Dirac Observables and Spin Bases for NRelativistic Particles �A.LucentiDipartimento di Fisica, Universit�a di Milanoand INFN, Sezione di Milano, ItalyE-mail: lucenti@mi.infn.itL.LusannaINFN, Sezione di Firenze, Firenze, ItalyE-mail: lusanna@vax�.�.infn.itM.PaurizCenter for Philosophy of Science, Pittsburgh University, Pittsburgh, PA 15260,U.S.A.and INFN, Sezione di Milano, Gruppo Collegato di Parma, Parma, Italy.E-mail: pauri@pr.infn.itAbstractThe construction of the Dirac observables in the P 2 > 0 stratumfor a system of N relativistic free particles is carried out on thebasis of a quasi-Shanmugadhasan canonical transformation relatedto the existence of a Poincar�e group action. The explicit form ofthe Dirac observables is derived by exploiting an internal Euclideangroup having the Poincar�e canonical spin as generator of rotations.This procedure provides the symplectic version of the conventionalangular momentum composition.�This work has been supported partly by I.N.F.N., Italy Iniziativa speci�ca FI-41, and par-tly by the Network Constrained Dynamical Systems of the E.U. Programme `Human Capitaland Mobility'.zOn leave from Dipartimento di Fisica, Universit�a di Parma, Parma, Italy.

1 IntroductionIt is generally recognized after Dirac [7] that the Hamiltonian description of themost signi�cant classical systems is based on a pre-symplectic co-isotropicallyembedded submanifold of the standard phase space, i.e. a manifold equippedwith a degenerate two-form (in this paper we deal with 1st class constraintsonly). This entails that the generalized Hamiltonian formalism includes theconstraints which de�ne the pre-symplectic submanifold [8, 13, 12, 15, 41, 42,9]. In particular, the Lagrangians of all the manifestly covariant relativisticsystems are singular [32] and the singular Legendre transformation identi�esonly a submanifold of the cotangent bundle.As far as classical physics is concerned, the fundamental issue is the con-struction of the observables for the constrained system (the so-called Diracobservables), i.e. the functions on the pre-symplectic manifold that are inva-riant under the gauge transformations generated by the constraints and therebyconstant on the gauge orbits that foliate the pre-symplectic submanifold itself.After Shanmugadashan [39, 26] 2 a constructive method [26, 27] to �nda complete basis of Dirac observables has been based on the existence of aspeci�c set of canonical transformations. Within a local chart of the 2M-dimensional phase space with coordinates (qi; pi), covering a region of the pre-symplectic submanifold de�ned by m < M �rst-class constraints �u(q; p) �0; (u = 1; : : : ;m), canonical transformations (qi; pi) �! (Qu; Pu; Qv; Pv) ; (u =1; : : : ;m; v = 1; : : : ;M �m) always exist such that Pu � 0 de�ne the same re-gion . One says that (Pu) is a local Abelianization of the �rst-class constraints,that (Qu) are the associated local Abelian gauge variables (locally spanning theintersection of the gauge orbits in with the given chart), and that (Qv ; Pv) isa local complete symplectic basis of `strong' Dirac observables (i.e. not only ha-ving zero Poisson brackets with the constraints but also with the Abelian gaugevariables). Therefore, locally, one gets a separation between physical (labeled byv) and gauge (labelled by u) degrees of freedom. The local chart (Qv; Pv ; Qu)realizes a so-called Darboux chart on the pre-symplectic submanifold . Theexistence of a Shanmugadashan canonical transformation for �nite-dimensionalsystems is guaranteed by the Lie theory of function groups (fully developed byEisenhart [10]).The main issue concerning classical relativistic systems in general (particles,strings and �eld con�gurations) is whether they admit a class of Shanmugada-shan canonical transformations that are de�ned globally over the whole phasespace or, at least, in the neighbourhood of the pre-symplectic submanifold .If a class of such global transformations exists, and provided one forgets aboutthe issue of manifest Lorentz covariance, one gets a globally de�ned separation2Let us remark that the well-known papers [43] and [44] about �eld theory which containthe basic de�nition of path-integral measure in phase space, are based on the existence ofa guessed canonical transformation, which was shown by Shanmugadashan to be actuallyexistent, at least in a �nite number of dimensions.1

of the Dirac observables from the gauge sector of the theory, hence the possibi-lity of reformulating the description of the classical system in terms of physicalquantities alone.For many relativistic systems [27] in particular (particles [22], Nambu string[4, 5, 6], Yang-Mills �elds with fermions [29]) the above procedure is feasibleby exploiting the Poincar�e group and adapting the variables to its structure.Since this group is always assumed to be globally implementable, its associatedmomentum map entails precisely the existence of a global Shanmugadashancanonical transformation.In this paper we shall apply the Shanmugadashan technique to the relativi-stic kinematics of N free scalar particles described by N �rst-class constraintsp2!�m2! � 0, ! = 1 : : :N . The constraint pre-symplectic submanifold of everyrelativistic system is foliated in strata [1] associated with the four types of Poin-car�e orbits P 2 > 0, P 2 = 0, P 2 < 0, P � = 0. Each stratum must be analyzedseparately since di�erent little groups entail di�erent adaptation of variables.By restricting ourselves to the main stratum (P 2 > 0) of the total momentumP� = P! p�!, and exploiting the semidirect product structure of the Poincar�egroup, we de�ne a preliminary center of mass decomposition (see Section 3).The relative coordinates, once boosted at rest by the standard Wigner boosttechnique for the relevant Poincar�e orbits, are splitted into a rotational scalar(a relative time) and a Wigner vector, in agreement with the covariance proper-ties under the SO(3) little group of the orbits. This is the �rst step (adaptationof the coordinates to the Poincar�e group) towards a Shanmugadhasan cano-nical transformation. A second step is a non-linear canonical transformationadapted to N-1 suitable combinations of the N �rst-class constraints. As amatter of fact, this is a quasi-Shanmugadhasan canonical transformation, sinceonly N-1 of the new momenta vanish in force of the constraints. The rema-ning �rst class constraint may be put in the form of a polinomial equation inpP 2 of order 2N , whose solutions describe the 2N disjoint branches of the massspectrum of the system. Therefore 2N�1 di�erent Shanmugadhasan canonicaltransformations adapted to 2N�1 branch pairs are required. Note that this �nalcanonical transformation cannot be performed in general if interactions are pre-sent (except for the special cases of Liouville integrability. See [28] for a moredetailed description of the mass spectrum). At the above stage of the quasi-Shanmugadhasan transformation, one of the canonical variables is the squareroot of the �rst Poincar�e invariant, pP 2. For P 2 > 0 the second Poincar�e inva-riant is W 2 = �P 2~S2 (where ~S is the rest-frame Thomas canonical spin) andone should take into account the spin orbits corresponding to both ~S2 6= 0 and~S2 = 0. In the present paper, we shall deal with the main stratum of orbits~S2 6= 0 only, using the constructive theory of the canonical realizations of Liegroups [32, 33, 34, 35]. We look for coordinate charts simultaneously adaptedto the constraint manifold and to the group structure. The adaptation of thecoordinates is obtained by means of a homeomorphism between a subset of the2

