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About the Use of Real Dirac Matrices in 2-dimensional Coupled Linear Optics C. Baumgarten1 Paul Scherrer Institute, Switzerlanda) (Dated: 5 January 2012) 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 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 2 of a Dirac spinor. The equations of motion for a relativistic charged particle - the Lorentz force equations - 1 are shown to be isomorph to envelope equations of two-dimensional linear coupled optics. A universal and 0 straightforwardmethod to decouple two-dimensional harmonic oscillators with constant coefficients by sym- 2 plectic transformations is presented, which is based on this isomorphism. The method yields the eigenvalues n (i.e. tunes) and eigenvectors and can be applied to a one-turn transfer matrix or directly to the coefficient a matrix of the linear differential equation. J 4 PACS numbers: 47.10.Df, 41.75.-i, 41.85.-p, 05.45.Xt, 03.30.+p, 03.65.Pm ] Keywords: Hamiltonianmechanics,particlebeamfocusing,coupledoscillators,Lorentztransformation,Dirac h equation p - c c INTRODUCTION Hamilton function so that their treatment lacks general- a ity3. . s Even though there is continuous interest in this field The real Dirac matrices (RDMs) have been known for c (see, for instance,1–11), the treatment of coupled linear a long time as the Majorana representation, going back si optics in two (or more) degrees of freedom has not yet to a paper by Ettore Majorana13,14. In this article, we y reached the same level of generality, transparency and will use the RDMs in a very practical way that has - to h conceptual clarity as provided by the Courant-Synder the knowledgeofthe author-notyetattractedmuchat- p [ theory for one degree of freedom. tention. The RDMs form a basis of 4×4-matrices with Thisarticleisaboutcoupledlinearopticsasrequiredto remarkablepropertiesalso-andmaybeespecially-inthe 2 describe, for instance, the motion of charged particles in context of classical mechanics. The RDMs are essential v acceleratorsandionbeamoptics. Eventhoughionbeam ingredients for a formulation of a theory of symplectic 1 0 optics is in principle tree-dimensional, often symmetries flow in two dimensions. They enable to survey all possi- 0 can be used to reduce the problem to the treatment of blesymplectictransformationsinanelegantandstraight 2 two-dimensional systems. In accelerators like cyclotrons manner. . or synchrotrons, the beam circulates in the horizontal The use of real instead of the complex Dirac matri- 9 0 plane and the electric and magnetic fields are symmet- ces has several reasons: First, linear coupled optics is a 1 ric with respect to this so-called median plane. In this classical theory and the relevant terms in the Hamilton 1 case vertical motion is neither coupled to the horizontal functionarereal. Second,theRDMsareacompletebasis: : nortothelongitudinalmotion. Butthedispersionofthe Any real 4 4 matrix can be written as a linear combi- v × i bendingmagnetscoupleshorizontalandlongitudinalmo- nation of the RDMs. This in fact guarantees generality X tion. In other configurations, the transversal degrees of and completeness of the presented theory of linear cou- r freedomarecoupledwitheachotherbysolenoidmagnets, pled motion in two dimensions. And finally - the RDMs a butnotwiththelongitudinalmotion-orthelongitudinal are discriminable with respect to all relevant structural degreeoffreedomis“hidden”asthebeamisnotbunched properties: Each matrix is either symplectic or antisym- but continuous. Therefore we will treat coupling in two plectic, either even or odd, two RDMs either commute dimensions like most authors1–10. In an accompanying oranticommute. The use ofthe RDMs supportsanother papertheproblemoftransverse-longitudinalcouplingby clear distinction - real or imaginary. space charge forces in isochronous cyclotrons is treated Furthermore the introduction of RDMs into classical in linear approximation12. This special type of coupling mechanics may provide new insights into the relation- does not allow to apply the method of Teng and Ed- ship of Hamiltonian mechanics, special relativity and wards without modifications12. Other authors like Qin the Dirac equation. It is known for a long time that and Davidson