positive sign for positive definite second moments. For instance f 0 is according to the definition given by which is negative even though the corresponding second moment is positive. The normalization is done by multiplication with γ 2 k = ± 1, which equals −1 for k ∈ {0, 7, 8, 9}. This can be expressed by the multiplication with a quadratic diagonal matrixg that is an extended version of the "metric tensor" g µν : g = Diag(−1, 1, 1, 1, 1, 1, 1, −1, −1, −1) The transformed EQOM are: where the matrixB is explicitely given bỹ Eq. 101 is another way to express the envelope equations (Eq. 76 and Eq. 78). The explicite relation between the second moments σ ij and the expectation values is given in App. E.

A. Symplectic Electrodynamics
The upper left 4 × 4-block of the matrixB equals the electromagnetic field tensor, if we replace the "vector" (F 4 , F 5 , F 6 ) T by the electric field E and (F 7 , F 8 , F 9 ) T by the magnetic field B. In this section we investigate the equivalence of two-dimensional symplectic flow and relativistic electrodynamics. In the next section we show that rotations and Lorentz boosts in Minkowski space are formally identical to a subset of symplectic transformations in two-dimensional coupled linear optics.
If one writes the 4-potential Φ (4-current J, 4momentum P) as a 4-vector using the RDMs γ 0 . . . γ 3 according to and the electromagnetic fields E and B as then the Maxwell equations (MWEQs) can be written (remarkably compact) as: with the usual choice of units. The Lorentz force can also be expressed by RDMs. If the 4-momentum is defined by -where E is the energy and p the momentum -then the Lorentz force equations can be written as where τ is the proper time. In the lab frame time dt = dτ γ EQ. 108 yields (setting c = 1): which are exactly the Lorentz force equations. We use the formal identity of Eq. 78 and Eq. 108 to introduce the electromechanical equivalence (EMEQ) of 2-dim. symplectic motion and the motion of charged particles in electromagnetic fields as described by the Lorentz force equations. The statement of the EMEQ is, that the transformation properties of the elements of the S-matrix with respect to symplectic transformations are formally identical to the transformation properties of momentum P and electromagnetic field F in Minkowski space. The analogy allows to obtain an overview over the components of the force matrix F and to establish a meaningful nomenclature for the RDM-coefficients: The ten elements of the force matrix are associated with the energy and momentum of -and with electric and magnetic field components "seen by" -a relativistic charged particle: If the force matrix is split into the electrodynamical F ed and the mechanical components F m , the matrices are explicitely given by Note that there are ten force terms in symplectic motion and only six electromagnetic field components. Correspondingly there are ten symplectic transformations in 2dim. symplectic flow, but only six transformations known in Minkowski space -rotations about and boosts along three axis. One also finds that Eq. 101 and Eq. 102 correspond to the Lorentz force equations only, if F 0, F 1, F 2 and F 3 are zero. In this caseB is block-diagonal -the first block being the upper left 4 × 4-matrix and the second block the lower right 6 × 6-matrix. Coupled linear motion in two dimensions does not include such restrictions and is in this sense a richer theory. Nevertheless let us emphasise that the theory of symplectic motion in two dimensions does not allow scalar source terms, i.e. non-vanishing coefficients of γ 15 in the force matrix. If the MWEQs are derived from twodimensional symplectic motion, then the Lorentz gauge is included. To make this clearer, we look at the first part of EQ. 106, which includes a scalar expression ∂ t φ + ∇ A: where γ = (γ 1 , γ 2 , γ 3 ) T . Since the force matrix may be composed exclusively of symplices, the scalar term has to vanish, i.e.: which is the so-called "Lorentz gauge". In Sec. II F it will be shown, that the so-called duality rotation also has properties which are incompatible to the presented theory of two-dimensional symplectic motion.
In the next section we give a survey of symplectic transformations using the EMEQ according to Eq. 110. This concept allows to obtain a very transparent and systematic treatment of two-dimensional coupled linear optics. Of course there is some danger of puzzling the formally introduced electromagnetic components of the EMEQ with the fields of the accelerator due to the similarity of notation.

