Spin and exotic Galilean symmetry

Physics Letters B 547 (2002) 306–312

www.elsevier.com/locate/npe

Spin and exotic Galilean symmetry

C. Duvala, P.A. Horváthyb

a Centre de Physique Théorique, CNRS Luminy, Case 907, F-13288 Marseille Cedex 9, Franceb Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, Parc de Grandmont, F-37200 Tours, France

Received 25 September 2002; accepted 4 October 2002

Editor: L. Alvarez-Gaumé

Abstract

A slightly modified and regularized version of the non-relativistic limit of the relativistic anyon model considered by Jackiwand Nair yields particles associated with the twofold central extension of the Galilei group, with independent spin and exoticstructure. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Strange things can indeed happen in the plane:for example, a particle (called an anyon) can havefractional spin and intermediate statistics [1]. Anotherone is that the planar Galilei group admits a non-trivial two-parameter central extension [2] leading toan “exotic” model (equivalent to non-commutativemechanics) [3,4].

Soon after its introduction, Jackiw and Nair [5] red-erived our “exotic” model in [3] by taking the non-relativistic limit of their relativistic spinning anyon[6]. This may suggest that spin and exotic structureare related: one can trade one for the other, but onecannot have both. Here, we argue that, at least non-relativistically, spin and “exotic” structure can coex-ist independently. We show this, first, by reviewingthe most general group-theoretical construction asso-ciated with the twofold extended Galilei group, next,

E-mail address: [email protected] (P.A. Horváthy).

by slightly modifying the contraction considered byJackiw and Nair [5]. Our abstract construction is illus-trated by the acceleration-dependent model of [7] andby Moyal field theory [8,9].

2. Exotic models

The free “exotic” particle model of [3] consistsof a five-dimensional “evolution space”T ∗R2 × Rdescribed by position,x, momentum,p, and time,t(see [10]), which is endowed with the presymplectictwo-form

ω= −dpi ∧ dxi − κ

2m2εij dpi ∧ dpj + dh∧ dt,

(2.1)with h= p 2

2m,

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)02783-1

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C. Duval, P.A. Horváthy / Physics Letters B 547 (2002) 306–312 307

wherei, j = 1,2 andm,κ are given constants inter-preted as mass and “exotic” parameter.1 The equationsof motion are then given by the null foliation ofω.

The manifest Galilean invariance of the two-form(2.1) provides us, via the (pre)symplectic version ofNœther’s theorem [10], with conserved quantities,most of which take the standard form. The angularmomentum,j , and the Galilean boost,g, contain,however, new terms, namely,

j = q × p+ κ

2m2 p 2,

(2.2)gi =mxi − pit + κmεijpj .

Those terms change the Poisson brackets of theGalilean boosts, which now satisfygi, gj = −κεijrather than commute, as usual.

The system isbosonic: the quantum angular mo-mentum operator is, in the momentum representa-tion, = −iεjkpj ∂pk [2,3,11]. Owing to the inherentambiguity of planar angular momentum, an arbitraryconstant,s0, representing anyonic spin, can be freelyadded to the angular momentum, though. Let us there-fore discuss all related classical models.

All “elementary” systems associated with thedoubly-centrally extended Galilei group were deter-mined by Grigore [11] who, following Souriau [10],identifies them with coadjoint orbits of the group, en-dowed with their canonical symplectic structure. (Herethe word “elementary” means that the action of theGalilei group should be transitive; at the quantum levelthis means that the representation be irreducible.)

These orbits are 4-dimensional, and depend onfour parameters denoted bys0, h0, m and κ . Theirsymplectic structure are pulled-back, on “evolutionspace”, precisely as the closed two-form (2.1). Thislatter only depends onm and κ ; the two otherparameters only show up in the way the Galilei groupacts on the orbit, i.e., in the associated conservedquantities. These latter are found to be the linearmomentum,p, the boost,g, in (2.2), together with theangular momentum and energy, viz.,

j = x × p+ κ

2m2 p 2 + s0,

1 As a rule, in the non-relativistic theory, all spatial indicesi, j, kare lower indices. Ordinary convention is, however, adopted for theposition of the relativistic indicesλ,µ,ν.