quasi-Shanmugadhasan chart and the orbits of the co-adjoint action of the Poin-car�e group (see Section 4). In particular, the canonical transformation of therelative variables is constructed in such a way that j~Sj becomes one of the newcanonical momenta. In this way, the Darboux chart contains a typical chart onthe orbits of the co-adjoint representation of the Poincar�e group and both ofthe Poincar�e invariants appear in the �nal canonical basis.Let us stress that there is a number of motivations for constructing thiscanonical basis. First of all in the case of special-relativistic systems, the pre-symplectic manifold equipped with the closed degenerated two-formmust alwaysadmit a global pre-symplectic implementation of the Poincar�e group. Unfortu-nately, the studies on pre-symplectic geometry turn out not to be fully developedas those on other structures: for instance, Poisson manifolds (i.e. the duals ofpre-symplectic manifolds) have been deeply explored because of their relevancein many sectors of mathematical physics. Therefore, there is a surviving needto develop all the mathematical tools that can be instrumental to the analysisof special-relativistic physical systems with constraints.As shown in [28], there are several related physical motivations for deepeningthe understanding of these canonical bases: i) The need to take into accountfrom the beginning the decomposition of the space of solutions of the physicalsystem in the strata corresponding to the various types of Poincar�e orbits; notethat this decomposition is well-known in �eld theory but it is never really usedin the applications. ii) The unsolved problem of relativistic bound-states: sincethe states of Fock space are tensor products of single-particle states, there isno control on the relative times of the in-out particles, a fact that is re ectedin the spurious solutions of the two-particle Bethe-Salpeter equation. Thesesolutions are excitations of the relative energy, viz. the variable conjugated torelative time (incidentally, this was the main motivation for �nding the canonicaltransformation (15)). iii) The actual impossibility of formulating 3 (or N > 3)�rst class constraints describing interactions among 3 (orN > 3) scalar particleshaving the cluster decomposition property in closed form [46, 47] (again a bigobstacle in presence of relative time). Finally: iv) A related problem is theimpossibility of �nding the branches of the mass spectrum for N > 3 (eitherfree or interacting) particles, since one needs to solve algebraic equations ofhigher order. A thorough dicussion of these kinematical issues can be foundin [28]. There, N free scalar particles are �rstly analyzed along the lines of ourSection 2. Then, this system is reformulated on space-like hypersurfaces foliatingMinkowski space-time (following Dirac [7]): this reformulation forces one tochoose the sign of energy for each particle since the position of a particle onsuch an hypersurface is identi�ed by three numbers only (all the particles sharethe same time of the hypersurface). Hence, there are 2N di�erent Lagrangianscorresponding to all possible choices. After that, the description is restrictedto space-like hyperplanes and, �nally, the con�gurations belonging to the mainstratum P 2 > 0 are described on the special family of hyperplanes orthogonal3

to the total momentum P �. Each hyperplane is intrinsecally determinated bythe physical system as its rest frame. The �nal outcome of this analysis is anew kind of instant form, actually the rest-frame instant form, characterized byWigner covariance. This realizes the relativistic separation between the center-of-mass and relative motions and allows to determine the form of the branchesof the mass spectrum. In [28] it is also shown that the equal-time variables forthe N particles (equal with respect to the Lorentz-scalar rest-frame time) arethe same as the subset of relative variables that one gets in the usual N-timesformulation, after the canonical transformations leading to our equation (21).Finally, it is shown there that: i) in the rest-frame hyperplane, an Euclideankinematical group naturally appears (the Lorentz boosts are implemented asWigner rotations) and, above all: ii) action-at-a-distance interactions can beintroduced in such a way that the problem of cluster decomposition reduces tothe analoguos problem in Newtonian mechanics. Furthermore, a system of Ncharged scalar particles plus the electromagnetic �eld is described on space-likehypersurfaces and then reduce to the rest-frame instant form.In conclusion, the mathematical tools developed in the present paper canbe hopefully exploited in this new rest-frame instant form of dynamics in or-der to investigate unexplored aspects of spin dynamics. As a matter of fact,our procedure provides the symplectic version of the conventional angular mo-mentum composition. The method of using a quasi-Shanmugadhasan canonicaltransformation to obtain Dirac observables in a stratum of the Poincar�e groupseems to be a powerful and new technique. As we shall see, these new canonicalbases may be called `spin bases' legitimately, since they describe, at the sametime, the algebraic and the geometric aspects of the composition of classicalspins (angular momenta in the center-of-mass frame). In particular, our choiceof the Poincar�e invariants P 2 and W 2 (i.e. the total invariant mass and angularmomentum) as canonical variables (see for example Table 5) is the only choiceof Poincar�e invariants that can survive in the interacting case where individualparticle Poincar�e invariants are no longer constants of the motion. Finally, these`spin bases' seem likely to provide an important tool in the study of the Nambustring [4] and in classical �eld theory.After some preliminary Sections, in Section 5, starting from the quasi-Shan-mugadhasan transformation, we rearrange the physical relative variables for atwo-particle system in a typical form [32] of the canonical realizations of the E(3)group, which turns out to be instrumental for the extension of our constructionto systems with a number of particles higher than 2. By adapting the chart ofthe physical relative variables ~� and ~� to the E(3) group, we are automaticallyled to introduce the second Poincar�e invariant as a canonical variable and we�nally get the main result: a chart that embodies both the Poincar�e groupstructure and the constraint structure.In Section 6, starting again from the quasi-Shanmugadhasan transforma-tion, we build up the same construction for a three-particle system. Due tothe greater number of deegres of freedom, the procedure is, of course, more4

cumbersome. The geometry of the physical relative variables is characterizedby four Wigner vectors ~�! and ~�! (! = 1; 2). By independently transformingthe two pairs (~�1 , ~�1) and (~�2 , ~�2) as in Section 5, we get a typical form ofE(3) based on the vector ~S = ~S1 + ~S2 as generator of the `internal' rotations.The presence of ~S as a generator is actually the key factor, since in this waywe succeed in adapting the physical relative variables to the choosen `internal'E(3) group, and, at the same time, to the Poincar�e group, by enrolling the spininvariant as a canonical coordinate. Actually, this method provides the basisfor the construction of the `spin bases' for a system of N particles. In Section 8the constructive method is �rst naively applied to a system of N particles star-ting from the quasi-Shanmugadhasan transformation given in Section 3 andlooking for a chart adapted to the Poincar�e group. Instructed by the resultsof Section 6, we obtain chains of canonical transformations matching a set ofcanonical subrealizations of the E(3) group, labelled by particle indexes. Thecanonical transformation that leads to the chart adapted to the Poincar�e groupclearly depends on the combinatorial character of the matching (in some sense,this can be said to be the classical analogue of the spin composition in quantummechanics). On the other hand, in Section 7, it is shown how to invert thecanonical transformation that adapts the chart to the Poincar�e group and thegeometric meaning of the relative canonical variables is made fully transparent.2 N particle constraint theoryAs shown by Komar [17, 18, 19], a system of N relativistic scalar particles can bedescribed via constraint theory by introducing eight, instead of six, phase-spacevariables for each particle, namely (� = 0 : : : 3 ; ! = 1 : : :N):(x�!; p�!) ; (1)satisfying the usual Poisson algebrafx�! ; p�!0g = ��� �!!0 : (2)In the case of free particles, the N �rst{class constraints (c = 1):�! � p2! � m2! � 0 ; (3)de�ne a pre-symplectic submanifoldMc (co-isotropically embedded in the phasespace [13]). The constraint vector �elds Y! = f� ; �!g generate both the dy-namics and the gauge orbits on Mc since the canonical Hamiltonian is identi-cally zero. The dynamical evolution is ruled by the Dirac Hamiltonian HD =P! �!(�)�! (with �!(�) arbitrary multipliers). The Hamilton{Dirac equationsimply, for each particle:~p! = cost: ; p0! = �pm2! + ~p2!x�! = x�!(0) + p�! R �0 d� 0�!(� 0) : (4)5

