31 December 1998

Ž .Physics Letters B 445 1998 123–128

Massive spinning particles and the geometry of null curves

Armen Nersessian a,1, Eduardo Ramos b,2

a Joint Institute for Nuclear Research, BogolyuboÕ Laboratory of Theoretical Physics, Dubna, Moscow Region, 141980, Russiab Dept. de Fısica Teorica, C-XI, UniÕersidad Autonoma de Madrid, Ciudad UniÕersitaria de Cantoblanco, 28049 Madrid, Spain´ ´ ´

Received 4 September 1998Editor: L. Alvarez-Gaume

Abstract

We study the simplest geometrical particle model associated with null paths in four-dimensional Minkowski space-time.The action is given by the pseudo-arclength of the particle worldline. We show that the reduced classical phase space of thissystem coincides with that of a massive spinning particle of spin ssa 2rM, where M is the particle mass, and a is thecoupling constant in front of the action. Consistency of the associated quantum theory requires the spin s to be an integer orhalf integer number, thus implying a quantization condition on the physical mass M of the particle. Then, standardquantization techniques show that the corresponding Hilbert spaces are solution spaces of the standard relativistic massivewave equations. Therefore this geometrical particle model provides us with an unified description of Dirac fermionsŽ .ss1r2 and massive higher spin fields. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

The search for a classical particle model thatunder quantization yields the Dirac equation and itshigher spin generalizations has a long history. By farthe most popular approach is to supersymmetrize thestandard relativistic particle model, whose action isgiven by the proper time of the particle’s path. It isnot difficult to show that the system possessingNs2 s extended supersymmetry corresponds, afterquantization, to a massive spinning particle of spins. Nevertheless, the search for geometrical particlemodels of purely bosonic character, which may leadto similar results, is interesting on its own. The

1 E-mail: nerses@thsun1.jinr.ru2 E-mail: ramos@delta.ft.uam.es

reasons for this are quite simple, and they may beexemplified by Polyakov’s results on Fermi-Bose

w xtransmutation in three dimensions. In 1 , Polyakovwas able to show how, in the context of three-dimen-sional Chern–Simons theory, the presence of a tor-sion term in the effective action for a Wilson loopwas responsible for the appearance of Dirac fermionsin an otherwise apparently bosonic theory – thusopening the question of which is the natural counter-part, if any, in the four-dimensional case.

The purpose of this work is to show that there is ageometrical particle model, based purely on the ge-ometry of null curves in four-dimensional Minkowskispacetime, that does the job, i.e., that under quantiza-tion yields the wave equations corresponding to mas-sive spinning particles of arbitrary integer or half-in-teger spin. The action for such a model is given by

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01408-7

( )A. Nersessian, E. RamosrPhysics Letters B 445 1998 123–128124

the pseudo-arclength, the simplest among all thegeometrical invariants associated with null curves.

A few words are due now in order to motivate,from a more ‘intuitive’ point of view, our approachto this problem.

How is that a particle model based on null curves,which should correspond a priori with massless par-ticles, may be of relevance in the massive case? Themain clue that this can be indeed the case comesfrom the Zitterbewegung associated with Dirac’sequation. It is a well-known result that a measure-ment of the instantaneous speed in the relativistictheory of Dirac must lead to a result of "c – thatthis is in no contradiction with Lorentz invariance is

w xexplained, for example, in 6 – hence making plau-sible that light-like paths may play a role in thecorresponding particle system.

The second observation is a direct consequence ofthe one above: due to the null character of the pathany sensible local action should depend on higherorder derivatives. It is clear that a model for spinningparticles should contain extra degrees of freedom inorder to accommodate the ones associated with spin.Although some Lagrangian models have been intro-duced in the literature where the extra degrees of

Ž w xfreedom are added by hand see, e.g., 4 and refer-.ences therein , it would be highly desirable if they

were provided by the geometry itself. It is wellknown by now that higher derivative theories mayprovide the required phase space. As we alreadycommented, that is the case in three dimensions, andmoreover it has also already proven to work in fourdimensions, for massless particles, with a Lagrangiangiven by the first curvature of the particle’s world-

w xline 5 . Our choice has, then, been the obvious one,this is to consider the simplest geometrical invariantassociated with null, or light-like, paths. Therefore,and with no further delay we will begin to introducethe necessary geometrical background to work outthe model at hand.