restrict themselves to special forms of the the mathematical formalism of Twiss-parameters and Courant-Snyder invariants can be applied to quantum systems15. But no attempt has yet been made to apply the tools of quantum mechanics in classical mechanics. a)Electronicmail: [email protected] Nevertheless we emphasize that a lot of the presented 2 formalism, i.e. practically all equations that do not re- the elements in ψ and corresponding systems of RDMs fer to other matrices than γ , may be applied in arbi- as listed in Tab. III. 0 trarydimensions,ifγ isextendedcorrespondingly. This We identify the Dirac matrix γ with the symplectic 0 0 holds especially for the concept of the symplex, that we unit matrix, which is usually denoted by I, J or S. In introduce to identify the components of the “force ma- case of Eq. 1, the “time direction” γ is 0 trix”. We willdemonstratethe significanceofthe RDMs 0 1 0 0 for the treatment of coupled linear motion, in the con- 1 0 0 0 text of transfer matrices and eigensystems. This leads γ0 = −0 0 0 1 , (3) in a natural way to the two-dimensional generalization 0 0 1 0 of the Courant-Snyder theory. In the second part we  −    demonstrate how the RDMs can be used in the context in case of Eq. 2, the form is of the secondmoments using Poissonbrackets. We show that both, the Maxwell-equations and the Lorentz force 0 0 1 0 equations, can be formulated casually in terms of the 0 0 0 1 η = . (4) RDMs. This brings up an analogy which we call the 0  1 0 0 0 − “electromechanical equivalence”. Effectively we use the 0 1 0 0  −  isomorphismof symplectic transformationswith Lorentz   boosts and rotations in Minkowski space to introduce a γ0 is a skew-symmetric matrix that squares to the nega- physical nomenclature of symplectic transformations in tiveunitmatrixγ02 =−1,whileintheusualdefinitionof two dimensions. Based on this equivalence and on the the Dirac matrices one has γ02 = 1. The other three ba- distinctionofevenandoddmatriceswefinallydevelopea sic matricesγ1...γ3 aredefined bythe requirementthat generalalgorithmthatallowstodeterminethesymplectic theymustanticommutewithγ0 andwitheachotherand decoupling transformation. Furthermore the algorithm that they square to the opposite sign of γ2. The signa- 0 enables to compute the eigenvalues and the eigenvectors ture of the metric tensor gµν, hence, is ( 1,+1,+1,+1) of stable two-dimensional symplectic systems. - instead of (+1, 1, 1, 1)16,17: − − − − g = Diag( 1,1,1,1) µν I. REAL DIRAC MATRICES IN COUPLED LINEAR = γµγν+−γνγµ . (5) 2 OPTICS The other matrices that we use to form the symplectic basis are Thepositionanddirectionofachargedparticlewithin a beam is usually described by its coordinates and an- 0 1 0 0 − gles relative to the reference trajectory. In case of two 1 0 0 0 transversalcoordinates,thepositionoftheparticleisde- γ1 =−0 0 0 1  (6) scribedbyx,x,y,y ,wherex isthe horizontalandy the 0 0 1 0 ′ ′   vertical (“axial”) coordinate. The dashes represent the   derivativewithrespecttothepathlengthsalongtheref- 0 0 0 1 erence orbit. In case of transverse-longitudinalcoupling, 0 0 1 0 a common choice of the coordinates is x,x,l,δ where l γ = (7) ′ 2  0 1 0 0 is the longitudinal coordinate with respect to the bunch 1 0 0 0 centerandδ = p−p0p0 istherelativemomentumdeviation,   where p is the momentum of a specific ion and p is the 0 average momentum. 1 0 0 0 − Since the formalismis relatedto classicalHamiltonian 0 1 0 0 γ = (8) mechanics,weprefertowriteq andp forthedynamical 3  0 0 1 0  i i − variables and we collect them in a vector ψ: 0 0 0 1     ψ =(q ,p ,q ,p )T . (1) The remaining matrices are defined by 1 1 2 2 γ = γ γ γ γ ; γ = 1 Wecouldaswellchangetheorderingofthevariablesand 14 0 1 2 3 15 γ = γ γ ; γ = γ γ γ =γ γ use for example 4 0 1 7 14 0 1 2 3 γ = γ γ ; γ = γ γ γ =γ γ 5 0 2 8 14 0 2 3 1 ψ =(q1,q2,p1,p2)T . (2) γ6 = γ0γ3; γ9 = γ14γ0γ3 =γ1γ2 γ = γ γ = γ γ γ 10 14 0 1 2 3 This may (of course) have no influence on the physical γ = γ γ = γ γ γ 11 14 1 0 2 3 situation and its description, but it leads to a different γ = γ γ = γ γ γ 12 14 2 0 3 1 ordering of the RDMs. Four elements can be ordered in γ = γ γ = γ γ γ 13 14 3 0 1 2 4! = 24 different ways, but since we do not distinguish the indices, thereare4!