(2.3)h= p 2

2m+ h0,

supplemented with the Casimir invariantsm and κ .The parameters0 is, hence, interpreted asanyonicspin, andh0 is internal energy. Let us insist thats0and the exotic parameterκ are, a priori, independentquantities. Plainly, the model of [3] realizes (2.3) withs0 = h0 = 0.

Nonvanishing spin and internal energy also arisefor the acceleration-dependent system considered byLukierski et al. [7]. Here, phase space is 6-dimensionalwith coordinates X, p, Q, and endowed with thefollowing symplectic structure and Hamiltonian

Ω = −dpi ∧ dXi − θ2εij dpi ∧ dpj

− 1

2θεij dQi ∧ dQj ,

(2.4)h= p 2

2m− Q2

2mθ2 ,

respectively. This system is also invariant with respectto the planar Galilei group; the associated conservedquantities, namely, the linear momentum,p, the boost,g, the energyh in (2.4), augmented with the angularmomentum,

(2.5)j = X× p+ θ2

p 2 + 1

2θQ2,

provide us with a more general moment map (2.3) withinternal energyh0 = − Q2/(2mθ2) and anyonic spins = Q2/(2θ). Once again, the exotic parameterκ =−m2θ is non-trivial. Let us mention that the spin canbe made totally independent from the exotic parameterby adding a suitable term, whereas the constraints =−mθh0 is also relaxed. See the third reference in [7].

It is worth pointing out that the appearance ofthese two models is consistent with Souriau’s general“décomposition barycentrique” (Theorem (13.15) in[10]). Let us indeed consider the planar Galilei groupG. The translations and the boosts span an invariantabelian subgroupG of G (with Lie algebrag ⊂ g),so that the quotient group,G0 = G/G, is the directproduct of rotations and time translations.

Let us consider a classical system represented bya manifoldM endowed with a symplectic two-form,Ω , upon which the (planar) Galilei groupG actsby symmetries, i.e., in a Hamiltonian fashion. Thenthe direct productG × G0 is also a symmetry [10].

308 C. Duval, P.A. Horváthy / Physics Letters B 547 (2002) 306–312

Now, if dim M = 4, and the symplectic action istransitive, thenM can be identified with an affine-coadjoint orbit endowed with its canonical symplecticstructure. In fact,M is the dual of the invariantabelian Lie subalgebrag, viz M ∼= g∗. In our case,we get precisely the free exotic particle model (2.1),considered in [3]. If, however, dimM 6, then thesystem decomposes into the direct product of our“free exotic particle” phase space and a symplecticmanifold,M0, which describes the internal motions(this latter is distinguished by the vanishing of the“standard” conserved quantities), namely

(2.6)M ∼= g∗ ×M0.

The extended groupG×G0 respects this decomposi-tion: the rotations and time translations act indepen-dently on the internal space, contributing (anyonic)spin and internal energy.

This is exactly what happens for the extendedmodel of [7], whose phase space is decomposed intothat of our “free exotic particle” times the symplecticplane. (An additional constraint links the rotationswith the time translations, as highlighted by the linkedextra terms in (2.4) and (2.5).)

A field theoretical model with anyonic spin andnon-trivial exotic structure is provided by the Moyal–Schrödinger field theory [8] given by the non-localLagrange density

(2.7)LNC = i2

(ψ # ∂tψ − ∂t ψ # ψ

) − 1

2m∇ψ # ∇ψ

with the Moyal star-product associated with the defor-mation parameterθ . As shown recently [9], the theoryis Galilei-invariant, and the boosts

(2.8)Gi =m∫xi(ψ # ψ)d

2x − tPi − 1

2mθεijPj ,

(where P = ∫ d2x, with = [ψ( ∇ψ)−( ∇ψ)ψ]/(2i)is the conserved momentum) satisfy the “exotic” com-mutation relationG1,G2 = −κ ∫ |ψ|2 d2x with κ =−m2θ . Note that the angular momentum

(2.9)J =∫ [

x × − θ2| ∇ψ|2 + s0|ψ|2

]d2x

is anyonic, cf. (2.3).

