Classical Relativistic Spin Particles

Relativistic spin particles are approached from the stand point of Hamil-tonian actions of Poincaré group on itself. The several possible solutions are classified and realizations are given in terms of Dirac's constraints formalism.


Introduction
In 1] the dynamics of classical spin particles was approached on the basis of simple kinematical postulates that were some extension of rigid body kinematics.The role associated in other approaches like 3] with the so-called \inner space" of the classical particle was embodied in three mutually orthogonal spacelike axes belonging to the comoving space.So the phase space was a submanifold of the cotangent space of the Poincar e group that was de ned by three nonholonomous constraints, which basically require the momentum of the particle to be proportional to the time column of a Lorentz matrix.
Intuitive as it may be, this formulation is too restrictive.Indeed, as a consequence of the constraints, only positive mass particles can t into it and both massless particles and tachyons are rejected from the very beginning.
Furthermore, in 1] there was rather wide class of dynamics permitted by the requirement that a spin particle is described by a Poincar e invariant Hamiltonian system.It was only reduced by assuming the condition of spherical symmetry (which was introduced by hand) in such a way obtaining the expected result, namely, that the Hamiltonian should only depend on the two Casimir functions 5] of the Poincar e group.
In the present paper we shall change a little bit our perspective and not use constraints.The aim is to encompass also massless particles and tachyons.The underlying ideas, to be shaped in mathematical form, are the following: (A1) The con guration space is Poincar e group, G.Each point consisting of to c J.Llosa components (x; ), the rst one gives the position of the particle as seen by some given inertial frame, S, and the second (i.e., the Lorentz matrix ) gives the components of a spacetime orthonormal tetrad of vectors (the \body" axes) referred to the basis of S. Within this framework coordinate transformations from one inertial frame S to another, S 0 , are represented by the left action of Poincar e group (the symmetry group of relativity) on itself (the con guration space).On the phase space T G the corresponding lift must be considered (see 1] for details).According to the principle of relativity, physical laws must be the same in all inertial frames, (A2) the dynamics of a spin particle in the phase space T G should be invariant under the above left action.
Let us now consider the right action of the element g = (a; L) 2 G of Poincar e group on any point (x; ) 2 G of con guration space.
(x; ) ?! (x; ) (a; L) = (x + a; L) It results in a shift of the point x in Minkowski spacetime to x + a and in a change of the \body" axes, from to L. The right space rotation does not change 0 , the time \body" axis.Were the phase space de ned by some constraints involving 0 , as it happened in 1], it would be invariant by right space rotations.Furthermore, if spherical symmetry is assumed, there is no preferred triad of body axes (i.e., all of them are dynamically equivalent).Hence, (A3) the dynamical system describing a classical spin particle should be invariant under the right action of space rotations.
The right action of a boost does change the time axis of the \body" frame, 0 , and takes the point (x; ) o the phase space (in case the latter is de ned by some constraints involving 0 ).(A3') The right action of boosts can be used to extend, in a right invariant way, the dynamical system from the phase manifold to the whole T G.
It is necessary to stress the di erent roles assigned here to left and right actions of G. Indeed, by left action of (a; L) we change the reference frame, whereas by right action we are changing the \axes attached to the body" when L is a rotation.
In consequently with the above kinematical arguments, the dynamics of a classical spin particle will be de ned by a Hamiltonian system on T G, with its canonical symplectic structure 2], that is invariant under both actions (namely, left and right) of Poincar e group G on T G.
The present contribution could be taken as an exercise to chapter 4 in 2].The notation and most notions used here are largely de ned and fully developed there.Sections 2 and 3 are devoted to the general study of Hamiltonian systems on the cotangent space of a Lie group which are invariant under both, the left action and the right action of the group on itself.Then the right action invariance is used to reduce the phase space.In section 4 the particular case of Poincar e group, which is relevant to relativistic spin particles, is considered and all possible dynamics are classi ed.The nal results are given in the form of Dirac constrained Hamiltonian systems ( 4]).

Hamiltonian systems on the cotangent space of a connected
Lie group G is a connected Lie group, G its Lie algebra and G its dual vector space.