These equations provides the manifestly covariant description of the (!)th indi-vidual world{line. In order to reduce the remaining seven degrees of freedom foreach particle to the six Cauchy data, one has to further reduce the phase spaceby taking the quotient of Mc with respect to the gauge foliation generated bythe vector �elds Y! . On the other hand, within the constraint manifoldMc, therelative time among the particles is not restricted and each particle retains sevendegrees of freedom, one of which is related to a temporal variable (x0!). The�!'s in eqs. (3) are indeed the generators of the re-parametrization invariancetransformations on each world-line. From a physical point of view, this meansthat we have the freedom to de�ne either the `individual clocks' measuring thetime of each particle or a global time and N-1 `relative clocks'. This situation re-minds of the time-di�eomorphism invariance in general relativity where, again,the gauge freedom corresponds to the choice of local clocks. Finally, the assumedglobal action of the Poincar�e group on M is generated byP� =P! p�! ; J�� =P!(x�! p�! � x�! p�!); (5)while the Poincar�e invariants areP 2; W 2 = �P 2S2; (6)where S2 � j~Sj2 = �W�W�P 2 ;W � = 12�����J��P� . Mc is a strati�ed manifoldwhose strata are identi�ed by the allowed Poincar�e orbits de�ned by P 2, and has2N disjoined components (each particle having its past and future hyperboloids).We shall study the main stratum P 2 > 0.3 The Shanmugadhasan transformationFor a single-particle with coordinates (x�; p�) one can perform the followingcanonical point (in p�) transformation [20] (i = 1; 2; 3):8><>: � = �M ; T = � p � xMki = � piM ; zi = �M �xi � pip0x0� ; (7)where � = �;M = pP 2, so that the constraint � � p2 �m2 � 0 becomes� = �2 �m2 � 0 : (8)In this case, the manifold Mc has two disjoined components and therefore,setting � �! �0� = � � m, two Shanmugadhasan canonical transformations(SCT) can be de�ned. The physical meaning of the variables in eqs. (7) is thefollowing: 6

� T is the time measured in the rest frame (center-of-momentum) of theparticle;� ki is the spatial part of the four{velocity k� = � p�M (with k0 = (1+k2)1=2or k2 = 1);� zi, modulo a mass factor, are the Cauchy data of the initial position.The canonical pair fT ; �g = �1 describes the gauge sector. The constraint (8)�xes the value of � on the physical manifold, while the T variable remainsarbitrary. Its gauge �xing corresponds to the freedom of arbitrarily choosingthe time scale (the gauge �xing is equivalent to a de�nition of the overall clock).The Poincar�e generatorsP� = p�; Jk = 12�ijkzikj ; J i0 = k0zi ; J ij = zikj � zjki ; (9)are functions of the variables of the physical sector (observables). Under thePoincar�e transformation (a;�), the variables de�ned in eqs. (7) transform as(see [20]):8>>><>>>: T 0 = T + k�(��1a)� ; �0 = �z0i = �ij � �i�k��0�k� �0j! zj + ���i� � �i�k��0�k��0�� (��1a)�k0� = ���k� : (10)Note that zi are in fact the (non-covariant) Newton-Wigner position variables[31].For a system of N particles, a SCT must be carried out for each disjoinedcomponent of Mc. While this can be easily done for N free particles, the fea-sibility of the transformation in presence of an interaction depends upon theLiouville integrability of the interacting system.. It is possible, however, to de-�ne an intermediate SCT (called henceforth quasi-Shanmugadhasan canonicaltransformation) characterized by the fact that the gauge sector is separatedout, even though the incomplete resolution of the constraints forbids an explicitinversion of the transformation (see [22, 26, 27, 28]).The �rst step towards the construction of the quasi-SCT is the de�nitionof a preliminary canonical transformation of the form (! = 1; : : : ; N ; a =1; : : : ; N � 1):8>><>>: x� = 1pN X! ! x�! ; P � = X! p�!R�a = pNX! a! x�! ; ��a = 1pN X! a! p�! ; (11)7

where = 1pN 0@ 1: : :1 1A, and a = ( a!) constitute an orthonormal basis of theN-dimensional Euclidean space ( 2 = 1 ; 2a = 1 ; � a = 0; a � b = �ab ; ! !0 +Pa a! a!0 = �!!0). The inverse transformation is (!; !0 = 1; : : : ; N ; a =1; : : : ; N � 1): 8>><>>: x�! = x� + 1pN Xa a!R�ap�! = 1N P� + pNXa a!��a ; (12)so that the constraints of eq. (3) can be replaced by8>>>>>>><>>>>>>>: � = NX! �! = P 2 � NX! (m2! �NXa;b a! b! �a ��b)�a = pN2 X! a!�! = P ��a �pN2 X! a!(m2!�NXb;c b! c! �b � �c) : (13)The Poincar�e generators becomeP�; J�� = L�� + S��L�� = x�P � � x�P�S�� =Pa(R�a��a �R�a��a) : (14)A second step, which simpli�es the expression of the constraints, consists inperforming a further canonical transformation as follows3:8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

x� = x� +P! 12P 2 (m2! � (p! � !P ) � (p! � !P ))0@x�! � x� � X!0 2!0P � p!0!�1X!00 2!00P � p!00 �P � r!!00P � p! (p�!++ !P�)�X!0 P � r!0!00P � p!0 !0p�!0!!P� = P! p�!R�a = pNX! a! x�! + (X!0 2!0P � p!0 )�1X!00 2!00P � p!00P � r!!00P � p! � (p�! + !P�)���a = 1pNX! a! �P�! + P�2P 2pN(m2! � (p! � !P ) � (p! � !P ))�(15)3Remember that ! are equal to 1pN . 8

where r�!!0 = x�! � x�!0 . Then the N-1 constraints �a (see eq. (13)) can berewritten in the expressive form:�a = P � �a � 0 : (16)Note that, for N > 2, the transformation (15) cannot be inverted explicitly(see [22, 28]) (for N particles one would have to solve an algebraic equation oforder 2N�1). Now, the Poincar�e generators areP�; J�� = L�� + S��L�� = x�P � � x�P�S�� =Pa(R�a��a � R�a��a) : (17)The �nal step in the construction of the quasi-SCT is achieved in the follo-wing way:i) �rstly, the relative variables are boosted to the center of mass rest-frame(~P = 0) (a = 1; : : : ; N � 1):8<: rAa = LA�( 0P ; P )R�aqAa = LA�( 0P ; P )��a ; (18)by means of the Wigner boost with 0P�� (�pP 2; 0; 0; 0):L��(P; 0P ) = ��� + 2P� 0P �P 2 � (P+ 0P )�(P+ 0P )�(P+ 0P ) � P ; (19)ii) secondly, the center-of-mass coordinates are modi�ed as:�x� = x� � 1�(�+ P0) �P� S�� + ��S0� � S0� P�P��2 �� ; (20)and then transformed as in eq.( 7).9

Explicitly, the quasi-SCT is given by (see [22]) (a = 1,...,N-1):8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:T = 1N Xa (Ta) = �P � �xM ; � = Pa �a = �Mzi = �M ��xi � pip0 �x0�ki = � piM�a = �P � RaM ; �a = �P � �aM :�ia = Ria � P iM �R0a � ~P � RaM + �P0!�ia = �ia � P iM ��0a � ~P � �aM + �P0! :

(21)The variables (21) are reproduced in the scheme of Table 1: canonically conju-gated variables are aligned in the same column. The gauge sector lies to theright of the physical sector. On the other hand, the center-of-mass variables ap-pear in the upper part and the relative variables in the lower one. The Poincar�egenerators can be rewritten in the form8>><>>: P� = �k�J ij = zikj � zjki + Pa �Sija ; �Sija = �ia�ja � �ja�iaJ i0 = k0zi +Pa �Sika kk1 + k0 : (22)The quasi-SCT, summarized in Table 1, de�nes a chart adapted to the constraintstructure of the theory. The constraints, expressed in the new coordinates, takethe form (a = 1; : : : ; N � 1):� � = F �P 2 = �2; m! ; ~�a � ~�b� � 0�a = ��a � 0 ; (23)where F (�) is a polynomial of order 2N�1 in P 2 whose explicit form can beexhibited only if the canonical transformation (15) can be inverted in a closedform (essentially, one has to solve the system of the N-1 equations P � �a =P ��a� pN2 X! a!0@m2! � NXb;c b! c! �P ��b P ��cP 2 � ~�b � ~�c�1A ; (a =1; : : : ; N � 1); to get P � �a in terms of P � �a and of ~�b � ~�c. Here ~�a is thespace part of ��a de�ned in equation (11), taken after having boosted it to thecenter-of-mass frame (~P = 0)). Note, in addition, that the Wigner boost splitsthe relative variables in rotational scalars �a, �a and in Wigner vectors ~�a, ~�a,(see [28] for a more detailed discussion of this kinematical setting; the variables10