/2=12differentpermutationsof (9) 3 Type γx E.M. γx2 γxT γ0 γ14 γ10 s/a S/A γ˜0 e/o have the familiar form Vt γ0 φ −1 − − + + s S y e q˙ = ∂H Vs ~γ A~ +1 + + + − s A n (e,o,e) i ∂pi (13) B γ0~γ E~ +1 + + − + s A n (e,o,e) p˙i = −∂∂Hqi , B γ γ ~γ B~ −1 − − − − s S y (o,e,o) 14 0 or, in vector notation, At γ14γ0 φm −1 − + + − a A y o As γ14~γ A~m +1 + − + + a S n (o,e,o) ψ˙ =γ0 ψH, (14) P γ −1 − + − + a A y o ∇ 14 S γ =1 +1 + − − − a S n e where the dot represents the time derivative. It is well- 15 knownthatthe JacobianMatrixM ofa canonical trans- TABLEI.Propertiesoftherealγ-matrices. Thetypeencod- formation is symplectic, i.e. that18 ing is V for vectors, B for bi-vector, A for axial vector, P forpseudoscalarandS forscalarandreferstothesymplectic Mγ0MT =γ0. (15) transformationproperties. Thesubscriptst(s)indicatetime- (space-) like components of a four-vector, respectively. The A matrix M is antisymplectic, if column labeled γa gives the sign s that fulfills the following equation: γaγxγa = sγx. The column labeled “s/a” indi- Mγ0MT =−γ0. (16) cates,whetherγxisasymplexoranantisymplex,respectively. Accordingsly,thecolumnlabeled“S/A”tells,ifγxissymplec- Eq. 15 includes also ticorantisymplectic,respectively. Thecolumnlabeled “e/o” indicates,whetherthecorrespondingγ-matrixisevenorodd. MT γ M = γ 0 0 Even matrices are non-zero only in the block-diagonal com- MT = γ M 1γ (17) 0 − 0 ponents, odd matrices are zero in the block-diagonal compo- − M 1 = γ MT γ . nents. Finally,thecolumnlabeledγ˜ denotes,ifabasisexists − 0 0 0 − such that γx appears as the “time direction”, i.e. if a basis Theproductofsymplecticmatricesissymplectic. But exists in which γx plays the role of γ0. This is only the case for antisymmetric matrices, which are either both a symplex note: Also the product of two anti-symplectic matrices and symplectic or none of it. is symplectic (see also Tab. I). Notethatthedefinitiondeviatesfromthe conventionsin B. The Force Matrix and the Definition of a Symplex particle physics, where the product γ γ γ γ is usually 0 1 2 3 labeled γ5. The matrices are explicitly given in App. B. FromEq.12andEq.14onederivesthefollowinglinear If Ais anarbitraryreal-valued4 4matrix,thenA can EQOM: × be written as a linear combination of RDMs ψ˙ =γ Aψ =Fψ, (18) 15 0 A= a γ , (10) k k where F is the (generally time-dependent) force matrix. Xk=0 The antisymmetryofγ andthe symmetryofAyield: 0 where the RDM-coefficients a are given by the scalar k product FT = (γ A)T 0 = AT γT (γ )2 Aγ +γ A 0 a =A γ = k Tr( k k ), (11) = Aγ k · k 4 2 − 0 (19) = γ γ Aγ 0 0 0 andTr(X)isthetraceofX. TheRDMsformagroupand = γ Fγ 0 0 arethe basisofa vectorspace. The associatedalgebrais the real Clifford algebra Cl(3,1)16,17. In the following we call a matrix F that fulfills EQ. 19 a symplex(not“simplex”). Symplicesaresometimescalled A. The Hamiltonian ”infinitesimally symplectic” or ”Hamiltonian”19, but the author prefers a unique and short name. A matrix F a Linear coupled optics is characterized by a Hamilto- that holds nian function of the classical harmonic oscillator form FT = γ F γ H = 1ψT Aψ, (12) a − 0 a 0 (20) 2 where the superscript “T” denotes a transposed vector iscalledanantisymplex. γ itselfisasymplexasitisan- 0 or matrix. A is a (generally time dependent) symmet- tisymmetricandsquarestothenegativeunitmatrix. By ric matrix. The Hamilton equations of motion (EQOM) definition the basic matrices γ ...γ are also symplices. 1 3 4 Themostimportantpropertyofsymplicesisthesuper- The result of the operation position principle. Accordingtothisprinciplethesumof two symplices is a symplex: F γ Fγ 1 9 F = ± a a = f (γ γ γ γ ), (26) a k k a k a 2 2 ± (F1+F2)T = FT1 +FT2 =γ0F1γ0+γ0F2γ0 (21) kX=0 = γ0(F1+F2)γ0. is a projection. For a=14 for instance one has The superposition principle includes scalability: A sym- 9 3 plex multiplied by a scalaris still a symplex. Given that 21 fk(γk+γ14γkγ14) = fkγk theproductoftwosymplicesF andF isalsoasymplex, k=0 k=0 (27) 1 2 P9 P9 then one finds: 12 fk(γk−γ14γkγ14) = fkγk, k=0 k=4 γ (F F )γ = (F F )T P P 0 1 2 0 1 2 that is, γ separates the “vector”-components from the = FT FT 14 2 1 “bi-vector” components. = γ F γ γ F γ 0 2 0 0 1 0 (22) = γ F F γ 0 2 1 0 − F F = F F D. The Transfer Matrix 1 2 2 1 − F F +F F = 0. 