2. Frenet equations for null curves in M4

Let us start by constructing the Frenet equationsassociated with null curves in four-dimensional

Minkowski space-time. Our conventions for the sig-Ž .nature of the metric are q,y,y,y , i.e., time-like

vectors have a positive norm.If we denote by x the embedding coordinate of

the curve, the fact that the curve is null implies thatx 2 s0. Let us denote by e sx the tangent vector to˙ ˙qthe curve. From e2 s0 it follows that there is aqspace-like vector e such that1

e ss e , 1Ž .˙q 1

with e e s0, and e2 sy1. Although, a priori, aq 1 1

term proportional to e may appear in the right handqside of the above equation it is always possible toreabsorb it by a redefinition of e .1

It will now show convenient to chose aparametrization for which ss1. This is equivalentto demanding that xxsy1, and this can always be¨¨achieved by a change of parametrization unless xxs¨¨0, this last case being trivial will be excluded fromour considerations. This parametrization correspondsto choosing as our time parameter the pseudo-arclength, which is defined as

t 1r4X X Xs t s yx t x t dt . 2Ž . Ž . Ž . Ž .Ž .¨ ¨H

t0

We now pass to obtain Frenet equations in thisparametrization. From the definition of e it follows1

that e e s0 and e e s1, hence˙ ˙1 1 1 q

e sAe q f qCg , 3Ž .˙1 q y 2

where the two new vectors g and f are chosen to2 yobey that g 2 sy1, f 2 s0, g e sg e sg f s2 y 2 1 2 q 2 ye f s0, together with e f s1. From the fact that1 y q yŽ .e ,f ,e ,g span the tangent space at any point onq y 1 2

the curve the above equation trivially follows.But notice that one can choose a basis

Ž .e ,e ,e ,e withq y 1 2

1 2e s f qCe q C e , 4Ž .y y 2 q2

e sg qCe 5Ž .2 2 q

that, while preserving the same orthogonality rela-tionships, simplifies the above equation to yield

e sk e qe , 6Ž .˙1 1 q y

with k sAyC 2r2. The remaining Frenet equa-1Ž .tions associated with the Frenet frame e ,e ,e ,eq y 1 2

( )A. Nersessian, E. RamosrPhysics Letters B 445 1998 123–128 125

follow from the orthogonality relations, and the wholeset reads

xse , 7Ž .˙ q

e se , 8Ž .˙q 1

e sk e qe , 9Ž .˙1 1 q y

e sk e qk e , 10Ž .˙y 1 1 2 2

e sk e . 11Ž .˙ 2 2 q

Notice that in this case there are only two indepen-dent curvature functions k and k . This is intu-1 2

itively obvious due to the extra constraint on nullcurves coming from xxs0.˙˙

3. The classical model

We will consider the simplest geometrical actionassociated with null curves, this is, its pseudo-arclength:

Ss2a ds . 12Ž .HIt is convenient, although not strictly necessary, towrite the action in first order form. One then canwrite

Ss dt 2a ye e qp xye ye e( (˙ ˙ ˙ ˙ ˙H ž /q q q q qžyle2 . 13Ž .q /

Notice that the equation of motion for the Lagrangemultipliers p and l implies that

1r4ye e s yxx , 14( Ž . Ž .˙ ˙ ¨¨q q

thus proving the equivalence of both actions.It is now straightforward to develop the Hamilto-

nian formalism following Dirac’s description for sin-gular Lagrangians. We will skip the details, whichare completely standard. After a little work onearrives to a symplectic form

Vsdpndxqdp nde , 15Ž .q q

endowed with the following set of primary con-straints

22f sp q pe y2a , 16Ž . Ž .1 q q

f se2 , 17Ž .2 q

and secondary ones

f sp e , 18Ž .3 q q

f spe ya , 19Ž .4 q

f spp . 20Ž .5 q

The Hamiltonian is of the form

22 2 2HsÕ p qp e q pe y2a , 21Ž . Ž .Ž .q q q

where Õ should be regarded as an arbitrary functionthat is only fixed by choosing a particularparametrization. If one chooses to parametrize thepath by pseudo-arclength one gets that Õsy1r2a .

There is only one first class constraint, the genera-tor of reparametrizations. The dimension of the re-duced phase space is therefore 10.