x�; R�a ; ��a ; �x�; T; ~�a; �a; ~�a; �a of the present paper are denoted in [28] byx�; R�a ; Q�a ; ~x�; T ; ~�a; TRa; ~�a; �Ra, respectively).Since the evolution in an arbitrary parameter � is ruled by the Dirac Hamil-tonian HD = �(�)� +Pa �a(�)�a, it is seen that only �x� has its four-velocityparallel to the total four-momentum: dd� �x� = f�x� ; HDg = 2�(�) @F@P 2P�. Onthe other hand, from dd� x� = �(�) fx� ; �g+Xa �a(�) fx� ; �ag and dd� x� =�(�) fx� ; �g +Xa �a(�) fx� ; �ag, it follows that x� and x� exhibit a mo-tion with respect to �x�, which we shall denominate pseudo-zitterbewegung. Ingeneral, this phenomenon naturally arises from the conjunction of the requi-rements that a `position' dynamical variable be at the same time canonicaland relativistically covariant. As well known, only three di�erent `position'dynamical variables can be de�ned within the irreducible realizations of thePoincar�e group (namely the Moeller, the Fokker and the Pryce-Newton-Wignerthree-vector centers of mass) but none of these is at the same time canonicaland covariant. If, however, one is not restricted to Poincar�e algebra genera-tors alone and extra degrees of freedom are allowed, an inde�nite number of`position' dynamical variables can be constructed (see the relevant Referencesin [37, 38]). In particular, a Dirac-like canonical and covariant three-vectorposition variable ~x(t) can be constructed (in the instant form) in the case ofthe reduced two-particle model (pole-dipole of [37, 38]) in terms of the Poincar�egenerators and of an additional inner angular degree of freedom canonicallyconjugated to the magnitude of the intrinsic angular momentum. The timederivative of this vector is not parallel to the total momentum so that it under-goes an helicoidal motion which can be safely called a classical zitterbewegung.On the other hand, in the context of the present paper, the extra degrees offreedom belong to the gauge sector, so that the canonical and covariant po-sition vector ~x (see Section 3) is, necessarily, only partially adapted to theconstraints. Relative to the Fokker three-vector ~XF = � ~KP0 + ~�S ^ ~P�P0 , interpre-ted as a geometric (covariant but non-canonical) center of mass, ~x undergoesa motion that can be seen as `internal' with respect to the particle system asa whole. The terminology of pseudo-zitterbewegung in this case appears to bejusti�ed by the usage already established in the literature 4 and by the factthat ���~x� ~XF ��� has a typical relation with the invariants of the \spin" Lorentzgroup (precisely, in the case of two particles, in the rest frame ~P = 0, one has 54See for instance the terminology adopted in [45] in connection with the position operatorfor the �eld con�gurations in the Klein-Gordon case.5See Section 5 for the de�nition of the variables.11

j~x� ~XF j~P=0 = 1� ( �T 2R�2+��2R[(~� � ~�� )2+S2�2 +2 �TR��R ~� � ~�� �)1=2 = pS2 � I1� , whereI1 is the Lorentz invariant 12S��S�� ). Finally, it is obvious by construction thatall the position variables we take into account, in the non-relativistic limit andwhen evaluated at the same time, reduced to the standard Newtonian center ofmass. These issues will be exhaustively discussed elsewhere.In force of the quasi-SCT, the center of mass coordinates have been alsoautomatically adapted to the co-adjoint orbits of the Poincar�e group and theinvariant P 2 appears now among the canonical variables. However, the secondinvariant of the Poincar�e group, namely the Pauli-Lubanski invariant (see (9)),W 2 = �P 2~S2 �Si = �ijkPa �Sjka � ; (24)is not enrolled among the relative canonical variables and the canonical reali-zation of the Poincar�e group has not assumed, as yet, a typical form 6. Thecompletion of our program demands adapting all of the coordinates to the co-adjoint action of the Poincar�e group: this requires in turn that the relativevariables be adapted to the SO(3) group, which is the little group of the orbitswe are analysing.The basic tool for realizing this step is provided by the constructive theoryof the canonical realizations of Lie groups sketched in the following section. Itwill appear that exploiting the geometrical and group properties of the relativephysical variables needs the implementation of a second-rank group containingSO(3) as a proper subgroup, like E(3), SO(3,1) or SO(4).4 Canonical Realizations of a Lie GroupLet M be a symplectic manifold and � the action of the Lie group G on M:� : G�M !M : (25)It can be shown that it is possible to construct a peculiar class of local chartson M by exploiting the local homeomorphism existing between a submanifoldof M and the co-adjoint orbits on G� (dual of the Lie algebra G of G) [1, 32, 40].Each chart belonging to the above class, as characterized by its being adaptedto the group structure, is called typical form.According to the general theory [32], a canonical realization K of a Liegroup Gr (of order r) can be characterized in terms of two basic schemes: A)the scheme A which depends entirely on the structure of the Lie algebra LGr(including its cohomology) and amounts to a pseudo-canonization of the gene-rators, in terms of k invariants J1 : : :Jk and h = (r � k)=2 pairs of canonicalvariables (irreducible kernel of the scheme) functions of the generators; B) the6see Section 4 12

scheme B (or typical form) which is an array of 2n canonical variables Pi,Qi,de�ned by means of a canonical completion of the scheme A.Locally, the schemeB allows to analyze any given canonical realization of Gr and to construct themost general canonical realization of Gr.A generic scheme A can be usefully visualized by the following table:P1 : : : Ph J1 : : : JkQ1 : : : Qh (26)where variables belonging to the same vertical pair are canonical conjugated,and variables belonging to di�erent vertical lines commute. The quantities Jiclearly commute with all of the generators and are the invariants of the reali-zation. Of course, any set of k functional independent functionsJ 01(J1; � � � ;Jk); � � � ;J 0k(J1; � � � ;Jk) of the invariants are good invariants as well.A scheme A is called singular if, due to some functional relations, already exi-sting or imposed, among the invariants, some canonical pairs become singularfunctions of the generators and must be omitted from scheme A itself which,then, has to be re-determined from the beginning. In this case, the number ofcanonical pairs may result m < h.The scheme B, i.e. the canonical completion of scheme A, is accomplishedin general according to one of the following possibilities:Type 1: no new variable is added to the scheme A and k independentfunction of the canonical invariants are put identically equal to constants. 2htypical variables Pi, Qj are identi�ed with the variables of the irreducible kernel.The explicit expression of the canonical generators, in terms of the Pi,Qj of thescheme B, is obtained by inverting the functions of the scheme A. Then, anarbitrary �xed (with respect to the group parameters) canonical transformationS gives the expression of the generators in terms of 2h generic canonical variablepi, qj . Since, in this case, the phase space contains no submanifolds invariantunder the action of Gr, these realizations are called irreducible (i.e. transitive),for any k-uple of allowed constant values of the functions of the invariants.Type 2: a certain number l (l � k) of canonical variables Qs (called supple-mentary variables) turn out to be coupled or are axiomatically coupled, to l ofthe invariants Ji, building up l new canonical pairs, while k�l independent fun-ctions of the invariants are put identically equal to constants. Then, the genera-tors, as functions of the typical variables Pi,Qi,Ps (i; j = 1; : : : ; h; s = 1; : : : ; l),are obtained by inversion as before but do not depend on the supplementary va-riables Qs. The phase-space, in this case, is 2h+2l dimensional and, containingthe submanifolds Jt(p; q) = const: (t = 1; � � � ; k� l) as invariant submanifolds,corresponds to non-irreducible (i.e. intransitive) realizations for any (k� l)-uple13

