Wigner–Souriau translations and Lorentz symmetry of chiral fermions

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Wigner–Souriau translations and Lorentz symmetry of

chiral fermions

Christian Duval, M Elbistan, P.A. Horváthy, P.-M Zhang

To cite this version:

Christian Duval, M Elbistan, P.A. Horváthy, P.-M Zhang.

Wigner–Souriau translations and

Lorentz symmetry of chiral fermions.

Physics Letters B, Elsevier, 2015, 742, pp.322-326.

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Wigner–Souriau

translations

and

Lorentz

symmetry

of

chiral

fermions

C. Duval

a

,

M. Elbistan

b

,

P.A. Horváthy

b,c,

∗

,

P.-M. Zhang

c a_{Aix-MarseilleUniversity,CNRSUMR-7332,Univ.SudToulon-Var,13288Marseillecedex9,France} b_{LaboratoiredeMathématiquesetdePhysiqueThéorique,UniversitédeTours,France}

c_{InstituteofModernPhysics,ChineseAcademyofSciences,Lanzhou,PRChina}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received27November2014

Receivedinrevisedform19January2015 Accepted27January2015

Availableonline30January2015 Editor:M.Cvetiˇc

Chiral fermions can be embeddedintoSouriau’s masslessspinning particle model by“enslaving” the spin, viewedasagaugeconstraint.Thelatterisnot invariantunder Lorentzboosts;spinenslavement can be restored,however,by aWigner–Souriau(WS)translation, analogous toacompensating gauge transformation. Thecombined transformationisprecisely therecentlyuncoveredtwisted boost,which we nowextend tofinite transformations.WS-translations are identifiedwith the stabilitygroup ofa motionactingontherightonthePoincarégroup,whereasthe naturalPoincaréactioncorrespondsto actionontheleft.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Semiclassicalchiralfermionsofspin 1₂ canbedescribed bythe phase-spaceaction S

=

 

(

p

+

e A

)

·

dx dt

−

h

−

a

·

d p dt



dt

,

h

= |

p

| +

e

φ (

x

),

(1.1) where a

(

p

)

is a vector potential for the “Berry monopole” in

p-space,

∇

p

×

a

=

_2|_pp_|2_,



p

=

p

|

p

|

, p

=

0. Here A

(

x

)

and

φ (

x

)

are[static]vectorandscalarpotentialsande istheelectriccharge [1–6]. A distinctive feature is that spin is “enslaved” to the mo-mentum,i.e.,

s

=

1

2



p

.

(1.2)

An intriguing aspect of the model is its lackofmanifest Lorentz symmetry.Recently[5],itwas shown, though,that modifyingthe dispersion relationin(1.1) ash

= |

p

| +

e

φ (

x

)

+

₂p_|·_pB_| yields a the-ory which is covariant w.r.t.Lorentz transformations. Turning off theexternalfield,theirexpression#(6)reducesto

δ

x

= β

t

+ β ×



p

2

|

p

|

,

δ

p

= |

p

|β,

δ

t

= β ·

x

,

(1.3)

*

Correspondingauthor.

E-mailaddresses:duval@cpt.univ-mrs.fr(C. Duval),elbistan@itu.edu.tr

(M. Elbistan),horvathy@lmpt.univ-tours.fr(P.A. Horváthy),zhpm@impcas.ac.cn

(P.-M. Zhang).

where

β

is an infinitesimal Lorentz boost. This formulahas also been found, independently [6], by relating the chiral fermion to Souriau’smodelofarelativisticmasslessspinningparticle[6,7].

ThechiralandtheSouriausystems[7]haveseeminglydifferent degrees of freedom:the first one has“enslaved” spin, while the latterhas“unchained”spinrepresentedby avector s,whichmay not satisfy (1.2). However, the freedom specific to the massless case ofapplyingwhat we call a Wigner–Souriau (WS) translation,

Eq.(2.7)below,1_{allowsonetoeliminatetheadditionaldegreesof}

freedom oftheSouriauframework, sothat thetwo systemshave identical spacesof motions[6,7].This enabledusto “export” the naturalLorentzsymmetrytothechiralsystem,yielding(1.3).