Symplectic structure and group actions
T G is endowed with the canonical symplectic form 2 2 (T G), that is obtained as the di erential of the Liouville form 2 1 (T G): = ?d: ( The Liouville form is related to the projection map : T G ! G, by h where 2 T G, X 2 T (T G) and T is the corresponding tangent map.
We now have two actions of G on T G, namely, left translation, L g , and right translation, R g , which are de ned as the lifts to T G of the left translation and the right translation, respectively, on G: ) where T L g and T R g mean the respective cotangent maps.Since both are lifts of actions of G on the con guration space G, they both preserve the Liouville form and, in addition, are symplectic.Hence, for all 2 G, the corresponding in nitesimal generator XA of the action A (A=L,R) is Hamiltonian relatively to , with an associated Hamiltonian function J A ( ) 2 0 (T G) such that dJ A ( ) = i( XA ) = dh ; XA i ?$( XA ) (3) where $ means Lie derivative.
The correspondence !J A ( ) is called the momentum mapping for the action A (see 2], de nition 4.2.1).Since is preserved by both actions, L and R , the last term in (3) vanishes.Hence, a momentum mapping for the action A is:

Body coordinates
The cotangent space T G of a Lie group is a trivial bre bundle.Indeed, it admits a global chart: ) that is known as body coordinates (see 2], section 4.4) evoking an obvious similitude with rigid body kinematics.
Another global chart could also be used, corresponding to the so called space coordinates, that is de ned by : T G ?! G G g ?! (g; T R g ( g )) ) However, owing to reasons that will later on become clear, throughout this paper we shall only use body coordinates.
Let us now write the actions A , A=L,R, in terms of body coordinates.To this end we de ne ~ A by commutatively closing the following diagram: In order to have an explicit expression for h B ; X (h; ) i, we split X (h; ) 2 T (h; ) (G G ) into its two components X (h; ) = (v h ; ), with v h 2 T h G and 2 T G ' G .

The reduced phase space
As we have already pointed out in section 1, left invariance and right invariance for the systems under consideration have neatly di erent meanings.Whereas left invariance has a physical interpretation in terms of relativistic invariance, right invariance has been introduced to enlarge the phase space and so to avoid the need of constraints.
We shall now consider a R invariant Hamiltonian system on G G and eliminate this `unphysical' symmetry by reducing the phase space.The techniques are those developed in 2] (section 4.3) and the method basically consists in the use of the integrals of motion associated to the latter symmetry (that are in involution with respect to Poisson brackets) to eliminate some degrees of freedom.Now, theorem 4.3.1 in 2] can be applied.Indeed, we have a symplectic manifold (G G ; B ), on which the Lie group G acts symplectically | namely, ~ R | and an ad -equivariant momentum mapping JR .Furthermore, as it can be easily checked: 1) any 2 G is a regular value of JR (i.e., T (h: ) JR is surjective), 2) the momentum mapping JR de ned by ( 14) is ad -equivariant under R , that is, 8g 2 G, JR ~ R g = ad (g ?1 ) J R (see de nition 4.2.6 in 2]).
3) JR?1 ( ) = f(h; ); h 2 Gg is di eomorphic with G, and the isotropy group of , namely G = fg 2 G; ad (g) = g, acts properly and freely on JR?1 ( ).Hence, P G=G has a unique symplectic form, ( ) , such that: where and i are, respectively, the canonical projection and the inclusion: Therefore, given any 2 G we have a reduced phase space, i.e. a symplectic manifold (P ; ( ) ).

A realization of P
This reduced phase space P can actually be realized as a submanifold of G (this is a consequence of the Kirillov-Kostant-Souriau theorem, see 2], example 4.3.4(v)).Indeed, let us consider the coadjoint representation of G, and de ne: : G G ?! G (g; ) ?! ad (g ?1 ) (g) g ( ) For any given 2 G , we shall have an orbit: : G ?! G and, for any given 2 (G) G , ?1 ( ) is a coset in P = G=G .Therefore, P is di eomorphic with ?
(G), the orbit of 2 G by the coadjoint representation of G.The di eomorphism is ^ : P ?! ?hG ?! ad (h ?1 ) Thus ?G is a realization of P .
The symplectic form on ? is obtained from ( ) by inverse pullback: As a consequence of (15) and the fact that = ^ , we have that ^ ( ) is the only symplectic form on ?such that: where = ad (h ?1 ) .

The left action of G on ?
Since the following diagram G G i ?G ?! ?j j j with g and de ned in ( 16), is commutative, we have that the left action ~ L of G on G G is \projected" onto the coadjoint action of G on ?: g : ad (h ?1 ) ?! ad (g ?1 )ad (h ?1 ) = ad ((gh) ?1 ) By its very construction, acts transitively on ? .Hence, at any 2 ?, the set of its in nitesimal generators spans the whole tangent space T (? ), that is: Since TR h ( ) 2 T h G is an in nitesimal generator for the left action L, and the diagram (20) is commutative, we have that: V for any h 2 G and 2 G such that = ad (h ?1 ) .
In particular, taking h = e and = , we obtain: V ( ) = T ( ) = ?Ad ( ) 2 T (? ) where Ad is the coadjoint representation of G on G .