B. A Survey of Symplectic Transformations
Symplectic transformations have been introduced by EQs. 18 and 28. It was shown that the exponentials of symplices are symplectic and that there are ten symplices (in two dimensions) corresponding to ten free parameters in symplectic matrices. This indicates a one-toone relation between the symplices and the corresponding symplectic matrix. In the following we will classify the transformations that are based on single γ-matrices. Six of these transformations are known as rotations and Lorentz boosts in 3-dimensional space, based on the six bi-vectors γ 4 . . . γ 9 , i.e. the "electric and magnetic field components". We call the four remaining transformations phase rotation (induced by γ 0 , i.e. by "energy") and phase boosts induced by the spatial basis vectors γ 1 . . . γ 3 ("momentum"), respectively. All transformations are controlled by a continuous parameter a such that the inverse transformation is given by the negative of this parameter and a vanishing parameter a = 0 yields the unit matrix: The parameter a has a specific physical meaning in the EMEQ. In case of rotations it is the angle, in case of Lorentz boosts it is the inverse hyperbolic tangent of β = v/c.
Before discussing the action of the individual γsymplices, we look again at the Hamiltonian in order to show exactly why the transformation matrix should be symplectic: and insert the equation for a symplectic transformation of the forces F ′ = R F R −1 , then we find (usingψ ≡ ψ T γ 0 ) that the Hamiltonian is invariant under Lorentz transformations (LT) and under the assumption that the spinors transform according to or -in other words: If a transformation matrix R is symplectic, then the form (and the value) of the Hamiltonian and of the EQOM is conserved. Note that if the force matrix is transformed according to Eq. 117, then we find for the transfer matrix: Symplectic transformation matrices have the form where the matrix F is a symplex. The effect of a basic symplex γ b is given by: where c and s are the (hyperbolic) sine-and cosinefunctions according to Eq. 120. The last term on the right vanishes if γ b and γ a commute. In this case the matrix γ a remains unchanged: = γ a cos 2 (ε) + sin 2 (ε) = 1 for γ 2 b = −1 cosh 2 (ε) − sinh 2 (ε) = 1 for γ 2 b = 1 (122) If the RDMs γ b and γ a anticommute, one finds: The elementary symplectic transformations (in 2 dimensions) are summarized in Tab. II. The "frequency doubling" is an indication that the transformed elements are according to their transformation properties second moments. This is usually taken into account by using a transformation with the "half-angle".
Note that even though we describe here the transformation of the matrices instead of the RDM-coefficients, this is equivalent to the transformation of the RDMcoefficients assuming that the RDMs have the same form in all coordinate systems. This is only a matter of notation.
c = cos ε cosh ε cosh ε cosh ε cosh ε cosh ε cosh ε cos ε cos ε cos ε s = sin ε sinh ε sinh ε sinh ε sinh ε sinh ε sinh ε sin ε sin ε sin ε γ ′ 0 = cγ 0 .. Nevertheless it is instructive to look at the rotation matrices with respect to the transformation of the spinor ψ. The matrix of the spatial rotation about the z-axis, induced by γ 9 , is explicitely given by: Obviously γ 9 induces a rotation in the plane of q 1 and q 2 . The canonical momenta are rotated accordingly. Note that the rotation angle in the associated space is ε, the corresponding angle in the plane of the canonical variables is half of that: ε/2. The rotation about the x-axis (γ 7 ) mixes the canonical coordinates and momenta q 1 and p 2 (p 1 and q 2 ): A phase rotation is a rotation in phase space, for instance a rotation in the plane of q 1 and p 2 as described by Eq. 125. Another example is given by the rotation about the y-axis as induced by γ 8 -a combined phase rotation in opposite directions: (126) In combination with the phase rotation matrix γ 0 , this gives: so that both degrees of freedom are here decoupled. Note that in this case, the rotation angle ε is not divided by two.
We call the remaining symplectic transformations phase rotation induced by γ 0 and phase boosts induced by γ 1 , γ 2 and γ 3 . This is not quite correct, since we saw that also γ 8 induces a phase rotation. Strictly speaking, there is only one "real" spatial rotation possible in two dimensions -and this is induced by γ 9 . All other "rotations" are rotations in phase space and are therefore phase rotations. Nevertheless the rotations induced by γ 7 and γ 8 are spatial rotations in the context of the EMEQ.

D. Phase Rotation
The symplectic transformation driven by γ 0 is a "rotation" between spacelike components and electric field components, represented by γ 0 γ. γ 0 and the magnetic components γ 7 , γ 8 , γ 9 are unchanged. Expressing this using EMEQ one may write: The rotation matrix is given by As listed in Tab. III, the matrices γ 0 , γ 4 , γ 5 , γ 6 form an alternative basis to γ 0 , γ 1 , γ 2 , γ 3 . The phase rotation produced by γ 0 changes the "mixing" angle of the standard basis and this first alternative basis.