of allowed constant values of the invariant functions that have been constrained.A particular case of Type 2 is the so-called complete realization, correspondingto l = k and (r + k)=2 canonical pairs. This realization is completely determi-ned, locally, by the group structure. In geometrical terms, the variables of theirreducible kernel, together with the k supplementary variables, de�ne a localchart on the orbits of the co-adjoint representation of Gr.Type 3: an arbitrary number v � n� h of pairs of canonical variables Qu,Pu (u = 1; : : : ; v) (inessential variables) turn out to exist, or are axiomaticallyadded, to the scheme A. These variables are not canonically coupled to inva-riants nor share any functional relation with the variables of the scheme A, sothat they commute with all the variables considered up to now. Then, one pro-ceeds, as for Type 1 and Type 2, by inverting the functional dependence ofthe typical variables on the generators and performing an arbitrary �xed cano-nical transformation which leads to the generic form of the realization in termsof 2n canonical variables pi, qj . Since the inessential variables de�ne invariantsubmanifolds in phase-space, these realizations are non-irreducible (i.e. intran-sitive). Note that Types 1-and-3, or2-and-3, are mutually compatible.A-priori, we are interested in the rotation group SO(3), the Poincar�e group, andthe rank-2 groups E(3), SO(3,1) and SO(4). Their schemes A have the followingstructure:SO(3):The Poisson bracket algebra (PBA) and the scheme A arefSi; Sjg = �ijkSk ; (27)and P1 = S3 J1 =p(S1)2 + (S2)2 + (S3)2 � SQ1 = arctan S2S1 (28)respectively.T4 _�SO(3,1) (Poincar�e group):The generators are: ~J rotations, ~K special Lorentz transformations, P � =(P0; ~P ) space-time translations. The PBA isfJi; Jjg = � kij Jk fKi;Kjg = �� kij JkfJi;Kjg = � kij Kk fKi; Pjg = ��ijP0fJi; Pjg = � kij Pk fKi; P0g = �PifJi; P0g = 0 fPi; P0g = 0fPi; Pjg = 0 ; (29)14

and the scheme A for the P 2 > 0 class can be written as~P = ~P P4 = S3 J1 = S J2 =M~Q = � ~KP0 + ~S ^ ~PP0(P0 +p�) Q4 = arctan S2S1 (30)where 8>>><>>>: M2 = P 20 � ~P 2Si = P0p�J i + ( ~K ^ ~P )ip� � ~J � ~Pp�(P0 +p�)P iS = p(S1)2 + (S2)2 + (S3)2 : (31)The rank-2 groups E(3), SO(3,1) and SO(4) have the following PBA's:E(3):�Si ; Sj = �ijkSk ; �Si ; Rj = �ijkRk ; �V i ; Rj = 0 ; (32)SO(3,1):�Si ; Sj = �ijkSk ; �Si ; Rj = �ijkRk ; �Ri ; Rj = ��ijkSk ; (33)SO(4):�Si ; Sj = �ijkSk ; �Si ; Rj = �ijkRk ; �Ri ; Rj = �ijkSk ; (34)while the corresponding schemes A can be given the unique formS3 S I1 I2tan�1 S2S1 � = tan�1 S (~S ^ ~R)3[~S ^ (~S ^ ~R)]3 (35)provided the invariants are de�ned asI1 = ~R � ~R; I2 = ~R � ~SI1 = ~S2 � ~R2; I2 = ~R � ~SI1 = ~S2 + ~R2; I2 = ~R � ~S ; (36)respectively, in the three cases.15

5 System of two particlesThe simplest model in which both the constraints and the canonical groupstructure can be signi�cantly exhibited is provided by a system of two particles.Table 2 summarizes the quasi-SCT in this case.As anticipated at the end of Section 3, the next step in the construction ofthe �nal basis consists in adapting the relative physical variables (~�; ~�) not onlyto the SO(3) little group, but rather to a larger second-rank group. In whatfollows we shall exploit the Euclidean group E(3) since its Abelian invariantsubgroup is particularly suited to be realized in terms of the `internal' relativevariables, and the role of the Abelian subgroup is a key factor to the wholeconstruction.First of all, the vectors ~� and ~� are replaced by the scalar radial variables 7� � p~�2 ; ~� � � ; (37)and the new vectors (i; j = 1; 2; 3)�i � �i� ; �i � ��i? � p~�2(�ij � �i�j)�j ; (� � ~�? = 0) ; (38)so that the canonical spin can be written:~S = ~� ^ ~� = p~�2~�? ^ � = ~� ^ � : (39)Then, we get an explicit expression of the irreducible kernel of the scheme (35)by realizing the vector ~R simply asRi = �i : (40)Since ��i ; �j = 0, ~S � � = 0 and � � � = 1, this automatically implementsa realization of the Euclidean group E(3), characterized by the �xed valuesI1 = ~R � ~R = 1, I2 = ~R � ~S = 0 of the invariants, and by a functional form of thevariable canonically conjugated to S given by 8:� = tan�1 1S (~�2 �3�3 � ~� � ~�) = tan�1 1S �3�3 : (41)Note that the following formulae hold true:@R(�)@� = R(�+ �2 ) = ~SS ^ R(�) ; (42)7For any vector ~V , the expression V means ~VV throughout the paper.8Of course the superscripts 3 in the following expressions means "third component of athree-vector". Its geometric meaning is shown in Figure 1.16

and 8>>>>>><>>>>>>: R1 = S2pS2 � (S3)2 sin� � S1S3SpS2 � (S3)2 cos�R2 = S1pS2 � (S3)2 sin� � S2S3SpS2 � (S3)2 cos�R3 = pS2 � (S3)2S cos� ; (43)so that the inverse transformation of the relative variables can be written:8<: �i = � Ri(~S; �)�i = ~� � ~�� Ri(~S; �) + S� (R(~S; �) ^ ~S)iS : (44)Finally, owing to the Poisson bracket relations:�Si ; � = �Si ; � � � = 0 ; f� ; �g = f� ; � � �g = 0 ; (45)and f� � � ; �g = 1 ; (46)it follows that the E(3) canonical realization has the radial variables as inessen-tial variables and therefore is a non-irreducible canonical realization of Type 3(see Section 4), summarized in Table 3. Since now the Poincar�e invariant S hasbeen enrolled as a canonical variable, the adaptation to the Poincar�e group iscomplete. Table 4 summarizes the typical form of the Poincar�e realization, di-splaying the same distinctions made before between gauge and physical variablesas well as between center of mass and relative variables.Note that the vectors ~S; ~R; ~S ^ ~R de�ne a right-handed orthogonal referenceframe (see Figure 1) and satisfy the following peculiar algebraic relations:�Si ; Sj = �ijkSk ; �Ri ; Rj = 0 ;�Si ; Rj = �ijkRk ;n(~S ^ ~R)i ; Sjo = �ijk(~S ^ ~R)k ;n(~S ^ ~R)i ; (~S ^ ~R)jo = �R2�ijkSkn(~S ^ ~R)i ; Rjo = RiRj �R2�ij : (47)6 The system of three particles as a canonicalrealization of the Euclidean group E(3)The simplest model in which both the constraint and the Poincar�e group structu-res are substantially exploited, and the Euclidean second-rank group plays ane�ective instrumental role, is provided by a system of three particles. The initialscheme is given in Table 1. According to our program, we have to adapt therelative physical variables (�ia; �ja); (a = 1; 2) to the E(3) co-adjoint orbits. Since17