The implicit equations for ?
? is characterized by some functions f on G de ning constraints that look like f( ) = constant on ?: (29) Now, since the in nitesimal generators (23) span the whole tangent space T (? ), the constraints (29) must satisfy: hdf; V ( )i = 0 ; 2 ?; 2 G (30) which implies that hAd ( ) ; (df) i = h ; ; (df) ]i = 0 : (31) Writing the latter condition in terms of one basis of G, (i.e., in a given parametrization of G) it reads: i C i jk @f @ j = 0 ; where C i jk are the structure constants of G in that parametrization.The solutions of equation (32) are related to the Casimir invariants of the group 6], 5].

The Poincar é group
To apply the above results to Poincar e group, it will be helpful to begin with a previous recall of some general results and notations.h = (x ; ), g = (y ; L ) denote any couple of elements in G, the Poincar e group.The group law is hg = (x + y ; L ) (33) A general element in the Lie algebra G is denoted by = (b ; V ) ; V + V = 0 ; and a general element in the dual space G is written as = (a ; W ) ; W + W = 0 : Indices are raised and lowered with Minkowski metrics = (?+ ++).
The commutation relations in G are those corresponding to the semidirect sum (b ; V ); (c ; T )] = (V c ? T b ; V T ?T V ) (34) and the dual product is: Other useful results are listed below: 1] 1.The coadjoint action: = (a ; W ) ?! ad (g ?1 ) = (a 0 ; W 0 ) with a 0 = a and W 0 = y a 0 ?y a 0 + L L W : Now, the point is to nd the simplest representative (a ; W ) for a given couple (c 1 ; c 2 ).It is important to realize that c 1 and c 2 are the Minkowski squares of two mutually orthogonal 4-vectors, namely, a and a W . Hence, if one of them is either timelike or lightlike, the other is necessarily spacelike.That is, c 1 < 0 ) c 2 > 0 and c 2 < 0 ) c 1 > 0 : To nd a simple representative (a 0 ; W 0 ) for ?(a;W) , we shall use that the skewsymmetric matrix W can always be written as: where the formulae relating E , b and to W and a depend on the class of the latter relatively to the Minkowski metric.
If a a 6 = 0, it follows obviously that: E = 1 a a W a ; = 1 a a a W ; and b = a : (43) In the case a a = 0, a lightlike vector b can be chosen such that b a = 1 and the vectors E and are given by: E = W b ; = a W : In this second case the decomposition(42) is not unique.It immediately follows from (42) and (36) that: ? (a;W) = ?(a;W 0 ) with W 0 = 1 2 b : Hence, the 4-vectors a and determine ?(a;W) .According to the discussion above, these two 4-vectors are orthogonal to each other and are obviously connected with the two Casimir functions of Poincar e group.Hence, only the cases listed below are possible: The di erent cases are analysed with detail in the appendices.As a summary, the several reduced phase spaces ?(a;W) are realized as the submanifolds of T (R 8 ) de ned by the constraints: In terms of these variables, the linear and angular momenta, (38) and (39), respectively, are p and J = x p ? x p + 1 2 (r l ?r l ) : (54) Now writing = 1 2 l in equations ( 54) and (52), the expressions (49), the elementary Poisson brackets (47) and the constraints (46) follow immediately.

3 . 4 . 1 .
The momentum mapping for the left action of G: JL (h; ) = ad (h ?1 ) = (p ; J ) The reduced phase space: explicit realizations Poincar e group has two Casimir functions, namely, the square of mass and the square of Pauli-Lubanski 4-vector.According to subsection 3.3, given any couple of possible values (c 1 ; c 2 ), we shall have a submanifold ?(c 1 ;c 2 ) and each connected component of it will be a realization of a ?, 2 G .The implicit equations are: 0, c 2 > 0 c 1 > 0, c 2 > 0 c 1 > 0, c 2 < 0 c 1 > 0, c 2 = 0 c 1 = 0, c 2 = 0 c 1 = 0, c 2 > 0 cases I, II, III, and IV, K = ?1=4 in the case V and K = 0 in the case VI, and L = 1 in the cases I, II, III, and V, L = c 1 in the case IV, and L = c 2 in the case VI.