1 2 2 1 The solution of Eq. 18 can be written by a symplectic The product of two symplices is a symplex, if (and only transfer matrix M(t,t0): if) the symplices anticommute. Since the four basic ma- tricesγ0...γ3 anticommutewitheachother,thesixpos- ψ(t)=M(t,t0)ψ(t0). (28) sible bi-vectors γ γ are symplices. ν µ If the force matrix is constant in time, then Since a symmetric n n-matrix A is described by ν × parameters with M(t,t )=exp(F(t t )). (29) 0 0 − ν =n(n+1)/2, (23) The time derivative of EQ. 28 is: theforcematrixF=γ Amustcontainthesamenumber 0 ψ˙(t) = M˙ (t,t )ψ(t ) of independent components. In case of n = 4 we expect 0 0 ν = 10 force components. These are the four basic ma- = Fψ(t) (30) trices,and the mentioned six bi-vectors γ γ =γ ...γ . = FM(t,t )ψ(t ) ν µ 4 9 0 0 Hence, a general force matrix in two dimensions has the form so that the following differential equation holds for M: 9 M˙ =FM. (31) F= f γ . (24) k k k=0 Comparison with EQ. 18 shows that the column vectors X of the transfer matrix M are solutions of EQ. 18. If n is the dimension of ψ, the complete transfer matrix can C. Symmetric Products and Projectors be obtained by integrating Eq. 18 n times, using the n euclidean unit vectors as starting conditions ψ(t ). 0 ThesymmetricproductofamatrixF1,whichiseither Ifthetransfermatrixisexpressedbyatime-dependent a symplex or an antisymplex, and a symplex F2 is again matrix Φ according to a symplex: M = expΦ= ∞ Φk (F1F2F1)T = FT1 FT2 FT1 k=0 k! (32) = ( γ F γ )(γ F γ )( γ F γ ) M˙ = Φ˙ + Φ˙ Φ+PΦΦ˙ + Φ˙ Φ2+ΦΦ˙ Φ+Φ2Φ˙ +... , ± 0 1 0 0 2 0 ± 0 1 0 2 6 = γ (F F F )γ . 0 1 2 1 0 (25) then if (and only if) Φ and Φ˙ commute, i.e. if Sinceallγ-matricesareeithersymplicesorantisymplices, any expression of the form ΦΦ˙ =Φ˙ Φ, (33) 15 Eq. 32 can be written as: a γ Fγ k k k k=0 M˙ = Φ˙ 1+Φ+ Φ2 + Φ3 +... X 2 6 with arbitrarycoefficients a is a symplex, if Fis a sym- = Φ˙ (cid:16)expΦ (cid:17) (34) k plex. Tab.Ishowstheresultofγaγxγa fora=[0,10,14]. = Φ˙ M 5 so that in this case one finds E. The Definition of Coupling F = Φ˙ Before starting an investigation on decoupling there t (35) Φ(t,t ) = F(t)dt=F¯τ, should be a clear definition of coupling. A possible (and 0 t0 typical)definitionreferstothestructureofthe forcema- R trix. As we defined two degrees of freedom q and q , 1 2 withτ =t t sothatthetransfermatrixcanbewritten − 0 the obvious form of decoupling is a block-diagonal force as matrix: M(t)=exp F¯τ . (36) A 0 F= , (43) If Φ and Φ˙ would anticomm(cid:0)ute, i(cid:1).e. if 0 B ! ΦΦ˙ +Φ˙ Φ=0, (37) where A and B are 2 2-matrices. × Therearelessobviousformsofdecoupling-considera then the square - and any even power of - Φ would be constantforcematrix. Thesecondderivativeofthestate constant and therefore one would find: vector ψ is: M˙ = ∞ (Φ2k2+k1Φ˙)! =sinh(Φ)Φ−1Φ˙ (38) ψ¨=Fψ˙ =F2ψ. (44) k=0 P If the square of a constant force matrix is (block-) diag- We define the matrices M and M according to s c onal, then the system can be regarded as decoupled in second order. The solutions can be found separately for M = M +M s c MMsc ≡≡ eexxpp((ΦΦ))−+22eexxpp((−−ΦΦ)) == MM+−22MM−−11 ==csionshh((ΦΦ)) (39) etdhaoecmhre-dlaebgtuirvteeeitpohdfaofsereessendbooetmtwd.eeetInenrmtthhienisedcitffaheseerefntuhtnecdtecigoorunepaellsinfoogfrmfifrxeeoes-f the solution. A force matrix of the Dirac-form is an example: so that Eq. 37 would yield F = Eγ +p γ +p γ +p γ M˙ = 0 0 1 1 2 2 3 3 (45) c (40) F2 = (E2 p2 p2 p2)1. M˙ = M Φ 1Φ˙ . − − 1− 2− 3 s s − Even though the odd component p γ is not block- In Sec. IH it will become clear that Eq. 37 has to be 2 2 diagonal,the secondorderdifferential equationis decou- rejected in the case of stable focused systems. pled. Another interesting example is a constant force Beam transfer lines and circular accelerators are typ- matrix X of the form ically composed of guiding elements that provide a constant force matrix for a certain time (or better: length). X = γ0−γ2−γ6−γ7 Examples are dipole- or quadrupole magnets, drifts and − 2 X2 = γ (46) bends. In this case, it is possible to express the transfer − 11 X4 = 1 matrix as a product of transfer-matrices for the individ- ual elements: Here one finds that neither X nor X2 or X3 are block- M(t ,t ) = M(t ,t )...M(t ,t )...M(t ,t ) diagonal. Nevertheless the fourth time derivative of ψ is n 0 n