E. Phase Boost
The transformations induced by γ 1 , γ 2 and γ 3 are called "phase boosts." These boosts are the exact analogue of the Lorentz boosts, but the transformation properties of E and p are exchanged, as can be seen from Tab. II. While Lorentz boosts are known to have the invariants E B and E 2 − B 2 , a phase boost has the invariants p B and p 2 − B 2 . Hence the phase boosts transform mass and electromagnetic energy into each other as the Lorentz boosts are transforming mass and kinetic energy. We will come back to this in Sec. III.

F. The Duality Rotation
The duality rotation can be expressed by the matrix γ 14 , but it is not a symplectic transformation and has some surprising and puzzling features. Since γ 14 commutes with all electric and magnetic field components (see Tab. IV), the usual form of the transformation can not transform the fields, but only the particle properties of energy and momentum. Using the nomenclature of Eq. 111, one finds: The last line of Eqs. 130 is usually referred to as duality rotation, explicitely given by: The duality rotation fails to fulfill all reasonable requirements in the context of the EMEQ. First, it is not symplectic. Second, the transformation is not a similarity transformation and finally, it has different forms for the mechanical and the electrodynamical components. If the MWEQs are deriveable from the EMEQ and symplectic flow, then there is no magnetic charge or current density, but the MWEQs are forced to have the "classical" form with To conclude -the duality rotation does not fit into in the concept of the EMEQ.

G. Real Dirac Matrices and Elements in Coupled Linear Optics
In the following we give some examples of force matrices as used in linear optics of ion beam physics 20,24 . We use the transversal coordinates (x, x ′ , y, y ′ ) as the pairs of conjugate variables (q 1 , p 1 , q 2 , p 2 ). In ion beam physics there is the convention to use the path length s along the reference orbit as the independent variable. The force matrix of a drift (force free motion) is given bẏ The transversal terms in a solenoid field: where K = BS 2 (B ρ) with the solenoid field B S , charge q and bending radius ρ. The bending radius depends on the momentum p and is given by This can be written using RDMs as: A radially focussing quadrupole is described by: where K 2 = |g| (Bρ) with g = ∂By ∂x = ∂Bx ∂y so that -written again with RDMs -one finds: The axially focusing quad is given bẏ A horizontal bending magnet: expressed by RDMs: Obviously the scale of the momentum coordinates has to be adjusted to all individual element types and changes even with their excitation (and the beam energy), while the scale of the position coordinates is fix. The scaling transformations that are required have the following form which allows to scale coordinates and momenta separately. This matrix -let us call it R -can be written as and it is symplectic, if or -with the equivalent definition Hence the phase boost in the direction γ 3 can be understood as a transformation that changes the relative scale of the classical coordinates and momenta. The other scaling transformations are induced by the other diagonal RDMs. In the chosen basis these are γ 4 , γ 12 and γ 15 = 1. Besides γ 3 , only γ 4 is a symplex and is a legitimate candidate for a symplectic scaling transformation: while γ 12 and γ 15 do not generate symplectic transformations: exp (γ 12 ε) = Diag(e −ε , e −ε , e ε , e ε ) exp (γ 15 ε) = Diag(e ε , e ε , e ε , e ε ) (147) We note that there are symplectic transformations to independently change the scales between q 1 and p 1 and between q 2 and p 2 , respectively. But there is no symplectic transformation to change the scales between the directions (q 1 , p 1 ) and (q 2 , p 2 ). Nevertheless, the rescaling that is required to compare force matrices of quadrupole, solenoids, dipols, etc. to RDMs can be done by symplectic transformations, namely by R 3 = exp (γ 3 ε).
III. DECOUPLING 2-DIM. HARMONIC OSCILLATORS Consider a system of two coupled degrees of freedom as it is common in many parts of physics. In the general case, the force matrix will be time dependent and so will be the symplectic transformation matrix R: Decoupling now means that the transformed force matrix is either a SYSY or block-diagonal. It may still be timedependent. Eq. 149 can also be written as 25 Given that the transformation matrix has the form where G is constant, then one obtainṡ IfF = const, then it follows from Eq. 152 and from the fact that R and G commute, thaṫ Obviously, the form of the Lorentz force equation (Eq. 108) is very common in coupled linear optics and can be obtained with a few simple assumptions for symplectic transformation matrices as in Eq. 149 or for symplices as in Eq. 153 or for second moments (i.e. the S-matrix, respectively) as in Eq. 76. In the following we present a straightforward recipe for decoupling constant force matrices. It will be shown that this method can also be applied to transfer matriceshence can be used to compute the properties of matched beams even in cases where the force matrix itsself is not constant.
At the end of this section we scetch some problems related to the general case of time-dependent forces. But a comprehensive treatment of the time-dependent case is beyond the scope of this article.