In this Note we derive (1.3) in another, rathermore intuitive way,namelyby embedding thechiralfermion modeldirectlyinto theSouriau model,namelyby viewingspinenslavement,(1.2),as agaugecondition withWStranslationsandviewedasgauge trans-formations.ThenourclueisthatanaturalLorentzboostdoesnot leavetheconstraint(1.2)invariant;the“gauge”condition(1.2)can, however, be restored by a WS translation, yielding, once again, (1.3).Thisisreminiscentofwhathappensingaugetheories,where a space–time transformation can be a symmetry when the vari-ation of the vector potential can be compensated by a suitable gaugetransformation[12].

1 _{Wigner–Souriautranslations(called“ Z -shifts”in[6])wereknown toWigner}

[8]andtoPenrose[9].Theiruseinthesemiclassicalframeworkwasadvocatedby Souriau[7].Theywereoverlookedbymostpresentauthorswiththenotable excep-tionofStoneetal.[4],whoalsosuggestedviewingthemasgaugetransformations.

http://dx.doi.org/10.1016/j.physletb.2015.01.048

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

FurtherinsightisgainedinSection5whichclarifiesthe geom-etryhiddenbehind:whilethenaturalPoincaréactioncorresponds theleft-actionofthePoincarégrouponitself,WStranslations cor-respondtotheright-actionofthestabilitygroupofamotion.

WeconcludeourLetter bysome remarks about non-commut-ativity.

2. Symplecticdescriptionofthechiralandthemasslessspinning models

Both the chiral and the Souriau models can conveniently be described within Souriau’s framework [6,7], where the classical motionsare identified withcurves orsurfacesin some evolution space,andaretangenttothekernelofaclosedtwo-form;see[6] fordetails. Welimit ourconsiderationsto thefreecase;coupling toexternalfieldshasbeendiscussedintheliterature,see[6,13].

We first consider the [free] chiral model (1.1). It has been shown[6] that the associated variational problem admits an al-ternative geometric formulation. To that end, we introduce the seven-dimensional evolutionspaceV7

=

T

(

R

3

\ {

0

})

× R

described bytriples

(

x

,

p

,

t

)

,andendowitwiththetwo-form

σ

definedby

σ

=

ω

−

dh

∧

dt where

ω

=

dpi

∧

dxi

−

1 4

|

p

|

3



i jk_p idpj

∧

dpk

,

h

= |

p

|.

(2.1)

The two-forms

ω

and

σ

are closed, since

∇

p

·

b

=

0. The ker-nelof

σ

definesan integrabledistribution,whoseleaves[integral manifolds] can be viewed as generalized solutions of the varia-tionalproblem. Here, the kernel is one-dimensional anda curve

(

x

(

t

),

p

(

t

),

t

)

istangenttoitifftheequationsofmotion,

dx dt

= 

p

,

d p

dt

=

0

,

(2.2)

are satisfied [6]; the solution is plainly p

=

p0

=

const, x

(

t

)

=

x0

+ 

pt, x0

=

const, i.e.,themotionisinthe



p direction withthe

velocityoflight.

The Souriau model admits a similar description [6]. Restrict-ing ourselves againto the free case, the evolution spacehere is nine-dimensionalandisdescribedby

V9

=



R

,

P

∈ R

3,1

,

S

∈ o(

3

,

1

)



PμPμ

=

0

,

P4

>

0

,

SμνPν

=

0

,

SμνSμν

=

1 2



.

(2.3)

Theevolutionspaceisendowedwiththeclosedtwo-form

σ

= −

d Pμ

∧

dRμ

−

2 dSμλ

∧

SλρdSρμ

.

(2.4)

A

(

3

+

1

)

-decompositioncanbeintroducedbywriting,inaLorentz frame, R

= (

r

,

t

)

, and P

= (

p

,

|

p

|)

where p

=

0.The components

Sμν ofthespin tensorcaninturnbesplitintospaceandspace– timecomponents,

Si j

=



i jksk

,

Sj4

= Σ

j where

Σ

= 

p

×

s

,

(2.5) the3-vector s beinginterpretedasthespininthechosenLorentz frame.Notethat

Σ

isnotanindependentvariable,sothatapoint of V9 _{can be labeled by}

₍

_r

_,

_t

_,

_p

_,

_s

₎

_{. Then the [free] equationsof}

motionassociatedwiththekernelof

σ

in(2.4)areexpressedas,2

2 _{These equations can also be obtained in a Wess–Zumino framework [10],}

namelyas the variationalequations ofthe Lagrangian #(3.1) of[10]; the lat-terisinfacttheone-form(B.1)in[6],whoseexteriorderivativeisthetwo-form

d(μ0·g−1_dg ).