n 1 k k 1 1 0 = exp(F τ−)...exp(F τ−). “decoupled”. n n 1 1 (41) We refer to systems as decoupled in n-th order, if the The transfer matrix is symplectic, if Φ is a symplex: n-th order EQOM have the form n d Mγ MT = ∞ Φk γ ∞ (ΦT)k ψ =Bψ, (47) 0 k! 0 k! dτ (cid:18)k=0 (cid:19) (cid:18)k=0 (cid:19) (cid:18) (cid:19) = P∞ Φk!k γ0 P∞ (−)k−1γ0 Φk!k γ0 where B is block-diagonal. In case of time-dependent (cid:18)k=0 (cid:19) (cid:18)k=0 (cid:19) force matrices, the second derivative is = P∞ Φk!k ∞ P(−kΦ!)k γ0 (cid:18)k=0 (cid:19) (cid:18)k=0 (cid:19) ψ¨ = F˙ ψ+Fψ˙ = exPp(Φ) exp(−PΦ)γ0 = (F˙ +F2)ψ (48) = γ 0 Hence we consider systems with time-dependent force (42) matrices to be decoupled to second order, if the expres- The exponential of a symplex is symplectic. sion F˙ +F2 is block-diagonal. 6 F. Symplectic Symplices The time derivative of EQ. 18 is Tab.IlistsallRDMswiththeirmainproperties. Each ψ¨ = F˙ ψ+Fψ˙ (54) RDMisteitherasymplexorananti-symplex,symplectic oranti-symplectic. FourRDMsareboth-symplecticand symplex. If F is a symplectic symplex (SYSY), then the The formal difference between Eq. 53 and Eq. 54 is that combination of EQ. 42 and EQ. 19 yields: UissymplecticandFisasymplex. IfUisalsoasymplex (orifFissymplectic),thentheseequationsthroughlight Fγ FT = Fγ γ Fγ = F2γ =γ onthe structural equivalence betweena (time) derivative 0 0 0 0 − 0 0 (49) F2 = 1. andthemultiplicationwithaSYSY.Ifthedynamicsofa ⇒ − systemisdescribedbyaSYSY,thenthe(time)derivative Asymplexissymplectic,if(andonlyif)itssquareequals isitsselfasymplectictransformation. Onecouldsaythat the negative unit matrix. But note that symplectic ma- SYSYsareoperators,whichareequivalenttoderivatives. trices 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, G. Eigenvalues and Eigenvectors then it has the natural unit of a frequency or wavenum- berifthedotisinterpretedastimeorpathlengthderiva- Eigenvalues and eigenvectors play an important role, tive, respectively. In practical problems of accelerator if the system has some kind of self-feedback. Circular physics,itwillalwaysbepossibletofindatypicallength acceleratorslike storagerings are a simple example for a or time interval that can be used to redefine the dot- systemwith self-feedback. Another example are systems derivative in such a way that the force matrix is unit- with a constant force matrix, so-called “constant focus- less. Inconsequencethismeansthatforcematriceswhich ing channels”. If λ is the diagonal matrix containing fulfill F2 = Ω21 with a constant Ω are equivalent to the eigenvalues of F and E is the matrix of columnwise − SYSYs: eigenvectors,then ψ˙ = dψ =Fψ dt FE=Eλ. (55) d ψ = Fψ (50) ⇒ dτ Ω dτ =Ωdt If E can be reversed, then In this respect the Dirac-operator (Eq. 45) is a SYSY. F=EλE 1. (56) − From Eq. 49 and Eq. 29 it is quickly derived that the transfer matrix of a constant SYSY F is given by In the simplest case of a constant force matrix one finds that M = 1 cost+F sint. (51) M(τ,0) = exp EλE 1τ − Eq. 51 is known in the Courant-Synder theory of 1-dim. ionbeamoptics. Theunitlessparametertiscalledphase = ∞ (cid:0)(EλEk−!1τ)k(cid:1) advance. If t represents the phase advance for one turn, k=0 then it is called tune. In the 1-dim. theory, the matrix = P∞ Eλkkτ!k E−1 (57) F has the form k=0 = EPexp(λτ)E 1 − α β = EΛ(τ)E−1, F= , (52) γ α! − − where where α, β and γ are the so-called Twiss-Parameters20. Λ(τ)=exp(λτ), (58) AccordingtotheconceptoftheSec.IE,aconstantSYSY is decoupled in second order and Eq. 51 shows that this is the diagonal matrix of the eigenvalues of M. definition is meaningful. Besides this a SYSY also has The trace of the product of an antisymmetric and equal eigenfrequencies - as will be shown in Sec. IG - a symmetric matrix is zero. Hence F has zero trace. andinconsequencethereisacommonphaseadvancefor As similarity-transformations preserve the trace and the both degrees of freedom. eigenvalues, we find Tr(λ) = Tr(F) = 0 and hence the LetUbeatimedependentsymplectictransformation, sum of all eigenvalues is zero. The eigenvalues are either then we obtain for the transformed “spinor” ψ˜: real or (two) pairs of complex conjugate values. A symplectic transformation is said to be strongly stable, if all ψ˜ = Uψ eigenvaluesofM aredistinct andlie onthe unit circlein ψ˙˜ = U˙ ψ+Uψ˙ (53) the complex plane18. This means that for stable (oscil- latory) solutions the eigenvalues of F are two conjugate 7 pairs of imaginary values: the diagonalized transfer matrix can be computed from Eq. 58 and Eq. 59. We introduce the abbreviations λ = Diag(iω , iω ,iω , iω ) 1 1 2 2 − − = −iω1+2ω2 γ3−iω1−2ω2 γ4 (59) Σc = cos(ω1τ)+2cos(ω2τ) =cos(ω¯τ) cos(∆ωτ) Σ = sin(ω1τ)+sin(ω2τ) =sin(ω¯τ) cos(∆ωτ) s 2 Eq. 56 yields: ∆s = sin(ω1τ)−2sin(ω2τ) =cos(ω¯τ) sin(∆ωτ) ∆c = cos(ω1τ)−2cos(ω2τ) =−sin(ω¯τ) sin(∆ωτ), F2 = Eλ2E 1 (63) − = −E(ω12+2ω22 1+ ω12−2ω22 γ12)E−1 (60) where = −ω12+2ω22 1− ω12−2ω22 Eγ12E−1. ω¯ = ω1+ω2 2 This shows that F is (isomorphic to) a SYSY, if ω12 = ∆ω = ω1−2ω2 (64) ω2. SYSYs are degenerate. The absolute values of all 2 eigenfrequencies of a SYSY are equal. sothatthe diagonalmatrx Λ(Eq.57andEq.58)canbe written as H. The Form of the Transfer Matrix Λ=Σ 1 iΣ γ i∆ γ ∆ γ , (65) Eq.36allowsspecificfunctionalformsf(τ)forthema- c − s 3− s 4− c 12 trix elements - depending on the dimensionality and the properties of the force matrix. The force matrix of a and the one-turn-transfer matrix of strongly stable sys- strongly stable system has eigenvalues that are grouped tems is according to Eq. 57 given by: in two pairs of imaginary values - the eigenfrequencies. Ring-accelerators always have this property - but not in M = Σ 1 iΣ Eγ E 1 each section. Ion beam transport systems are usually c − s 3 − i∆ Eγ E 1 ∆ Eγ E 1 (66) composedofsectionswithseparateelementslike dipole-, s 4 − c 12 − − − quadrupole or sextupole magnets, drifts, solenoids, and so on. These elements are characterized by their indi- vidualtransfermatricesandarenotnecessarily“stable”. Eq. 66 is the generalization of the Twiss-matrix for 2- Only the product of the transfer matricesof all elements dimensional systems21. It shows that transfer matrices in a ring-accelerator has to be stable. Eq. 36 is actually usually have - in contrast to symplices - a scalar compo- computed as a series: nentaswellascomponentswhichareantisymplices. The second and the third term of Eq. 66 have the same form M=exp(F¯τ)= ∞ (F¯τ)k . (61) asaforcematrix(thoughdifferenteigenvalues). Thelast k! term vanishes in case of a degenerate system with equal k=0 X eigenfrequencies. In this case the one-turn transfer ma- In case of pure RDMs, i.e. F¯ = γ with k [0...9], trix has the form of Eq. 51. k ∈ the functional form of the elements of the corresponding In order to split the transfer matrix into the compo- transfermatricesareexponentialsofτ -ifweinclude(hy- nents according to Eq. 66, we make use of the method perbolic) sine- and cosine forms. In the case ofarbitrary ofprojections,modified for this purpose, asfollows: The symplices,otherformsarepossible: AsquarematrixFis inverseofasymplectic(transfer-)matrixMhasthesame callednilpotent,ifFq =0forsomepositiveintegerq >1. eigenvectors as M: A simple example is the “force” matrix of a drift, which is given by (see Eq. 133 in Sec. IIG below): M 1 = EΛ 1(τ)E 1 − − − (67) F = γ0+γ6 = EΛ( τ)E−1 =E exp( λτ)E−1, 2 − − M = ∞ (F¯τ)k (62) k! so that using Eq. 17 yields: k=0 = 1P+F¯τ. Sincethemaximal(non-zero)powerofan nmatrixFis 12(M±M−1) = EΛ(τ)±2Λ(−τ)E−1 (68) × = 1(M γ MT γ ). n 1,polynomialsuptothirdorderaswellasproductsof 2 ∓ 0 0 − polynomials and exponentials are also possible solutions of Eq. 61. If we recall Eq. 39, then we can determine the RDM- In case of a constant force or in case of a transfer ma- coefficients of the transfer matrix and the matrices M s trix for a complete turn in a stable circular accelerator, and M according to Eq. 10 and Eq. 11. Let m be the c k 8 resulting coefficients, then insertion into Eq. 68 yields: Another possible parameterization is given by a density functionρ(ψ,τ)=ρ(q ,p ,q ,p ,τ)whichshouldbenor- 1 1 2 2 9 malized such that M = 1(M+γ MT γ )= m γ s 2 0 0 k k k=0 = EΛ(τ)−2Λ(−τ)E−1 P Z ...Z ρ(q1,p1,q2,p2,τ)dq1dp1dq2dp2 =1. (74) = iΣ Eγ E 1 i∆ Eγ E 1 s 3 − s 4 − In this case one writes − − 15 (69) M = 1(M γ MT γ )= m γ c 2 − 0 0 k=10 k k σ = ... ρψψT dq1dp1dq2dp2. (75) = EΛ(τ)+2Λ(−τ)E−1 P Z Z = Σ 1 ∆ Eγ E 1 Butindependentofthespecificpracticalmethodofcom- c c 12 − − putation, we assume that the matrix of second moments σ = ψψT is well-defined and has a non-vanishing de- h i terminant. From Eq. 