A. Eigenvalues of the force matrix
The eigenvalues of the force matrix as expressed by Eq. 111 are given by The system is stable, if the eigenvalues are purely imaginary, i.e. if the eigenfrequencies are real. This is the case, if the following conditions are fullfilled: The force matrix can only be a SYSY, if the eigenfrequencies are equal, i.e. if K 2 = 0. This is for instance the case, if E = B = 0. If the vectors p, E and B are orthogonal to each other such that p = α 2 E × B with some constant α, then it follows where E = | E|, B = | B| and p = | p|. In this case the force matrix is a SYSY, if E B = E p. We will show in the following that the standard form of the block-diagonal force matrix as it can be obtained by symplectic transformations, is given by The Hamilton matrix A = −γ 0 F d corresponding to this force matrix is diagonal: The RDM-composition of F d is given by The vectors p, E and B are orthogonal to each other so that Eq. 156 holds, but with a reversed sign for the triple product. Obviously one can exchange E with B y and p x with E z without changing the eigenfrequencies. This proves that stable systems are possible with B y > E and/or with E z > p x , i.e. with | B| > E and /or | E| > | P |.
We define a stable system to be regular or massive, if all parameters α, β, γ and δ have the same sign (are positive), i.e. if the decoupled Hamiltonian function in the new coordinatesH =ψ TÃψ is positive definite: In a regular system the coefficients of the force matrix according to Eq. 157 yield If one pair, either (α, β) or (γ, δ) is negative, then the Hamiltonian function contains a negative kinetic energy term and is indefinite. We then call the system irregular or magnetic and it is easily shown that in this case (162)

B. Decoupling a Constant Focussing Channel in two Dimensions
The RDMs and the presented theory of symplectic transformations together with the EMEQ facilitates the task of decoupling -since we have already the geometrical and physical understanding that provides the optimal strategy to solve the problem. In Sec. I E we discussed the meaning of coupling and it turned out that constant symplectic force matrices (SYSYs) do not require decoupling. Instead the next derivative yields a decoupled second-order equation. In the following we would like to discuss decoupling of EQ. 18 in cases where the force matrix is not a SYSY. Then decoupling is the task to find a transformation matrix R such that Since products of symplectic transformations are again symplectic, we may find the solution in several steps. One distinguishes even and odd matrices, where odd matrices couple the two degrees of freedom while even ones do not 26 . In the chosen basis eight matrices, namely γ 0 , γ 1 , γ 3 , γ 4 , γ 6 , γ 8 , γ 12 , γ 15 are even and the remaining eight matrices are odd. Since the transformation is required to be symplectic, the anti-symplices γ 10 . . . γ 15 can be excluded. There are six even (γ 0 , γ 1 , γ 3 , γ 4 , γ 6 and γ 8 ) and four odd matrices (γ 2 , γ 5 , γ 7 and γ 9 ) remaining. This can directly be seen from the force matrix, which is given by The odd components are p y , E y , B x and B z corresponding to the odd matrices γ 2 , γ 5 , γ 7 and γ 9 . Products of even matrices result even matrices. Hence the transformation to block-diagonal form can be done by the exclusive use of odd force matrices. Note that the odd symplices all anti-commute with each other. Two of them (γ 2 and γ 5 ) square to +1, the other two to −1. Hence they are a basis of the Clifford-algebra Cl(2, 2).
Even matrices may be used for convenience, but exclusively odd matrices are required to bring a regular force matrix into block-diagonal form. The transformations have the form of Eqs. 117 and 120, where in the latter we replace the parameter τ with ε b 2 : After each transformation, the coefficients have to be updated according to Eqs. 10 and 11.
Then the recipe for regular systems is as follows: 1. Use the rotation-matrices R 7 (about x-axis with ε 7 = arctan pz py ) and R 9 (about z-axis with ε 9 = − arctan px py ) to align the momentum p along the y-axis.
3. Use again R 7 and R 9 (with ε 7 = arctan Bz By and ε 9 = − arctan Bx By ) to align the magnetic field along the y-axis. If this is done, we have B x = B z = 0.
Ez ) to rotate about yaxis and make E x vanish. Note that this step is not required to obtain block-diagonal form. But it is required for a complete diagonalization.