−

p

· ˙

r

+ |

p

|˙

t

=

0

,

p

˙

=

0

,

˙

s

=

p

× ˙

r

,

(2.6) andcanbededucedfromEq.(3.6)in[6].Aparticularsolutionof (2.6)isobtainedbyembeddingthechiralsolutionaboveintothe spin-extendedevolutionspaceV9 by identifying r with x in (2.2) and completing[trivially] withaconstant spinvector, p

=

p₀

=

const,

r

(

t

)

=

r0

+ 

pt, r0

=

const, s

(

t

)

=

s0

=

const. Aremarkablefeature

ofEqs.(2.6) isthat, foranarbitrary3-vector W ,the transforma-tionwerefertoasaWigner–Souriau (WS)translation[4,6–9],

r

→

r

+

W

,

t

→

t

+ 

p

·

W

,

p

→

p

,

s

→

s

+

p

×

W (2.7) takes a solution of (2.6) into another, equivalent one. The ker-nel of (2.4) is invariant under WS-translations and is in fact three-dimensional, swept by the images of the embedded solu-tions.

The spinvector here, s,isnot necessarily“enslaved”,i.e., may not beparallel to themomentum, p. Thespin constraintin(2.3) impliesneverthelessthat theprojection ofthespinonto the mo-mentumandtheperpendicularcomponent

Σ

satisfy,

s

· 

p

=

1

2

,

Σ

= 

p

×

s

,

(2.8)

respectively, see (2.5). From the WS action above we infer that

Σ

→ Σ +

p

× (

p

×

W

)

.Itfollowsthat

Σ

canbeeliminated: choos-ing W

= Σ/|

p

|

carries

Σ

tozero.

The Poincarégroup actsnaturally on V9_{,namely accordingto}

[6,7]

⎧

⎪

⎪

⎨

⎪

⎪

⎩

δ

r

=

ω

×

r

+ β

t

+

γ

,

δ

t

= β ·

r

+

ε

,

δ

p

=

ω

×

p

+ β |

p

|,

δ

s

=

ω

×

s

− β × Σ,

(2.9)

where

ω

,

β,

γ

,

ε

areidentifiedwithinfinitesimalrotations,boosts, translations and time-translations; their action duly projects to Minkowskispace–timeasthenaturalone.Inwhatfollows,we fo-cusourattentionatboosts;WS-translationswillbestudiedfurther inSection5.

3. Embeddingthechiralsystemintothemasslessspinning model

Now we embed the evolution spaceof the chiral model, V7, intothatofthemassless spinningparticle, V9.Wenote firstthat, by(2.8),spinenslavement,(1.2),isequivalentto

Σ

= 

p

×

s

=

0

,

(3.1)

which, viewed as a constraint, defines a seven-dimensional sub-manifoldof V9 _{thatwe parametrizewithr}

_,

_p

_,

_{t anddenote(with}

asuggestiveabuseofnotation)still V7_{.Eqs.(2.6) and(2.8) imply}

that

˙Σ = |

p

|(

pt

˙

− ˙

r

)

.Requiring

Σ

=

0 isthereforeconsistentwith the dynamics: the motions ofthe chiral systemlie in the inter-section of the three-dimensionalcharacteristic leavesof V9 _with

the surface V7 defined by spin enslaving; they remain therefore motionsalso fortheextendeddynamics.3 _{Achiralmotionis}

em-beddedintoV9 _{byrespectingthegaugecondition(1.2),namelyas}

γ

(

t

)

= (

r

(

t

)

=

x

(

t

),

p

(

t

),

s

(

t

)

=

1₂



p

(

t

))

.

4. Lorentzboostactions

WeconsidernowaninfinitesimalLorentzboost,

β.