variables with di�erent particle labels commute, Table 1 can be rewritten as the`direct product' of two E(3) realizations constructed as in Section 5. In fact, wehave now a pair of commuting reference frames de�ned by the two sets of E(3)generators, namely:R1; ~S1; S1 ^ ~R1 and R2; ~S2; S2 ^ ~R2 ; (48)the canonical spin being ~S = ~S1 + ~S2 : (49)The two irreducible kernel of the E(3) realizations areS31 S1tan�1 S21S11 �1 S32 S2tan�1 S22S12 �2 (50)We will construct now, starting from the two previous single-particle realizationsof E(3), a global canonical realization of a single E(3) with ~S given by (49) andwith �xed values of the invariants (as in Section 5). De�ning a unit vector Rsuch that: R2 = 1; R � ~S = 0; nRi ; Rjo = 0 ; (51)and making the ansatz R = a R1 + b R2 ; (52)eqs. (51) are satis�ed by the following values of a and b:ab = � R2 � ~S1R1 � ~S2 ; b = 1 + ( R2 � ~S1R1 � ~S2 )2 � 2(R1 � R2)( R2 � ~S1R1 � ~S2 )! : (53)Then, the irreducible kernel of the E(3) realization in terms of the relative phy-sical variables �ia; �ja is still given by the scheme of eq. (35), if ~R is replaced by R(see eq. (52)). This naturally de�nes the normalized common reference frame:S � ~SS ; R; � � S ^ R : (54)A canonical completion of the typical form requires the construction of fournew inessential variables which, together with the single-particle radial variables(�a; �a �~�a), build up the right number of deegres of freedom. The new variableshave to be guessed by exploiting the geometry of the reference frames (48)whose relative orientation is not constrained: actually, the inessential variablesto be constructed do supply precisely the geometrical information expressingthe arbitrary relative orientation. 18

First of all, the following scalar functions can be shown to commute withthe angle �, de�ned in the scheme A of eq. (35):R1 � R ; R1 � � ; R1 � �2 ;R2 � R ; R2 � � ; R2 � �1 ;R1 � ~S1� (R1 � R2)2 + 12S ; R2 � ~S R2 � ~S1R1 � ~S2! ((R1 � R2)2)� 1 � 12S ;~S1 � �R1 � ~S ; ~S2 � �R1 � ~S ;~S1 � ~S + 12S2 R2 � ~S1R1 � ~S2!2 ; ~S2 � ~S + 12S2(1� R2 � ~S1R1 � ~S2!2) :(55)

Then, a possible choice of the remaining inessential canonical variables, re-specting the particle-label simmetry, is the following:p1 = R1 � ~R ; p2 = R2 � ~R ;q1 = A1(p1; p2) (~S1 � ~S2) � (R1 ^ R2)2�1� (R1 � R2)2� � B1(p1; p2) R1 � �R2 ;q2 = A2(p1; p2) (~S1 � ~S2) � (R1 ^ R2)2�1� (R1 � R2)2� � B2(p1; p2) R1 � �R2 ; (56)whereA1 = �12 �R1 � R2�A2B1 = ��R1 � R2� R2 � ~S1R1 � ~S2!2 + 2 R2 � ~S1R1 � ~S2!�1 + �R1 � R2�2���R1 � R2�A2A2 = � 1 R2 � ~S1R1 � ~S2!+ �R1 � R2�B2 = 1� 2�R1 � R2�2 � � R2 � ~S1R1 � ~S2!21� �R1 � R2�2 A2 :(57)

19

Clearly, the involved appearance of the quantities qi; pj of eq (56) depends uponthe choice made in eq.(51), which in turn depends upon the requirement thatthe E(3) invariants have the �xed value ~R � ~R = 1 and ~R � ~S = 0. Giving up theseconditions, allows to get a simpler expression of the vector ~R of the global E(3)in terms of the single-particle vectors R1 and R2 and to achieve thereby a clearergeometric description of the symplectic composition of angular momenta. Weput: ~R �! ~N = 12 �R1 + R2� ;~� = R1 � R2 (58)so that the invariants of E(3), now enrolled as canonical variables, become:I1 = ~N � ~N = 12 (1 + R1 � R2) � N 2I2 = ~N � ~S = 12 (R1 � ~S2 + R2 � ~S1) : (59)In this situation, the canonical completion of scheme A of (35) amounts tobuilding a scheme B of the complete realization of E(3) ( Type 2 with l =k = 2), so that we have to construct two supplementary variables coupled tothe invariants. Furthermore, the presence of the `Abelian' vector ~N entails thatthis complete realization of E(3) contains two sub-realizations of SO(3) which arejust those generated by the `external' or `inertial' and the `body' componentsof the intrinsic angular momentum, respectively (see [33]: eqs (39),(41)), (ingroup-theoretical terms left and right translations). Actually, in the presentcase, if (~k3; ~k1; ~k2) are `external' unit axes, their `body' counterparts are N = ~NN ,� = ~�� = �R1 � R2�q2(1� R1 � R2) and N^�, respectively. Then the `body' componentsof the angular momentum are:�S3 = ~S � N ; �S1 = ~S � � ; �S2 = ~S � N ^ � : (60)This gives us a hint to the construction of the supplementary variable canonicallyconjugated to ~S � N , because the searched variable must have formally the samegeometrical structure as the variable conjugated to S3. By canonical completion,we get in fact the following two pairs of canonical variables:p1 = ~S � N = �S3 = I2 ; p2 = pN �N = N = I1q1 = tan�1 ~S � (N ^ �)~S � � = tan�1 �S2�S1 ; q2 = (~S1 � ~S2) � (N ^ �)� : (61)The di�erence ~S1 � ~S2 between the particle spin vectors appears in the secondsupplementary variable ( q2 in eq. (61)). It is easy to check that, in geometrical(not in group-theoretical !) terms, q2 has the same relation to N , as � to S.20

In conclusion, the complete realization of E(3) can be summarized as follows:S3 S ~S � N Ntan�1 S2S1 tan�1 S (~S ^ N)3[~S ^ (~S ^ N)]3 tan�1 ~S � (N ^ �)~S � � (S1 � S2) � (N ^ �)� (62)The �rst three canonical pairs contain a simultaneous canonical description ofthe `external' and the `body' intrinsic angular momentum. The geometric mea-ning of whole set of canonical variables (in particular that of the supplementaryvariables) will be made transparent presently. Finally, let us remark in thisconnection that the geometry of our coordinates should also provide some hintsfor solving the problem of extracting the physical degrees of freedom of theelectromagnetic �eld via di�erential operators, exploiting the natural directionS instead of the usual radiation condition (see [42]).The �nal form of the whole set of canonical variables, which are now adaptedboth to the constraint structure and to the action of the Poincar�e group, issummarized in Table 5.7 Relative variables referred to the intrinsicframe of a canonical realization of E(3):geometric meaning of the variablesThe geometric interpretation of the canonical variables can be made transparentby exploiting the inverse transformation of the one that brings from the quasi-SCT to the typical form. To this aim, it is instrumental to decompose ~�a and~�a in terms of the vectors S, R and S ^ R (a = 1; : : : ; N � 1):8><>: ~�a = Aa ~SS + BaR + Ca ~SS ^ R~�a = Da ~SS + EaR + Fa ~SS ^ R : (63)The basis vectors and the coe�cientsAa; Ba; Ca; Da; Ea; Fa have to be expressedas functions of the variables appearing in the typical form (see in particulareq. (43)).For simplicity, we shall work out the calculation for the case N=3. First ofall, we de�ne the vectors (see scheme (62)):~N = R1 + R22 ; ~� = R1 � R22 ;~S = ~S1 + ~S2; ~W = ~S1 � ~S2 ; (64)21

which satisfy 8>>>>>>>>><>>>>>>>>>:~� � ~� = 1� ~N � ~N~N � ~� = 0~N � ~S = �~� � ~W~N � ~W = �~� � ~S ; (65)and the Poisson bracket relations 9 :�N i ; N j = 0��i ; �j = 0�W i ; W j = �ijk Sk : (66)Eqs. (64) provide the relations among ~N; ~�; ~S and ~W and the single-particlevectors Ra and ~Sa. Therefore, since eq. (44) gives the expression of the single-particle relative variables ~�a and ~�a as functions of the vectors Ra and ~Sa,the inverse transformation can be simply obtained by re-expressing the vectors~N; ~�; ~S and ~W in terms of the variables of the typical form.The relevance of the geometric decomposition obtained by using the or-thonormal frame S, R and S ^ R follows from the fact that these vectors arefunctions of the irreducible kernel of the little group of the massive orbits (E(3)has SO(3) as subgroup). Let us put (see scheme (62))8>>>>>>>>>>>>><>>>>>>>>>>>>>:

� = tan�1 S (~S ^ N)3[~S ^ (~S ^ N)]3� = tan�1 ~S � (N ^ �)~S � �� = tan�1 S2S1� = (~S1 � ~S2) � (N ^ �)� ; (67)and de�ne the new angle: = ~N � ~SN S = N � S : (68)9Note that these vectors generate the 2-rank groups � ~N; ~S� � E(3), �~�; ~S� � E(3),� ~W; ~S� � SO(4) 22

The geometric setting corresponding to the above de�nitions is displayed inFigures 1 and 2, in which the frame S; R; S ^ R refers to (49) and (52), whilethe frame N; �; N ^ � is de�ned by (58) and subsequent formulae. Finally, weget: ~N = N cos S + N sin R~� = �1�N 2�1=2 �sin cos� S � cos cos� R + sin� S ^ R�~W = SN (1�N 2)1=2 sin cos cos� + � sin sin�! S+ � SN (1�N 2)1=2 cos� sin2 + � sin� cos ! R+ � cos� + sin� cos SN(1�N 2)1=2! S ^ R : (69)These expressions, together with eqs. (44), (63) and (64) provide the desiredinverse transformation.8 System of N particlesFinally, the construction of the typical form for the system of three particles, gi-ves us a clue for dealing with a system of N particles. We will exploit the `directproduct' of N -1 sub-realizations of E(3) in order to obtain a single realizationwhose typical form has the correct number of inessential variables.Since the irreducible kernel of any E(3) realization has two `deegres of fre-edom', the construction of a single global realization requires the preliminarychoice of an initial label-pair within the N -1 possible ones. Then the irreduciblekernel of this initial two-label realization has to be combined with the kernelof any of the remaining single-particle realizations. The variables of the initialtwo-label realization that do not belong to the irreducible kernel become inessen-tial variables of a three-label realization, which is being constructed in the sameway as the two-label realization is built up from the single-particle realization.It is now clear how to iterate the procedure until a single global realization ofE(3) with a (N � 1)-label irreducible kernel is constructed. Note, �nally, thatthis procedure gives rise to �N�12 � di�erent chains of canonical transformationscorresponding to �N�12 � �nal realizations of E(3).We shall carry out explicitly the construction of a typical form for a systemof four particles. We begin as usual with Table 1 and rewrite the variables ofthe irreducible kernel in the `direct product' form 1010The inessential variables of the single-particle realizations are not-listed for simplicity.23

S3(a) S(a)tan�1 S2(a)S1(a) �(a) S3(b) S(b)tan�1 S2(b)S1(b) �(b) S3(c) S(c)tan�1 S2(c)S1(c) �(c)The explicit procedure is as follows. First of all, two labels are selected and the�rst irreducible kernel is constructed (see Table 6)S3(ab) S(ab)tan�1 S2(ab)S1(ab) �The form of � obviously depends on the choice of the vector R(ab) of eq. (43),which now satis�es conditions (51).Then, the irreducible kernel of the three-label realization is constructed byde�ning the two vectors 11 (see eq. 58)~N((ab)c) = 12 �R(ab) + R(c)� ; (70)and ~�((ab)c) = 12 �R(ab) � R(c)� : (71)This leads from Table 7 to Table 8. The angle � of the irreducible kernel is now:�((ab)c) = tan�1 S h~S ^ ~N((ab)c)i3h~S ^ h~S ^ ~N((ab)c)ii3 : (72)The typical form clearly depends on the choice of the chain of labels (see Table 8).Corresponding to each choice of labels, a basic reference frame in the <3 spaceof Wigner vectors is de�ned. The vector ~S is shared by any of the typicalforms corresponding to any possible di�erent chain, while the unit vector Rdepends on any given chain according to eq (43), with � given by eq. (72). The�4�12 � possible reference frames di�er from one another by a spatial rotationof the vectors ~R e ~S ^ ~R around the ~S direction. The rotation angle relatingtwo typical forms is simply given by the di�erence between the two variablesconjugated to S. The geometry is as follows: if ~a and ~b (like Ra and Rb) aretwo vectors with �ai ; aj = �bi ; bj = 0, from the expressions�a = tan�1 S (~S ^ ~a)3[~S ^ (~S ^ ~a)]3�b = tan�1 S (~S ^~b)3[~S ^ (~S ^~b)]3 ; (73)11Again, the remaining degrees of freedom are realized by inessential variables.24

we get the angle �a � �b = tan�1 ~a � (~S ^~b)~b � �~S ^ (~S ^~b)� : (74)Of course, eq. (74) does not exhaust the description of the di�erence betweenany pair of typical forms: the transformation between the inessential variablesis needed too.9 AcknowledgmentsMassimo Pauri would like to express his deep appreciation and thanks to theCenter for Philosophy of Science, for the warm and stimulating intellectualatmosphere experienced there and the generous partial support obtained duringthe preparation of the present work at the University of Pittsburgh.

25

References[1] R. Abraham and J. E. Marsden, Foundations of Mechanics (The Benjamin-Cummings Publishing Company, New York, 1978)[2] A. Barducci , C. Casalbuoni and L. Lusanna, Nuovo Cimento 54 A, 340(1979)[3] A. O. Barut and R. Raczka, Theory of Group Representations and Appli-cation (World Scienti�c, Singapore, 1986)[4] F.Colomo and L.Lusanna, Int. J. Mod. Phys. 7 A n.8, 1705 (1992)[5] F.Colomo, L. Longhi and L.Lusanna, Int. J. Mod. Phys. 5 A n.17,3347(1990)[6] F.Colomo, L. Longhi and L.Lusanna, Mod. Phys. Lett. 5 A, 17 (1990)[7] P.A.M. Dirac, Rev. Mod. Phys. 21, 392 (1949)[8] P.A.M. Dirac Lectures in Quantum Mechanics, Bolfare Graduated Schoolof Science, Monograph Series, Yale University, 1964.[9] B.A. Dubrovin,M. Giordano ,G. Marmo and A. Simoni, Int. J. Mod. Phys.8 n.21, 3747 (1993)[10] L. P. Eisenhart, Continuous Groups of Trasformations (Dover, NewYork,1969)[11] J.M. Evans and Ph.A. Turkey, Int. J. Mod. Phys. 8 n.23, 4055 (1993)[12] M.J. Gotay and J.M. Nester, in Group Theoretical Method in Physics (Au-stin, 1978), Lectures Notes in Physics 94, (Springer, Berlin, 1979).[13] M.J. Gotay, J. M. Nester and G. Hinds, J. Math. Phys. 19, 2388 (1978)[14] J. Gomis, K. Kanimura and J. M. Pons, Progr. Theor. Phys. 71, 1051(1984)[15] A. Hanson , T. Regge and C. Teitelboin, Constrained Hamiltonian Systems,Accademia Nazionale dei Lincei, Roma 22(1976)[16] Dr. E. Kamke, Di�erentialgleichungen, L�osungsmethoden und L�osungen(Akademische Verlagsgesellshaft Geest & Portig K.-G., Leipzig 1950)[17] A. Komar, Phys. Rev. 17 D, 1881 (1978)[18] A. Komar, Phys. Rev. 18 D, 1887 (1978)[19] A. Komar, Phys. Rev. 18 D, 3617 (1979)26