Use R 2 (ε 2 = artanh
Ey E ) to make the parallel electric field component E y vanish.
All but the last step should be clear. In the rest frame we have p = 0, so that rotation matrices can be applied without restrictions. The (odd) γ 2 -term (i.e. p y ) will remain zero. The alignment of B along the y-axis brings the odd terms B x and B z to zero, the rotation about the y-axis yields E x → 0. The only remaining odd term is γ 5 , i.e. E y : where we wrote m instead of E as the γ 0 coefficient to make clear that the transformation into the rest frame has already been done. The last step is required to achieve the block-diagonal form of Eq. 157 where wherem is given bym This shows that phase boosts are transformations which convert electromagnetic energy into mass and vice versa in the same sense as Lorentz boosts convert mass into kinetic energy or vice versa. Note that we could also exchange the order of the last two steps, i.e. of R 8 and R 2 . The result would be the same since γ 8 and γ 2 commute. If the force matrix is irregular (but stable), then the energy term might be too small to perform the Lorentz boost into the rest frame as suggested by the recipe. In this case, the magnetic components provide the focusing strength that stabilize the system (see Eq. 154).
In this case we have to prepare the force matrix in the following way: 1. In the irregular case, we have to consider the possibility that p 2 ≥ E 2 . In this case the first step is a phase rotation R 0 with ε 0 = 1 4 arctan 2 E P E 2 − P 2 to minimize p 2 .
2. Align B along the y-axis.
3. Rotate about y-axis to maximize p z (i.e. to make p x = 0).
4. Apply phase boost using γ 1 by ε 1 = 1 2 artanh( pz By ). If the energy is still to low, perform the following steps: 5. Align (again) B along the y-axis.
6. Rotate about y-axis to maximize E z (i.e. to make E x = 0). 7. Apply Lorentz boost using γ 4 by ε 4 = 1 2 artanh( Ez By ). Now the energy E should fulfill E 2 > p 2 , so that the usual recipe for regular matrices can be applied.
An inspection of Tab. III may help to explain these transformations. Magnetic force matrices are irregular with respect to the definition of the state vector, i.e. the choice of γ 0 as explained above with Eq. 1 and Eq. 2. If we inspect Tab. III, then we can see that in basis system 9), where γ 8 represents the energy and γ 3 represents p x , γ 1 is the driver of a Lorentz boost. Interpreted in the usual system this means that it is possible to minimize p z using R 1 , if B 2 y > p 2 z . The second suggested transformation corresponds to the same basis and a Lorentz boost along y (accordingly in basis system No. 9, this is driven by γ 4 ). The fact that we have to look at a different basis system to find the appropriate symplectic transformation, legitimizes us to claim that such systems are irregular. The examples given in Sec. II G have γ 0coefficients which are at least as strong as any other component. This fact seems to support the assumption that only regular systems exist in linear coupled optics. This assumption is wrong and in the following section we give an example for an irregular system.

C. Example for an Irregular System
In an accompanying paper we describe a simplified and idealized cyclotron model with space charge, which is an example for an irregular system 12 . The force matrix can easily be transformed into the form of Eq. 167. Using ψ = (x, x ′ , l, δ) T as the canonical coordinates, where x and l are the horizontal and longitudinal position of the ion, x ′ = dx ds is the horizontal angle and δ = p−p0 p0 is the relative deviation of the momentum p from the average momentum p 0 . The (constant) force matrix F is given by where γ is the relativistic factor, h = 1/r is the inverse bending radius of the magnetic field and k x = h 2 (1+n) = h 2 γ 2 is the horizontal restoring force. n = r B dB dr is the field index. K x and K z are the horizontal and axial space charge force 12 , respectively. The eigenfrequencies of the force matrix are The RDM-coefficients are The system is resonably simple as there are only two elements that are not block-diagonal. This allows to guess a transformation matrix instead of using the above recipe: Where the follwoing abbreviations are used: The transformed force matrix is theñ Since γ 2 K z > 0 and Γ > 0 is given, the transformed force matrix is irregular and the transformed HamiltonianH has the form: It is also an example for a classical system, that can not be diagonalized by the method of "symplectic rotation" of Teng and Edwards 12 .