Fors

=

1₂



p

wehave

δ

β

(

p

×

s

)

=

12

β

×

p,whichdoesnot vanishingeneral:spin

Fig. 1. EnslavingthespinamountstoembeddingthechiralevolutionspaceV7_into

that,V9_{,ofthemasslessspinningparticle.Achiralmotionthusbecomesthe}

in-tersectionofV7_{withthecharacteristicleafofkerσ(depictedasaverticalplane).}

Enslavementisnotinvariantundera(natural)boostb,butasuitablecompensating Wigner–SouriautranslationW allowsustore-enslavethespin.Thecombinationof thetwotransformationsyieldsthetwistedboostB in(1.3).

andmomentumdonotremainparallel eveniftheyweresoinitially:

embeddingthechiralsystemintotheSouriaumodelthroughspin en-slavementisnotboost-invariant: naturalboostsymmetryisbroken by

spinenslavement.However,asdictatedbytheanalogywithgauge symmetries[12],letusapplyaninfinitesimalWS-translation

δ

Wr

=

W

,

δ

Wt

= 

p

·

W

,

δ

Wp

=

0

,

δW

s

=

p

×

W with W

=

β

× 

p

2

|

p

|

.

(4.1)

Then

δ

Ws

=

1₂

(β

− 

p

(



p

· β))

, implying that the combined trans-formation

δ

= δ

β

+ δ

W doespreservespinenslavement,

δ(

p

×

s

)

=

0. Thetransformation of V7 generatedby

δ

isprecisely thetwisted boost(1.3).4

So far we studied infinitesimal actions only. But our strategy isvalidalsoforfinitetransformations,asweshowitnow. Firstly, fromAppendix Cof[6]weinfertheactionofafiniteboost b on

theevolutionspaceV9,namely

⎧

⎪

⎪

⎨

⎪

⎪

⎩

r

=

r

+ (

γ

−

1

)(

b

·

r

)

b

+

γ

tb

,

t

=

γ

(

b

·

r

+

t

),

p

=

p

+

γ

|

p

|

b

+ (

γ

−

1

)(

b

·

p

)

b

,

s

=

s

+ (

γ

−

1

)

b

× (

s

×

b

)

−

γ

b

× Σ,

(4.2)

whichisconsistentwiththeinfinitesimalaction(2.9).Then, start-ingwithenslavedspin,s

=

1₂



p,wefind,

Σ



=

p 

_×

_s

|

p

|

=

1 2

γ

2

|

b

| +

γ

2

−

1



b

· 

p



b

×

p

|

p

|

(4.3)

whichdoesnotvanishingeneral:spinisunchained.Atlast,the fi-niteWS-translation(2.7)withW

= Σ



/

|

p

|

restoresthevalidityof (1.2): spinisre-enslaved, s

=

1₂



p.The combinedtransformation forfiniteboosts,

r

=

r

+

γ

tb

+ (

γ

−

1

)(

b

·

r

)

b

+

Σ



|

p

|

,

(4.4)

4 _{Theorderisirrelevant,sinceWS-shiftsandboostscommute.}

completedwitht

=

t andp

=

pwheret

,

pand

Σ

aregiven in(4.2)and(4.3),respectively.Theinfinitesimalactionis(1.3)asit shouldbe.Inconclusion,Fig. 1isvalidalsoforfiniteboosts.

We nowturnto thespaceofmotions[6,7],definedasthe quo-tientoftheevolutionspacebythecharacteristicfoliationof

σ

;we denoteitbyM.Theequationsofmotionofthespin-extended sys-tem,(2.6),implythat

˜

x

(

t

)

=

r

(

t

)

− 

pt

+

Σ

|

p

|

(4.5)

isinfactaconstantofthemotion,dx

˜

/

dt

=

0.Itcanbeused there-foreto labelthemotion,i.e.,a characteristicleafin V9_.The

con-servedmomentum, p,isanothergoodcoordinate setonM,which issix-dimensional,andwhosepointscanthereforebelabeledby

˜

x

and p

=

0.