[20] G. Longhi and L. Lusanna, Phys. Rev. 34 D, 3707 (1986)[21] G. Longhi, L. Lusanna and J. M. Pons, J. Math. Phys. 30 n.8, 1893 (1989)[22] L. Lusanna, Il Nuovo Cimento 64, 65 (1981)[23] L. Lusanna, Rivista del Nuovo Cimento 14 n.3, 1 (1991)[24] L.Lusanna, Phys. Rep. 185 n.1, 1 (1990)[25] L.Lusanna, J. Math. Phys. 31 n.9, 2126 (1990)[26] L. Lusanna, Int. J. Mod. Phys. A 8, 4193 (1993)[27] L. Lusanna, Classical Observables of Gauge Theories from the Multitem-poral Approach, Talk given at the Conference Mathematical Aspects of theClassical Field Theories, Seattle, in Contemporary Mathematics 132, 531(1992); Dirac Observables: From Particles To Strings and Fields, Talk gi-ven at the Symposium on Extended Objects Bound States, Karuisawa (O.Hara, Ishida and S. Naka eds., Word Scienti�c, Singapore, 1993); Hamil-tonian Constraints and Dirac Observables, Talk given at the ConferenceGeometry of Constrained Dynamical Systems, Cambridge, Isaac NewtonInstitute (J.M. Charap ed., Cambridge University Press, Cambridge, 1995)[28] L.Lusanna, Int. J. Mod. Phys. A12, 645 (1997)[29] L.Lusanna, Int. J. Mod. Phys. A10, 3531 and 3675 (1995)[30] G. W. Mackey, Unitary Group Representation in Physics, Probability andNumber Theory (The Benjamin-Cummings P.Co., 1978)[31] T.D. Newton and E.P. Wigner, Rev. Mod. Phys. 34, 400 (1949)[32] M. Pauri and G. M. Prosperi, J. Math. Phys. 7, 366 (1966)[33] M. Pauri and G. M. Prosperi, J. Math. Phys. 8, 2256 (1967)[34] M. Pauri and G. M. Prosperi, J. Math. Phys. 9, 1146 (1968)[35] M. Pauri and G. M. Prosperi, J. Math. Phys. 16, 1503 (1975)[36] M. Pauri and G. M. Prosperi, J. Math. Phys. 17, 1468 (1976)[37] M. Pauri, Invariant Localization and Mass-spin Relations in the Hamil-tonian Formulation of Classical Relativistic Dynamics, Parma University1978[38] M. Pauri, Canonical (possibly Lagrangian) Realizations of the Poin-car�e Group with Increasing Mass-Spin Trajectories, in Group TheoreticalMethods in Physics, Lecture Notes in Physics n.135, (Kurt Bertrand Wolfed., Springer-Verlag, 1980) 27

[39] S. Shanmugadhasan, J. Math. Phys. 14 n.6, 677 (1973)[40] J. M.Souriau, Structure des Syst�emes Dynamiques (Dunod Universit�e, Pa-ris, 1970)[41] E.C.G. Sudarshan and N. Mukunda, Classical Dynamics : A Modern Per-spective (Wiley & Sons, New York, 1974)[42] K. Sundermayer, Constrained Dynamics (Springer-Verlag, Berlin, 1982)[43] L. D. Faddeev, Teor. mat. Fizika 1 3 (1969); translation in Theor. Math.Phys. 1 1 (1970)[44] L. D. Faddeev and V. N. Popov, Phys. Lett. B25 29 (1967)[45] H. Feshbach and F. Villars, Rev. Mod. Phys. 30 24 (1958)[46] H. Sazdjian, Ann. Phys. (NY) 136 136 (1981)[47] F. Rohrlich, Phys.rev. D23 1305 (1981)[48] J.Weyssenho� and A.Raabe, Acta Phys. Pol. 9 7 (1947)

28

Table 1: Quasi-SCT transformation for a system of N particles: chart partiallyadapted to the constraint structure.~k � = �pP 2~z T~�a �a~�a �aTable 2: Quasi-SCT for a system of two particles: chart partially adapted tothe constraint structure. ~k � = �pP 2~z T~� �R~� �RTable 3: System of two particles: relative physical variables adapted to theinternal E(3) group.S3 S R2 = 1 ~R � ~S = 0 �tan1� S2S1 � ~� � ~��Table 4: System of two particles: chart adapted to the constraint structure andto the Poincar�e group. k1 k2 k3 � = �pP 2z1 z2 z3 TS3 S � �Rtan�1 S2S1 � ~� � ~�� �R29

Table 5: System of three particles: chart adapted to the constraint structureand to the Poincar�e group (p1 and p2 are invariants of E(3) in the case of arealization with no �xed invariants).k1 k2 k3 �z1 z2 z3 TS3 S �1 �2 �1 �2tan�1 S2S1 � ~�1 � �1 ~�2 � �2 �1 �2p1 p2q1 q2Table 6: System of four particles (the �rst label choice is (ab); p1 and p2 areinvariants of E(3)).~k �~z TS3(ab) S(ab) S3(c) S(c) �(a) �(b) �(c) �a �b �ctan�1 S2(ab)S1(ab) �(ab) tan�1 S2(c)S1(c) �(c) ~�(a) � �(a) ~�(b) � �(b) ~�(c) � �(c) �a �b �cp1 (ab) p2 (ab)q1(ab) q2(ab)Table 7: System of four particles: chart adapted to the constraint structure(�rst label choice ((ab)c) ).~k �~z TS3 S �(a) �(b) �(c) �1 �2 �3tan�1 S2S1 �((ab)c) ~�(a) � �(a) ~�(b) � �(b) ~�(c) � �(c) �1 �2 �3p1 (ab) p2 (ab) p1 ((ab)c) p2 ((ab)c)q1(ab) q2(ab) q1((ab)c) q2((ab)c)30

Table 8: System of four particles: chart adapted to the constraint structure andto the Poincar�e group (�rst label choice ((ab)c) ).~k~zS3 Stan�1 S2S1 �((ab)c)�(a) �(b) �(c)~�(a) � �(a) ~�(b) � �(b) ~�(c) � �(c)~S(ab) � N(ab) N(ab)tan�1 ~S(ab) � �(ab)~S(ab) � (N(ab) ^ �(ab)) (~S(a) � ~S(b)) � (N(ab) ^ �(ab))�(ab)~S � N((ab)c) N((ab)c)tan�1 ~S � �((ab)c)~S � N((ab)c) ^ �((ab)c) (~S(ab) � ~S(c)) � N((ab)c) ^ �((ab)c)�((ab)c)31

List of Figures1 The � and � angles, and the S3 component. Note that, as regardsSections 6 and 7, ~R has to be replaced by ~N . : : : : : : : : : : : 332 The � and angles, the projection � � and the �S3 component. : 33List of Tables1 Quasi-SCT transformation for a system of N particles: chart par-tially adapted to the constraint structure. : : : : : : : : : : : : : 292 Quasi-SCT for a system of two particles: chart partially adaptedto the constraint structure. : : : : : : : : : : : : : : : : : : : : : 293 System of two particles: relative physical variables adapted to theinternal E(3) group. : : : : : : : : : : : : : : : : : : : : : : : : : 294 System of two particles: chart adapted to the constraint structureand to the Poincar�e group. : : : : : : : : : : : : : : : : : : : : : 295 System of three particles: chart adapted to the constraint struc-ture and to the Poincar�e group (p1 and p2 are invariants of E(3)in the case of a realization with no �xed invariants). : : : : : : : 306 System of four particles (the �rst label choice is (ab); p1 and p2are invariants of E(3)). : : : : : : : : : : : : : : : : : : : : : : : : 307 System of four particles: chart adapted to the constraint struc-ture (�rst label choice ((ab)c) ). : : : : : : : : : : : : : : : : : : 308 System of four particles: chart adapted to the constraint struc-ture and to the Poincar�e group (�rst label choice ((ab)c) ). : : : 31

32

S3

S R

S

k

k k

3

R

φ

1 2

α

Figure 1: The � and � angles, and the S3 component. Note that, as regardsSections 6 and 7, ~R has to be replaced by ~N .S

S

N

χ

βξ

W

^

^

N χ^

ψ

χ

3

Figure 2: The � and angles, the projection � � and the �S3 component.33