Yet another illustration is provided by the first-order non-local model given by the Lagrangian

(2.10)

ψ† #

(1

2(1+ σ3)∂t − σ · ∇ − im(1− σ3)

)ψ

,

whereψ =(Φχ

)is a two-component Pauli spinor. De-

spite the presence of the Moyal product, the associatedEuler–Lagrange equation is the “non-relativistic Diracequation” of Lévy–Leblond [12],

(∂1 + i∂2)Φ + 2imχ = 0,

(2.11)∂tΦ − (∂1 − i∂2)χ = 0.

These are Galilei invariant when the boosts are imple-mented as [12]

ψ(x, t)=(

1 0

−12(b1 + ib2) 0

)

(2.12)

× expim

[b · x − t b2/2]ψ(x − bt, t).

Using the same technique as in [9], we find that theMoyal star results in a new term in the associatedconserved quantity, viz.

(2.13)Gi =m∫xi(Φ #Φ)d2x − tPi − 1

2mθεijPj ,

where P = ∫ [ Φ( ∇Φ) − ( ∇ Φ)Φ]/(2i)d2x, cf. (2.8).TheGi ’s satisfy the “exotic” commutation relation.

3. Non-relativistic limit by group contraction

Let us now turn to the non-relativistic limit ofthe relativistic anyon model [6] considered by Jackiwand Nair in [5]. Expanding to first order in 1/c therestriction top0 mc2 + p 2/(2m), the relativisticsymplectic structure on 6-dimensional phase space

(3.1)ΩJN = −dpµ ∧ dxµ + Sc2

2εµνρ

pµ dpν ∧ dpρ(p2)3/2

(where the Greek indices range from 0 to 2) yieldsindeed our presymplectic two-form (2.1), when their“spin”, S, is identified with our coefficientκ .

The relativistic model admits, furthermore, thesymmetry with generators

(3.2)Jµ = εµνρxνpρ + Sc2 pµ√p2,


which satisfy the o(1,2) commutation relations rela-tions, Jµ,J ν = εµνρJρ . Then the quantitiesgi =εij Jj /c, satisfy, to leading order in 1/c, the exoticcommutation relation above, allowing Jackiw and Nairto identify theirgi with the Galilean boosts. In partic-ular,S becomes the exotic parameterκ . Their angularmomentum,

(3.3)J0 εij xipj + Sc2 + S

m2p 2,

does not admit, however, a well-defined limit asc→+∞, unless the divergent constantSc2 is removed.Only at this price are the exotic quantities, namely, theangular momentum and the boosts, (2.2), recovered.

Let us now present a mathematical constructionthat allows us to absorb the divergent terms into someextra coordinates. Extending the approach of Jackiwand Nair [5], our clue is to work with differential formsand group actions, rather than with the associatedconserved quantities.

To get an insight, let us first discuss the model ofa massive, spinless, relativistic particle in the plane.Following Souriau [10], it corresponds to a certaincoadjoint orbit of the Poincaré group (we still denoteby G), symplectomorphic toT ∗R2. The pull-back ofits canonical symplectic structure,ω, to G is indeeddα, whereα = pµ dxµ is a one-form onG. Usingphysical coordinates in a Lorentz frame, we obtain

(3.4)α = − p · d x +mc2√

1+ p 2

m2c2dt,

which is the “Cartan-form” [10] of the free relativisticLagrangian,L0 =mc2√1− v 2/c2.

Let us emphasize that the one-formα (as well as theLagrangianL0) diverges in the Galilean limitc→ ∞,as clearly seen by writing (3.4) as

(3.5)α = − p · d x +[mc2 + p 2

2m+O

(1

c2

)]dt.

The two-formdα has, nevertheless, a well-behavedlimit, namely the familiar presymplectic two-form ofan “ordinary” free, spinless, non-relativistic particle ofmassm, namely,

dα ≈ −dpi ∧ dxi + dh∧ dt(3.6)with h= p 2

2m,

where the notation “≈” stands for “up to higher-orderterms in 1/c2”.

In order to cure the pathology of (3.5), we considerthe trivial central extensionG=G× R of the planarPoincaré groupG. The new one-form to consider onGis the left-invariant one-form

(3.7)α = α + ε dτ,where τ is a coordinate on the centre(R,+) andε ∈ R. We now posit, for the sake of convenience,τ = t + u/c2 whereu is a new real parameter thatreplaces our oldτ . Also, let us introduce a new realconstant,h0, via

(3.8)ε = −mc2 + h0.