In[6],wederivedthetwistedboost(1.3)fromthePoincaré ac-tiononthespaceofmotions.Conversely,thelattercanbeobtained fromourconstruction here. Boostingin V9 _{accordingto (2.9) we}

findthattheprecedingtermscombinetoyield

δ

βx

˜

=

1 2

β

×



p

|

p

|

− 

p

(β

· ˜

x

).

(4.6)

Now WS-translationsactinturntriviallyonthespaceofmotions,

δ

Wx

˜

= δ

Wp

=

0,sincetheymove eachcharacteristicleave within itself.Completing(4.6) with

δ

βp

= |

p

|β

cf.(2.9),weendupwith

the (boost-)actiononthespaceofmotions,–Eq. (3.19) in[6].At last, applying(4.2)toeach termin(4.5)allows usto confirmthe finiteactiononthespaceofmotions,Eq.#(3.24)in[6].

5. ThegeometryofWigner–Souriautransformations

Do WS-translationsbelong to the Poincaré group? To answer thisquestion,let usfirstbriefly summarizeSouriau’s construction for“elementarysystems”[meaning thatthegroupacts symplecti-callyandtransitively][7].

The connected component of the Poincaré group, G, can be identified with the group of matrices g

=



L C

0 1



where L is

a Lorentz transformation, L

∈

H (the connected Lorentz group), and C a Minkowski-space vector. The Poincaré Lie algebra,

g

, is hence spanned by the matrices X

=



Λ Γ

0 0



where the infinites-imal Lorentz transformations and translations,

Λ

∈

so

(

3

,

1

)

, and

Γ

∈ R

3,1, respectively, are such that, in a given Lorentz frame,

Λ

=



_{j(ω) β} βT ₀



and

Γ

=



_γ ε



with

ω

,

β,

γ

∈ R

3 _{interpreted as an}

infinitesimal rotation, boost, and space-translation, while

ε

∈ R

stands foran infinitesimal time-translation.Here j

(

ω

)

isthe ma-trixofcrossproductin3-space, j

(

ω

)

x

=

ω

×

x.

ThePoincaré groupactsonitselfontheleft,g

→

hg,forh

∈

G,

and the quotient of G by the subgroup of Lorentz transforma-tions, G

/

H , can be identified with Minkowski space–time. This left-action of G onitselfprojects to thenaturalaction, infinitesi-mallythetwoupperlinesin(2.9),ofthePoincarégrouponspace– time.

The Poincaré group acts on the dual Lie algebra by the co-adjointrepresentation,definedby Coadg

μ

·

X

=

μ

·

g−1X g,forall X

∈ g

. We denote here by

μ = (

M

,

P

)

∈ g

∗ a “moment” of the Poincaré group, and by

μ ·

X

=

₂1Mμν

Λ

μν

−

Pμ

Γ

μ its contrac-tionwithX

∈ g

,see[7].

Contracting theMaurer–Cartan one-form, g−1dg, withand ar-bitrary fixed element

μ

0 of the dualof Liealgebra, yields a real

one-formonthegroupG;wedenoteby

σ

=

d

(

μ

0

·

g−1dg

)

its

ex-teriorderivative. Asageneralfact,thecharacteristicleavesofthe two-form

σ

,definedbyitskernel,areidentifiedwiththeclassical motions[11]associatedwith

μ

0.Indeed,thespaceofallmotions,

M, of an elementary system for the group G is interpreted by Souriau[7]astheorbitofsomebasepoint

μ

0 underthecoadjoint

action,M

=

CoadG

μ

0

≈

G

/

G0,whereG0 isthe stabilitysubgroup

of

μ

0.The coadjointorbit, M, ishencecanonicallyendowedwith

the symplectic structure defined by the Kostant–Kirillov–Souriau two-form

ω

,theimageof

σ

undertheprojection G

→

M.

Inourcaseandfollowing[7],thebasepointcanreadilybe cho-sen as

μ

0

= (

M0

,

P0

)

with M0

=

s



j(p0)0 0 0



and P0

= |

p0

|



p0 1



with



p₀

=

e3; here s

=

12 is the scalar spin and the positive

constant

|

p0

|

is the energy. Then a straightforward calculation

showsthatthePoincaré Liealgebraelement

(Λ,

Γ )

with parame-ters

ω

,

β,

γ

∈ R

3

_,

_ε

_{∈ R}

_{leavesthechosen basepoint}

_μ

0 invariant whenever

ω

= β × 

p0

+ λ

p0

,

γ

= −

1 2

β

× 

p₀

|

p0

|

+

ε



p0

.