The equations of motion, in the pseudo-arclengthparametrization, are given by

xse , 22Ž .˙ q

1e sy p , 23Ž .˙q q

a

12p s p e yp, 24Ž .˙q q

a

ps0. 25Ž .˙

Consistency of these equations of motion withFrenet equations imply that p sya e , psa e qq 1 yp2 e r2a , together withq

12k sy p . 26Ž .1 22a

The solution of these equations of motion are partic-w xular examples of null helices 3 .

The key observation, in order to arrive to a man-ageable expressions for the symplectic form on theconstrained surface, is that it is possible to define a

( )A. Nersessian, E. RamosrPhysics Letters B 445 1998 123–128126

standard free coordinate X out from the canonicalvariables of our model. Notice that

aXsxy p 27Ž .q2p

has the property that its time derivative is given bya a

Xsxy p s p. 28Ž .˙ ˙q2 2p p

¨And from this it trivially follows that Xs0.It is then natural to introduce the new coordinate

aE se y p 29Ž .q q 2p

so that the symplectic form V takes the simple form

Vsdpnd Xqdp ndE 30Ž .q q

on the constrained surface.The constraints may be equally expressed in these

variables, and they read

p2 qa 2 s0, 31Ž .q

p E s0, 32Ž .q q

a 22E q s0, 33Ž .q 2p

pp s0, 34Ž .q

pE s0. 35Ž .q

This constraint system suggests the introductionof the following complex coordinates

2(zsp q i p E . 36Ž .q q

In terms of z the constraints simply read

2 2z s0, zzq2a s0, and pzs0. 37Ž .Let us recall that the irreducible representations of

the Poincare algebra are labeled by the values of the´two Casimirs p2 and the square of the Pauli-Luban-sky vector

1m mnrsS s e p M , 38Ž .n rs2

with M the generator of Lorentz transformations.rs

In our particular case the explicit expression for Sm

reads

S se pn p r es , 39Ž .m mnrs q q

and one obtains that S2 sya 4, while there is norestriction on the possible values of p2. This implies

that our phase space is not elementary, i.e., thePoincare group does not act in a transitive way. In´order to obtain irreducible representations of thePoincare group under quantization – physical states´should always be decomposable into irreducible rep-resentations of the Poincare group – we will study´the elementary phase spaces defined through theextra constraint

p2 sM 2 , 40Ž .with M a free parameter with dimensions of mass. Apriori M 2 could be positive, negative or zero, wewill only consider the first case, because the othertwo correspond to unphysical irreducible representa-tions of the Poincare group, i.e., tachyonic and con-´tinuous spin representations, respectively.

Because of the null character of z it will showconvenient to introduce a spinor parametrization ofthe reduced phase space. Given an arbitrary complexfour-vector y it can be rewritten in spinor coordi-nates as follows:

1 0 3 1 2y qy y q iy˙A Ay s 41Ž .Ž . 1 2 0 3' ž /2 y y iy y yy˙ 1A A m nŽ .so that det y s g y y .mn2

Because of the two-to-one local isomorphism be-Ž .tween SL 2,C and the identity component of the

Lorentz group, one such Lorentz transformation on yis equivalently represented by the action of an

Ž .SL 2,C matrix acting on the undotted indices and itscomplex conjugate matrix on the dotted ones. Rais-ing and lowering of indices is mimicked in spinorlanguage by contraction with the invariant antisym-

Ž A B . Ž . 01metric tensors e s e , with e sq1, andA B

analogous expressions for the dotted indices.The null character of z now implies that˙ ˙A A A A'z s 2 aj h , 42Ž .

and from the second of our constraints it must followthat j Ah s1, or equivalently that j and h form anA

spinor basis. Notice, though, that this does not com-pletely fix the spinors j and h because one still hasthe freedom to rescale both as follows

1A A A Aj ™a j and h ™ h , 43Ž .

a

Ž .with a an arbitrary nonzero real number. Thisresidual freedom will be fixed as follows. Let us

( )A. Nersessian, E. RamosrPhysics Letters B 445 1998 123–128 127

consider the remaining constraint pzs0. In spinorcoordinates it reads

˙A Ap j h s0 44Ž .˙A A

or equivalently

˙ ˙ ˙A A 2 A A(p j sL p h sLMh , 45Ž .A

Ž .with L an arbitrary nonzero real number. One thenmay fix completely the freedom to rescale the spinors