18 one derives: A decoupled force matrix logically implies a decoupled transfer matrix. A comparison of Eq. 56, Eq. 59 and σ˙ = ψ˙ψT + ψψ˙T Eq. 69 shows that the force matrix and Ms differ only = hFψψiT +h ψψiT FT in the eigenvalues. Hence all the information that is re- h i h i = Fσ+σFT (76) quiredto compute the decoupling transformationcanbe = Fσ+σγ Fγ obtained either from the force matrix or from M . M 0 0 s c can be ignored in the context of decoupling. Even more thanthat: ThematrixofeigenvectorsEofthematrixM s We define the S-matrix using ψ¯ ψT γ by diagonalizesMc,too. We come backto this inSec. IIID ≡ 0 and Sec. IIIF, after the construction of the matrix of S σγ = ψψT γ = ψψ¯ , (77) 0 0 eigenvectors E. ≡ h i h i and obtain from Eq. 76 by multiplication from the right FromEq. 69one quickly derives that M is a symplex s with γ : while M is an antisymplex. As M and M share the 0 c s c sameeigenvectorsandsincealldiagonalmatrices,i.e. γ3, S˙ = FS SF γ , γ and the unit matrix commute, also M and M − (78) 4 12 s c commute: In ion beam physics Eq. 76 is called an envelope equa- M M M M =0. (70) tion, as the second moments define the envelope of an s c c s − ion beam. S is a symplex - as any symmetric matrix From Eq. 15 and Eq. 70 one derives multiplied by γ : 0 M2 M2 =1, (71) ST = γ0T σT c − s = γ σ − 0 (79) which includes that = γ σγ2 0 0 = γ Sγ . 0 0 1 = Σ2+∆2+Σ2+∆2 c c s s (72) Assuming a constantforce matrixF, Eq.78 tells us that 0 = Σ ∆ +Σ ∆ , c c s s the second moments are constant, if S and F commute. Using the eigenvector-decompositionEq. 56 of F gives in agreement with Eq. 63. 0 = FS SF − = EλE 1S SEλE 1 − − I. Second Moments and the Envelope Equations 0 = λE 1SE−E 1SEλ (80) − − − 0 = λS˜ S˜λ 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 Since λ is diagonal and S˜ commutes with λ, also in the following that the beam is centered, i.e. that the firstmomentsareallidenticallyzero. Ifthestatevectors S˜ =E−1SE=D, (81) ψ (τ) with i=1...n represent a family of n ions, where i mustbe diagonal(seeTab.IV),sothatthe matrixS has τ is the pathlength along the reference orbit, then the the form: matrix of second moments σ is given by S=EDE 1. (82) − n 1 σ = ψ ψT = ψψT . (73) TheforcematrixandtheS-matrixsharethesameeigen- n i i h i vectors - but will in general have different eigenvalues. i=1 X 9 J. Matching II. POISSON BRACKETS OF SECOND MOMENTS AND THE ELECTROMECHANICAL EQUIVALENCE Matching is aconceptofcircularaccelerators,i.e. systems with self-feedback, where eigenvalues and -vectors Thetotaltimederivativeofafunctionf(p,q,t)isgiven are well defined. A beam is matched, if the phase space bythePoisson-bracketswiththeHamiltonianfunction22: occupiedbytheionsinthebeamfitstothe“acceptance” of the machine. The practical consequence of mismatch- df(qi,pi,t) = ∂f + ∂f ∂H ∂f ∂H ing is an oscillation or “pumping” of the phase space dt ∂t i ∂qi∂pi − ∂pi ∂qi ∂fPn ∂H o distribution which typically leads to an increase of the ∂q1 ∂q1 beam emittance by filamentation20.  ∂f   ∂H  Wolskiformulatedthestateofmatchingforthegeneral = ∂∂pf1 γ0 ∂∂pH1 (87) case of periodic motion11. In this case, the restriction is  ∂∂qf2   ∂∂qH2  no more that F has to be constant, but that F must be  ∂p2   ∂p2  periodic: F(τ +C)=F(τ) for a given period C and any = ∂∂ft +∇q,pf(p,q,t)ψ˙ τ. Giventhe transfer matrix overone turn (or period) is = ∂f + f(p,q,t)Fψ. ∂t ∇q,p M, then the beam is matched, if We define the functions f by the “expectation values” k σ =MσMT . (83) according to Using Eq. 17, one quickly finds in analogy to Eq. 80: 1 f (p,q)= ψ¯γ ψ. (88) k k 2 σ = Mσγ M 1γ 0 − 0 − σγ = Mσγ M 1 fk(p,q)donotexplicitelydependontime. Evidentlythe 0 0 − f vanish for all non-symmetric matrices γ γ so that S = MSM 1 (84) k 0 k − there should be exactly n(n+1)/2 = 10 non-vanishing SM = MS functions f . The gradient = yields k p,q ψ 0 = MS SM ∇ ∇ − ∂ ψ =δ , (89) FollowingthesameargumentsasforEq.80onefindsthat ψi j ij the matrix of secondmoments is matched, if the transfer so that matrix overone period and the S-matrix share the same system of eigenvectors. The general form of the matrix ∇p,qfk = 21 γ0γkψ+ψT γ0γk S is againgiven by Eq. 82 where D has a form analogue = 21ψ(cid:0)T γ0γk+(γ0γk)T(cid:1) to Eq. 59 and is given by11: = 1ψT γ γ +γT γT 2 (cid:0) 0 k k 0 (cid:1) (90) = 1ψT γ γ γT γ D=Diag(iε1, iε1,iε2, iε2), (85) 2 (cid:0) 0 k− k 0(cid:1) − − = 1ψT γ γ +γ γ γT γ 2 (cid:0) 0 k 0 0(cid:1)k 0 where the εi are the emittances of the two degrees of = 1ψ¯ γ +γ γT γ . freedom. 