(5.1)

ThestabilityLiealgebra,

g

0,isthereforefour-dimensional,

param-etrized by

(β,

ε

,

λ)

, where

β

⊥ 

p0,

ε

∈ R

, and

λ

∈ R

represents

an infinitesimal rotation around



p₀. We note that the evolution space V9 _{is in fact the quotient of the Poincaré group by}

rota-tions around



p0, and(2.9) above isin fact the projection of the

infinitesimalleft-actionofthegroupG to V9

_≈

_G

_/

_SO

₍

₂

₎

_,whereas

thetwo-formd

(

μ

0

·

g−1dg

)

projectsas(2.4),asanticipatedbythe

notation.

Remember now that the Poincaré group also acts on itself

fromtheright, g

→

gh−1,for h

∈

G; its infinitesimal right-action is therefore given by matrix multiplication,

δ

Xg

= −

g X , where X

= δ

h ath

=

1.Choosingin particular X

∈ g

0,the actiononthe

group reads

δ

X



_{L C} 0 1



=



−LΛ−LΓ 0 0



and

Γ

=



_γ ε



, where

γ

and

ω

are as in (5.1). This 4-parameter vector field belongs, at each point g

∈

G,to thekernelofthe two-form

σ

;infact,itgenerates

its kernel [7]. Renaming the Poincaré translation C as R

= (

r

,

t

)

, a space–timeevent, showsthat theright-actionon G ofa vector

X fromthestabilityalgebra

g

0yields,

δ

Xr

=

1 2

β

× 

p

|

p

|

−

ε



p and

δ

Xt

= −

ε

.

(5.2)

Thistransformation satisfiesthe condition



p

· δ

Xr

= δ

Xt required foraninfinitesimalWS-translation;conversely,anyWS-translation

W isoftheform(5.2),with

β

perpendicularto p and

λ

arbitrary. Therefore(withaslightabuse)wewillreferto

g

0 actingfromthe

rightasaWS-translations.

Comparingnow(5.2)with(4.1)allowsustoconcludethatwhile aboost

β

actingontheleftunchainsthespin,thelatterisre-enslavedby aWS-translationwiththesameboost,

β

,actingfromtheright.

It is readily verified that the right-action of an X from the stabilityalgebra

g

0actsas

δ

Xp

=

0 and

δ

Xs

=

1₂



p

×(

p

×β)

, consis-tentlywiththeinfinitesimalaction.Thus,afterrotationsaround



p0

are factored out, the right-action of the stability subalgebra

g

0

projects to V9 as the WS translations. In conclusion, Wigner– Souriautranslationsare Poincarétransformations,—butwhichact on the right and not on the left as natural ones do. Projecting further down to the space of motions, M, they act trivially. The variousspacesareshown,symbolically,inFig. 2.

6. Relationtonon-commutativity

Thespacesof motionsofboth thechiralandthe Souriau sys-temscarrythesymplecticstructure

ω

in(2.1)[withx replacing

˜

x] [6].TranslatingintoPoissonbracketlanguage, wehave

{ξ

α

,

ξ

β

_{} =}

ω

αβ

_{(ξ )}

_where

₍

_ω

αβ

₎

_{= −(}

_ω

αβ

)

−1 are the coefficientsof the

in-verse of the symplectic form

ω

=

1

2

ω

αβ

(ξ )

d

ξ

α

∧

d

ξ

β. Therefore

thecomponents of the spaceof motions coordinates x do

˜

notto Poisson-commute,

Fig. 2. The evolutionspace, V9_{, is the quotientof the Poincarégroup, G}10 _by

rotations around p0. The space ofmotions isidentified with a coadjoint orbit M6_≈G10_/G

0 ofG10,whereG0,thestabilitygroupofthebasepointμ0.The

left-actionofG10 _{onitselfprojectstothenaturalPoincaréactiononboththespace}

ofmotionsandMinkowskispace–time,consistentlywithprojectingfirstto V9_and

thentoM6_andR3,1_{,respectively.}



˜

xi

,

x

˜

j



=

1 2



i jk pk

|

p

|

3

.