D. Diagonalization
Though the force matrix has been decoupled, there is motivation to continue the process of diagonalization by a few more transformations, since then the product of the transformation matrices equals the matrix of eigenvectors and the resulting diagonal matrix contains the eigenvalues. According to Eq. 56, 57, 82 and Eq. 85, the matched beam-ellipse can then be directly computed for any combination of the emittances ε 1 and ε 2 .
The intention is to bring F into the form of EQ. 56. Since we expect imaginary eigenvalues, the transformation matrices can not be real-symplectic. The first "direction" that we use is i γ 6 , which can formally be identified with a Lorentz boost with an infinite imaginary relative "velocity", so that the "angle" is given by The matrix is explicitely given by This matrix has some remarkable properties: It is unitary, symmetric and symplectic. The transformed force matrix is   (178) And the final transformation includes two RDMs, γ 0 and γ 8 . This combination allows for a relative phase rotation between both degrees of freedom as in EQ. 127, but now with imaginary angles: where so that In terms of RDMs the diagonalized force matrix is: Note that the last (double-) transformations is not necessary, if E z is already zero. The eigenvectors are now given by the product of all transformation matrices (in their order). Since R 8 is also symplectic, it turned out that the matrix of eigenvectors E as a product of (complex) symplectic matrices is (complex) symplectic.

E. Summary: Decoupling and Diagonalization
The force matrix F of stable systems was shown to have the form of Eq. 56 where the diagonal matrix λ has the form of Eq. 59. It turned out that there is an intermediate stage in the process which corresponds to a diagonalized Hamiltonian function. The similarity transformation R to reach this intermediate stage is symplectic and with Eq. 17 and Eq. 18 it follows that This equation can be reversed by multiplication with R (R T ) from the right (left), respectively: Note that R is in general not orthogonal, ie. R T R = 1. Hence a symplectic diagonalization can usually not be replaced by an orthogonal transformation. Nevertheless the symplectic decoupling diagonalizes the Hamiltonian.

F. The RDM-Coefficients of the Transfer Matrix and the Tunes
In the following we use the knowledge of the matrix of eigenvectors E to further investigate the general structure of the transfer matrix using Eq. 66 and Eq. 69. The "amplitudes" of M s are limited by the sine-function, but the structure is identical to the structure of the force matrix, i.e. M s is composed of symplices. The structural difference between transfer and force matrix is given by M c .
Note that the decoupling algorithm of Teng and Edwards for twodimensional systems is applied to the transfer matrix instead of the force matrix. But from Eq. 66 and Eq. 63 its it clear that the computation of the eigenvectors does not require to include M c . The decoupling can therefore be done with the force matrix F, -which is favorable in case of constant forces -or with M s . The latter enables the treatment of periodic systems with varying force matrix F. If the decoupling algorithm is applied to M s , then one obtains by comparison of Eq. 66, Eq. 182 and Eq. 167: The last missing value to determine the coupled tunes is ∆ c . We will argue in the following, that the decoupling of M s will casually diagonalize M c and hence yields ∆ c .
It is noteworthy that γ 12 is the only even matrix that anticommutes with all odd symplices (see Tab. V). This means that γ 12 plus the odd symplices γ 2 , γ 5 and γ 7 build an alternative basis of Cl(3, 1) -in which γ 9 is the pseudoscalar (see Tab. III).
In the following we use the diagonalization steps and reconstruct the components of M c . The scalar part can be ignored as it can not be transformed. The interesting part therefore is E γ 12 E −1 . The matrix E is the reverse product of all transformation matrices and hence we may analyze the structure of M c by applying all steps of the diagonalization in reversed order.
As can be seen in Tab. IV, the matrix γ 12 commutes with γ 0 , γ 6 and γ 8 , so that the reversed transformations of Eq. 178 and Eq. 181 have no effect on γ 12 . Hence we can skip the diagonalization steps and go back to the transformations of Sec. III B. The general structure of the resulting terms can be seen from Tab. IV. All transformations are symplectic, i.e. are launched by symplices. All non-vanishing commutators of RDMs in the range γ 10 . . . γ 14 with symplices yield matrices in the range γ 10 . . . γ 14 . Therefore the symplectic (back-) transformation of γ 12 exclusively yield axial-vector and pseudoscalar components. In the following we assume that the oneturn-transfer matrix is known and "off resonance", i.e. non of the coefficients of Eq. 63 are zero.
Then the first step of the back-transformation is The second step -the rotation about the y-axis using γ 8 again has no effect, since both, γ 12 and γ 14 commute with γ 8 . Hence the next steps are the rotations about z-and x-axis. Both rotations commute with γ 14 -so that the second term in Eq. 186 remains unchanged. But the first term is an (axial) vector term which rotates analogue to the vector term γ 2 . Hence the rotations create coefficients of γ 11 and γ 13 . The next step is the inverse lorentz boost in y-direction with γ 5 . Since γ 5 commutes with γ 11 and γ 13 , only the γ 12 -term transforms and launches a γ 10component. The remaining last two transformations are again rotations about the x-and z-axis. Rotations will neither change the pseudoscalar coefficient nor the time-like axial vector component γ 10 . Hence the pseudoscalar coefficient of the transfer matrix can be used to determine, if a phase boost is required for decoupling. According to the decoupling-recipe this is the case, if (and only if) the EMEQ-components of the force matrix fulfill E B = 0.
Hence one finds: If the decoupling transformation has been computed, then If τ is the repetition time (or length of the design orbit) of the accelerator, then the tune Q i equals Q i = ωiτ 2π . Note that the anticommutators γµγν +γν γµ 2 for µ, ν ∈ [10 . . . 14] form a diagonal 5 × 5-matrix and hence are the basis of a five-dimensional phase space (see Tab. V).