2 (cid:0)k 0 k 0 (cid:1) Sinceallγ-matricesareeit(cid:0)hersymplices(cid:1)orantisymplices, it is obvious, that K. Expectation Values and Scalar Product ψ¯γ for k [0...9] S-mThaetri“xexapcceoctradtiniogntov:alues” hψ¯γxψi are related to the ∇p,qfk =( 0k for k ∈∈[10...15] , (91) i.e. only symplices can have a non-vanishing expectation ψ¯γxψ = ψkγ0kiγxijψj value. For all symplices γk with k [0...9] the Poisson ijk ∈ brackets result: = P1 (ψkγkiγijψj +ψjγjkγkiψi) 2 ijk 0 x 0 x d ψ¯γ ψ = 21 P(γxijψjψkγ0ki+ψiψjγ0jkγxki) dτ 2k =ψ¯γkFψ. (92) ijk (cid:18) (cid:19) P On the other hand we find from EQ 18: = 1Tr (γijψjψkγkl+ψiψjγjkγkl) 2 x 0 0 x jk ! d ψ¯γ ψ = ψ˙T γ γ ψ+ψT γ γ ψ˙ = 1Tr(γPψψ¯+ψψ¯γ ) dτ k 0 k 0 k 2 x x = ψT FT γ γ ψ+ψT γ γ Fψ = 1Tr(γ S+Sγ ). (cid:0) (cid:1) 0 k 0 k 2 x x = ψT γ Fγ γ γ ψ+ψT γ γ Fψ (86) 0 0 0 k 0 k (93) = ψ¯Fγ ψ+ψ¯γ Fψ The RDM-coefficients of the S-matrix are - apart form k k − the sign - the expectation values of the RDMs (see = ψ¯(γkF Fγk)ψ − Eq. 11). 10 Combining EQ. 92 and 93 yields: where the matrix B˜ is explicitely given by d ψ¯γ ψ =ψ¯2γ Fψ =ψ¯(γ F Fγ )ψ, (94) F4 F5 F6 −F1 −F2 −F3 dτ k k k − k F4 F9 −F8 −F0 F3 −F2 (cid:0) (cid:1)  F5 −F9 F7 −F0 −F3 F1  so that F6 F8 −F7 −F0 F2 −F1 B˜ =  −F1 F0 F9 −F8 F6 −F5  0 = ψ¯2γ Fψ ψ¯(γ F Fγ )ψ  −F2 F0 −F9 F7 −F6 F4  0 = ψ¯(γkkF+F−γk)ψk − k (95)  −F3 −F3 FF20 F8 −−FF67 F5 F5 −FF94 −F8   F3 −F1 F6 −F4 −F9 F7   −F2 F1 −F5 F4 F8 −F7    AccordingtoEQ.24,theforcematrixFcanbewritten (102) Eq.101isanotherwaytoexpresstheenvelopeequations as (Eq. 76 and Eq. 78). The explicite relation between the 9 second moments σ and the expectation values is given ij F= Flγl, (96) in App. E. l=0 X so that EQ. 93 can be written as A. Symplectic Electrodynamics 9 f˙k = ddτ ψ¯γ2kψ = ψ¯γkγl−2γlγk ψFl The upper left 4 4-block of the matrix B˜ equals the 9 (cid:16) (cid:17) lP=0 (97) electromagnetic fiel×d tensor, if we replace the “vector” = GklFl. (F4,F5,F6)T by the electric field E~ and (F7,F8,F9)T by l=0 the magnetic field B~. In this section we investigate the P The matrixG is composedofthe expectationvaluesf equivalence of two-dimensionalsymplectic flow andrela- kl i and the indices of f are given by the upper left 10 10 tivisticelectrodynamics. Inthenextsectionweshowthat i part of the commutator table (Tab. IV). Note tha×t all rotationsandLorentzboostsinMinkowskispacearefor- commutators of symplices are either zero or again sym- mallyidenticaltoa subsetofsymplectic transformations plices. in two-dimensional coupled linear optics. Eq. 97 can be reorderedsuch that it has the form of a If one writes the 4-potential Φ (4-current J, 4- linear transformation of a 10-dimensionalvector fi: momentum P) as a 4-vector using the RDMs γ0...γ3 according to 9 f˙i = Bijfj. (98) Φ=φγ0+Axγ1+Ayγ2+Azγ3 =φγ0+~γA~, (103) j=0 X the 4-derivative D as The matrix G is antisymmetric. The reordering re- kl sults in a quite similar matrix Bij. Now we introduce D=∂tγ0 ∂xγ1 ∂yγ2 ∂zγ3, (104) − − − a normalization that assures a positive sign for positive definite secondmoments. For instance f0 is accordingto and the electromagnetic fields E~ and B~ as the definition given by F=E γ +E γ +E γ +B γ +B γ +B γ , (105) x 4 y 5 z 6 x 7 y 8 z 9 f =ψ¯γ ψ =ψT γ2ψ = ψT ψ, (99) 0 0 0 − then the Maxwell equations (MWEQs) can be written which is negative even though the corresponding sec- (remarkably compact) as: ond moment is positive. The normalization is done bky mul0ti,p7l,ic8a,t9io.n Twhitihs cγak2n=be±e1x,prwehssicehd ebqyuathlse−m1ulftoir- F = −DΦ (106) ∈ { } DF = 4πJ, plication with a quadratic diagonal matrix g˜ that is an extended version of the “metric tensor” gµν: with the usual choice of units. The Lorentz force can also be expressed by RDMs. If g˜ = Diag( 1,1,1,1,1,1,1, 1, 1, 1) the 4-momentum is defined by − − − − (100) g˜2 = 1 P= γ +p γ +p γ +p γ = γ +~p~γ, (107) 0 x 1 y 2 z 3 0 E E The transformed EQOM are: - where is the energy and p~ the momentum - then the E f˙ = Bf Lorentz force equations can be written as g˜f˙ = (g˜Bg˜)(g˜f) (101) dP q f˙˜ = B˜f˜, =P˙ = (FP PF) , (108) dτ 2m −