(6.1)

Puttingt

=

0 and

Σ

=

0 into(4.5)showsthatthechiralcoordinate

x satisfies the same commutation relation. Non-commutativity is thus a consequence of spin, cf.[15–18]; chiral fermions provide justanotherexampleofnon-commutativemechanics.

It is worth stressing that the non-commutativity is unrelated

to WS translations. Non-commutativity arises in fact due to the twisted symplectic structure ofthe spaceofmotions (modeled by

the phase space); but the WS translations act on the evolution spacebeforeprojecting,trivially,tothespaceofmotions.

In theoperator context a Bopp-shift transforms the commuta-tionrelationstocanonicalones[19].Inthe(semi)classicalcontext weare interestedinitamountstointroducingnew,canonical co-ordinates, eliminating spatial non-commutativity altogether. Now Darboux’stheorem[20]guaranteesthatthisisalwayspossible lo-cally.Indetail,weredefinethecoordinatesas



X

= ˜

x

+

a

(

p

),



P

=

p

,

(6.2)

wherea

(

p

)

isa“Berry”vectorpotential.Then,intermsof



P and



X , the symplectic form

ω

in (2.1) has the canonical expression

ω

=

d



Pi

∧

d



Xi, so that

{

Xi

,



Xj

} = {

Pi

,



Pj

} =

0 and

{

Pi

,



Xj

} = δ

ij. Spin seems thuseliminated. Thisis not possibleglobally,though, sinceany a issingularalongaDiracstring,asitiswell-known.

ThevariationoftheBerryvectorpotentialunderaboost

β

∈ R

3 is then found to be

δ

βa

= −β ×

2|pˆp|

− (

a

· β) ˆ

p

+ ∇

p

(

|

p

|β ·

a

)

. It follows that, in terms of the Darboux coordinates (6.2), the in-finitesimalactionofboostsisgivenby

δ

β



X

= −(β · 

X

)



P

|

P

|

+ ∇

P

|

P

|β ·

a



,

δ

β



P

= β|

P

|.

(6.3)

Redefining analogously the chiral coordinate as X

=

x

+

a and P

=

p allowsustoinfer

δ

βX

= β

t

− (β ·

a

) ˆ

P

+ ∇

P

|

P

| β ·

a



= β

t

+ |

P

|∇

P

(β

·

a

),

δ

βP

= β|

P

|,

(6.4)

which is reminiscent of but still different from the natural ac-tion(2.9).

7. Conclusion

In this Letter, we re-derived the twisted Lorentz symmetry (1.3) of chiral fermions by embedding the theory of Refs. [1–5] intoSouriau’smasslessspinningmodel[6,7]byspin enslavement, (1.2), viewed asa gauge fixing. The latter is not boost invariant, butenslavement canberestoredbya suitablecompensating WS-translation,whichisanalogoustoagaugetransformation[12].Our formula(4.4)extends(1.3)tofinitetransformations.

The motions of the extended model are not mere curves but 3-planes,sweptbyWS-translations.Themasslessrelativistic parti-cleisdelocalized[4,6–10]andbehavesratherasa3-brane.

Coupling to an external electromagnetic field breaks the WS “gauge”symmetry, sothat spincannotbe enslaved.Themotions thentake place alongcurves: theparticlegets localized[6].New results [21] indicate however that while a WS translation is un-observable for an infinite plane wave, it isobservable for a wave packet.Intuitively,formingandkeepingtogetherawavepacket be-havesasasortofinteraction.

Acknowledgements

We would like to thank Mike Stone for calling our atten-tion at chiral fermions and for enlightening correspondence. ME is indebted to ServicedeCooperationetCultureldel’Ambassadede FranceenTurquie and to the Laboratoirede Mathématiques etde PhysiqueThéorique of Tours University.P. Kosinski andhis collab-orators work ona related project[14].This work was supported bytheMajor StateBasicResearchDevelopmentPrograminChina (No. 2015CB856903)andthe NationalNaturalScience Foundation ofChina(GrantNos. 11035006and11175215).

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