G. Transformation to Equal Frequencies
Consider now a time dependent symplectic transformation as derived in Eq. 152 with with the corresponding matrices U = e i ω γ4 τ = Diag(e −i ω τ , e i ω τ , e −i ω τ , e i ω τ ) U −1 = e −i ω γ4 τ = Diag(e i ω τ , e −i ω τ , e i ω τ , e −i ω τ ) which does not affect the diagonalized force matrix F: but results in a frequency shift such that One may now chose ω such that Ω 1 = Ω 2 = Ω, that is: then we obtain the result that It is also possible to chose resulting in a frequency shift such that

H. Time Dependent Forces
The treatment of time dependent symplectic transformations goes beyond the scope of this article. Nevertheless we would like to make a few comments.
If the parameter ε of Eq. 120 is time dependent, it follows thatṘ Eq. 149 can then be written as: Consider for instance the case of a force matrix F with the simple periodic form Then the use of a symplectic transformation matrix results in a constant force matrix:ψ The general time dependent case is more involved. If we intend to use R b in order to let the corresponding odd component of the force matrix vanish, then Eq. 199 implies that we have to take the new (odd) term γ bε into account. Following the recipe from above, the first step is the alignment of the momentum along the y-axis using R 7 and R 9 , respectively. This introduces new (time dependent) terms in the coefficients of γ 7 and γ 9 -which is no problem at this stage. The second step is a boost along the y-axis using γ 5 with the intention to get rid of p y . Even this is not a problem, since γ 5 corresponds to E y , which has not yet been considered. Hence it is always possible to transform into the "rest frame" at the cost of additional "field" terms in B x , B z and E y .
Nevertheless the next steps include difficulties, since the alignment of B along the y-axis induces again additional terms in B x , B z . If the generators of the transformation and the terms to act on are identical, the straightforward approach fails. The rotation about the x-axis: More general one has

IV. SUMMARY
We introduced the real Dirac matrices and the concept of the symplex in two-dimensional coupled linear optics, i.e. classical Hamiltonian mechanics. Since these matrices form a basis for all real-valued 4 × 4-matrices, we can claim that our survey of symplectic transformations for such systems is complete and final. We have shown that a subset of these transformations is isomorphic to rotations and Lorentz boost as applied to Dirac spinors in quantum electrodynamics. We used this isomorphism to describe a general and straightforward decoupling algorithm for the case of constant forces and for transfer matrices in the context of periodic motion. The algorithm was tested numerically using random force matrices of stable systems and it never failed in case of stable systems.
There are six possible choices for the symplectic unit matrix γ 0 as listed in Tab. III corresponding to six antisymmetric RDMs. These choices are nothing but permutations of the dynamical variables in the state vector, and hence they do not differ in their physical content. For each possible γ 0 there are two choices for γ 1 , γ 2 and γ 3 , which can be transformed into each other by the phase rotation. Hence the choice of the system of RDMs is unique as soon as the order of the canonical coordinates and momenta is fixed.
We clarified the structure of the transfer matrix in two-dimensional coupled linear optics which we believe is the first universally valid generalization of the Courant-Snyder theory.