'by setting Ls"1r 2 , and then one may finallywrite the only remaining constraint in the followingform

M˙A Ap j j s" . 46Ž .˙A A '2

A direct computation now shows that the symplecticform in spinor coordinates reads

'2 s ˙A AVsdpnd X" i p dj ndj˙ž A AM

A Aqj dj nd p qc.c., 47Ž .˙/A A

where c.c. stands for complex conjugate of the previ-ous term, and s is a dimensionless parameter definedas

a 2

ss . 48Ž .M

4. The quantum theory

Ž w x.One may now check see 2 that the reducedphase space can be identified with the coadjoint orbitof the Poincare group associated with a representa-´tion of mass M and spin s. Quantization is then astandard exercise whose solution can be found, forexample, in the excellent book on geometric quanti-

w xzation by Woodhouse 2 . Nevertheless, because ofcompleteness, and the desire to simplify things alittle to the less mathematically oriented reader, wewill now sketch how the quantization procedure maybe carried away in spinor coordinates. In particular,we will show how Dirac equation may be obtainedin the ss1r2 case.

One should start by choosing a polarization, or insimpler words one should demand the wave functionto depend only on half of the canonical coordinates.Our choice will be that the wave function willdepend on p and j .

In order to quantize the system we should imple-ment the remaining first class constraints as condi-tions on the wave function. The first constraint p2 sM 2 implies that the wave function has support onthe mass hyperboloid. In order to implement the

Ž .second one 46 notice that, roughly speaking,Ap j and j are conjugate variables in the induced˙A A

symplectic form, therefore under quantization

EAp j ™ 49Ž .˙A A AEj

up to some numerical factors. A careful computationŽ .shows that 46 implies at the quantum level that

Ec p,jŽ .Aj s2 sc p,j , 50Ž . Ž .AEj

i.e., the wave function is a homogeneous function ofj of degree 2 s. Then single-valuedness of our wave

Ž .function under j™exp 2p i j requires s to be aninteger or half-integer number. Therefore one hasthat

c p,j sc p j A1j A2 PPP j A2 s , 51Ž . Ž . Ž .A A PPP A1 2 2 s

Ž .where c p has support on the mass hyper-A A PPP A1 2 2 s

boloid. It is now straightforward to check that itsFourier transform

3r21w x s c pŽ . Ž . A A PPP AHA A PPP A 1 2 2 s1 2 2 s ž / s2p HM

=eyi pxdt , 52Ž .

with H s the positive energy branch of the massM

hyperboloid, and dt its invariant measure, yieldpositive frequency solutions of the massive waveequation

Iw x qM 2w x s0. 53Ž . Ž . Ž .A A PPP A A A PPP A1 2 2 s 1 2 2 s

Notice that, for example, for ss1r2 this equationhas the same physical content that the Dirac equa-tion. This is so because the dotted component of the

( )A. Nersessian, E. RamosrPhysics Letters B 445 1998 123–128128

four component Dirac spinor, which will be denotedby x , may be defined trough the relationA

MA= w s x . 54Ž .˙ ˙A A A'2

Ž .This together with 53 implies that

MA= x s w , 55Ž .˙A A A'2

thus being equivalent to the standard Dirac equationŽ .for w,x .

Although this short discussion about the quantiza-tion of the system cannot make justice to this exten-sive topic, the interested reader may fill the gapswith the help of the current literature on the subject.

Acknowledgements

E.R. would like to thank J. Roca and J.M.Figueroa-O’Farrill for many useful conversations on

the subject. The work of A.N. has been partiallysupported by grants INTAS-RFBR No.95-0829, IN-TAS-96-538 and INTAS-93-127-ext.

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Publications, Oxford, 1991.w x Ž .3 W.B. Bonnor, Tensor, N.S. 20 1969 229.w x4 M.V. Atre, A.P. Balachandran, T.R. Govidarajan, Int. J. Mod.

Ž .Phys. A 2 1987 453.w x Ž .5 M.S. Plyushchay, Mod. Phys. Lett. A 4 1989 837. E. Ramos,

Ž .J. Roca, Nucl. Phys. B 477 1996 606.w x6 P.A.M. Dirac, The principles of quantum mechanics, Oxford,

Clarendon, 4th ed., 1958.