Then the divergence in the one-formα disappears (cf.(3.5)), and the latter simply reads

(3.9)α ≈ − p · d x +[ p 2

2m+ h0

]dt −mdu.

The one-formα hence converges in the limitc→+∞. Note that we also get the “internal energy”,h0, specific to Galilean Hamiltonian mechanics. It isworth mentioning that (3.8) is the most general ansatzfor a power series inc2 that guarantees convergenceof α while bringing non-trivial contribution to the non-relativistic limit.

Thus, the divergent term inα is absorbed into anexact one-form that involves the extra coordinate thatdrops out. The remaining part is regular and yields thecorrect conserved quantities.

The advantage of this approach is that triviallyextended Poincaré group now contracts nicely to thesingly extended Galilei group. The infinitesimal actionof the Poincaré groupG on spacetime reads, in fact,

(3.10)

δxi = ωεij xj + βit + γi,δt = β · x/c2 + ε,

with ω ∈ R generating rotations,β ∈ R2 boosts,γ ∈R2 space translations andε ∈ R time translations.

The infinitesimal action of the centre(R,+) of theextended groupG is readily written asδτ = η/c2 + ε′with η, ε′ ∈ R, so that the above definition ofu yields,in the limit c→ +∞, ε′ = ε and

(3.11)δu= − β · x + η,corresponding precisely to the infinitesimal action ofthe non-trivial 1-parameter central extension of the


Galilei groupG∞ on the “vertical” fiber parametrizedby u, while the usual Galilei action on spacetime isplainly deduced from (3.10) in the limitc → +∞,viz.,

(3.12)

δxi = ωεij xj + βit + γi,δt = ε.

4. Spin and exotic parameter

Let us now extend our improved contraction tospin. Starting with spinning massive particles movingin Minkowski spacetimeR2,1, a similar procedureleads to three items analogous to those previouslyhighlighted.

For a particle with spins ∈ R and massm> 0,dwelling in Minkowski spacetime, the one-form tostart with on the Poincaré group,G, is given [10]by α = pµ dxµ + se1µ deµ2 , wherepµ = meµ0 and(e0, e1, e2)x is an orthonormal Lorentz frame at thepoint x ∈ R1,2. This one-form retains, in an adaptedcoordinate system, the rather complicated-looking ex-pression

α = − p · d x +mc2√

1+ p 2

m2c2dt

+ s√

1+ p 2

m2c2

1+ ( p×u)2m2c2

[(1+ p 2

m2c2

)u× d u

+ p · um2c2

d u× p+ u× p( p · u) p · d pm4c4

(1+ p 2

m2c2

)(4.1)− u× p

m2c2d( p · u)

],

with u ∈ S1, an arbitrary unit vector in the plane.Note thatu × d u = dφ, whereφ is the argument ofu (actually the rotation angle of the SO(2) subgroupof the Lorentz group SO(1,2)). A tedious calculationyields furthermore a presymplectic two-form,ω = dα,similar to (3.1), whose behaviour to orderc−2 is

(4.2)ω ≈ −dpi ∧ dxi + dh∧ dt + s

m2c2dp1 ∧ dp2,

whereh is as in (2.1).Note that if one considerss as being independent

of c, then the two-formω in (4.2) plainly tends, as

c → +∞, to that of an “ordinary” non-relativisticparticle, (3.6). Our clue is to posit, instead, the Jackiw–Nair-inspired ansatz, cf. (3.8),

(4.3)s = κc2 + s0,whereκ ands0 are new constants. Then, in the non-relativistic limit we recover precisely our exotic two-form (2.1). This is just like in [5], up to a minordifference in the interpretation: it is ours, and not theirS = sc2 in (3.1) that should be called relativistic spin.The parameterS has indeed physical dimension[S] =[h/c2], and cannot represent spin, whose dimension,[h], is carried correctly by ours. The constantsκ in(4.3) (whose dimension is[h/c2]) is hence interpretedas the exotic parameter; alsos0 will turn out to beGalilean anyonic spin.