V. DISCUSSION AND OUTLOOK
Since the Dirac matrices are usually known in the context of relativistic quantum electrodynamics (QED), it is sort of inevitable to contemplate the relationship between QED and classical mechanics. It is known that the eigenvalues of stable systems lie on the unit circle in the complex plane 18 , so that the treatment of eigenvectorswhich are essential to the theory -requires the introduction of imaginary numbers. David Hestenes gave a geometrical interpretation of the unit imaginary in quantum mechanics 27,28 . In context of two-dimensional symplectic flow, the unit imaginary can be interpreted statistically, as the complex eigenvectors are able to replace an statistical ensemble of real-valued trajectories. This is to say, that the matrix σ = ψψ with a real-valued "spinor" ψ has a vanishing determinant, i.e. zero emittance -unless we introduce some kind of averaging -either over a statistical ensemble or over time. The introduction of complex-valued eigenvectors circumvents this problem as well-known in the Courant-Snyder theory.
There is a theorem that all symplectic systems of the same dimension are isomorphic 18 . If the reverse is also true then the existence of the EMEQ suggests that Minkowski space-time is "embedded" in a fourdimensional phase space and that special relativity can be "deduced" from classical symplectic motion. This deduction implies that the components of the relativistic momentum are not proper canonical variables in the classical sense, but instead energy and momentum appear to be second moments of the dynamical variables of a two-dimensional symplectic system. In fact, the invariance of mass is then formally and physically identical to the invariance of the phase space volume (emittance) in symplectic flow. This formal analogy seems to find its correspondence in the relation E = m c 2 =h ω .
There are still many open questions as for instance: why are there six field components in electromagnetic theory instead of ten as we should expect from twodimensional symplectic motion? The answer could be related to the dimensionality of the system under consideration. The phase-space dimension n of a classical system and the number ν of "valid" symplices are related by Eq. 23 and can be resolved for n: If we insert ν = 6, then we find n = 3. Classically the phase space dimension should be an even number, since Hamilton mechanics is based on pairs (q i , p i ). But a generalization of Hamilton mechanics for odd phase space dimensions has already been presented by Nambu utilizing several distinct Hamilton functions 29 . We believe that the description of odd-dimensional phase spaces can also be done if canonical pairs (q i , p i ) are used -if the dimensionality of a systems is defined by the number of available force components, i.e. the number of symplices. Relativistic electromagnetic theory knows six symplices (and hence six symplectic transformations), two-dimensional coupled optics has ten, in App. A we argue that complexification of the presented theory leads to a 15 valid symplices corresponding to a 5-dimensional phase space.
ACKNOWLEDGMENTS Mathematica R 5.2 has been used for some of the symbolic calculations. Addition software has been written in "C" and been compiled with the GNU c -C++ compiler 3.4.6 on Scientific Linux.

Appendix A: Complexification
If we consider complex spinors then it is quickly shown that the Hamilton equations are: Given the Hamiltonian function is expressed by with the matrix A then one obtains: With implicit summation we write If we take the adjunct of the second equation it becomes obvious that the matrix A must be self-adjoint or hermitian. If we switch back to the (q i , p i ) approach then we find additional restrictions compared to a "classical" system: The matrix A appears for both -coordinates and momenta. The complex notation of a 4-dimensional spinor is therefore not suitable for the general description of 4-dimensional classical harmonic oscillators. The symmetric matrix A in the real-valued 2-dim. description has been replaced by an hermitian matrix in the complex notation. It can be shown that the generalization of a symplectic transformation matrix U then is There are 15 complex Dirac matrices with zero trace that obey Eq. A7 so that the phase space is -according to Eq. 206 5-dimensional.

Appendix B: The γ-Matrices
To complete the list of the real γ-matrices used throughout this paper:

Appendix C: Arbitrary Hamiltonians
The concept presented above is far more general, as any arbitrary Hamiltonian H(Ψ) may be written as a Taylor serie up to second order: From Eq. C1 and Eq. 14 one finds the following EQOM: Leach formulated a time canonical transformation to transform a time-dependent Hamiltonian of the form given by EQ. C1 to a different (simpler) form, which we will summarize in the following. Given a symplectic time-dependent transformation matrix R, one writes for the spinor ψ in the transformed system: The new HamiltonianH should have the form and the new EQOM arė or -using EQ. C3 and EQ. C2: ψ =Ṙ Ψ + RΨ +ψ 0 =Ṙ Ψ + R (γ 0 ε + γ 0 A Ψ) +ψ 0 . (C6) Combining EQ. C5 and EQ. C6 yields: where the "coordinate-free and coordinate-dependent parts have been separated" 25 .
We are especially interested in a transformation which keeps the forces γ 0 A = γ 0Ã but setsε = 0 in the new system. One obtains with F = γ 0 A: If one considers the (rather trivial) situation that ε = 0, then the conclusion is that any spinor ψ can formally be replaced by ψ 0 + R ψ -or more generally: Given a set of vectors ψ k where each vector ψ k fulfills the EQOṀ then one finds that a superposition of the form where the symplectic matrices R k obey the EQOṀ also fulfills the EQOM: where a is the index of the row, b the index of the column and c the table entry.
holds for a block-diagonal matrix T: Note that the ordering is chosen such that the first six (last four) elements of x correspond to expectation values of even (odd) RDMs. The inverse T −1 has the same block-structure.