Presenting the spin term in Eq. (4.1) as

κc2dφ + s0 dφ+ κ

m2

[(3

2p 2 − ( p× u)2

)dφ

(4.4)

+ ( p · u) d u× p− u× p d( p · u)]

+O(

1

c2

)

yields

α ≈ − p · d x +[mc2 + p 2

2m

]dt

(4.5)+ κc2dφ + s0dφ + κ

2m2 p× d pmodulo a closed one-form. Thus, while the exteriorderivative dα behaves correctly, the one-form (4.5)contains instead two divergent terms, namelymc2dtandκc2dφ. Removing them and settings0 = 0 wouldyield the Cartan-form of the Lagrangian used in [3].

This time, regularization is achieved by the (trivial)double central extension G = G× R of the Poincarégroup, endowed with the canonical one-form:α =α + χ dθ where α is as in (3.7) andθ parametrizesthe new(R,+)-factor, the coordinateχ having thedimension ofh. The divergence associated with theenergy having been removed by the first trivial centralextension, let us posit2 χ = −κc2 andθ = φ −w/c2wherew ∈ R is the new parameter to consider in place

2 Adding ac-independent constant toχ would just modify theform of the (anyonic) spins0 by an overall additive constant.


of θ . Then, unlikeα in (4.5), the one-formα is well-behaved in the limitc→ +∞ and retains (modulo aclosed one-form), the expression

α ≈ − p · d x +[p2

2m+ h0

]dt −mdu

(4.6)+ s0 dφ + κ

2m2 p× d p+ κ dw.

In order to make sense of the group contractionG∞ = limc→+∞ G, we need to compute the infini-tesimal action of the Lorentz group on the rotations,parametrized byφ, which serve to define our secondextension coordinate,w. Using the known form of thematrices spanning the Lie algebra o(1,2), one writesthe infinitesimal Lorentz action on itself to obtain fi-nally

(4.7)δφ ≈ ω+ 1

2c2β × b.

The infinitesimal action of the extra central subgroup(R,+) of G being written asδθ = 9/c2 + ω′ with9,ω′ ∈ R, we finally get from the definition ofw, inthe limit c→ +∞, firstly ω′ = ω and, secondly, theexpression

(4.8)δw = 9+ 1

2β × v,

where we have putv = p/m for the velocity.Formula (4.8) gives the infinitesimal action of the

exotic Galilei groupG∞ on the new, exotic, extensionfiber with coordinatew. This latter can be viewedas providing an extended evolution space. Notice,though, that the action ofG∞ on this evolutionspace—whose infinitesimal form is given by (3.12),(3.11), and (4.8)—isnot the lift of an action on someextended spacetime, as it involves the velocityv in(4.8).

Let us mention that the conserved quantity associ-ated to a symmetry is obtained, in this framework, bysimply contracting the (invariant) one-formα on thegroup with the infinitesimal generator of the symme-try. Using (4.6), we recover in particular the generalconserved quantities in (2.3). No divergences occur.

Our modified contraction with the ansatz (4.3)yields hence a non-relativistic model with both anexotic structure and anyonic spin.

5. Discussion

The derivation of the exotic model presented byJackiw and Nair [5] has some subtle points: their“spin”, which becomes our exotic parameter, has notthe expected physical dimension, and their angularmomentum diverges asc tends to infinity. Such diver-gences are indeed familiar in the theory of group con-traction; our mathematical construction, here, allowsus to eliminate the divergent terms by resorting to adouble group extension of the Poincaré group and torecover, for free, the twice centrally extended Galileigroup of planar physics.

In a recent letter, Hagen [13] also discusses therelation of spin and exotic extension. He considersthe Lévy–Leblond equation (2.11) but with ordinaryLagrangian, i.e., (2.10) with ordinary product, whichhas no non-trivial second extension [12].

In conclusion, let us stress that fractional spin arisesdue to the commutativity of the planar rotation groupalone, independently of any further symmetry. This iswhy one can have relativistic as well as non-relativisticanyons. Although the commutative structure of planerotations plays a role for the “exotic” structure also,this latter clearly involves more symmetries, namelythe two-dimensional group-cohomology of the Galileigroup. On the other hand, this phenomenon definitelydoes not arise in the relativistic context since thePoincaré group has trivial group-cohomology.

Acknowledgements

We are indebted to Professors R. Jackiw andM. Plyushchay for their interest and correspondence.

References

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Spin and exotic Galilean symmetry