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Currents for Arbitrary Helicity Norbert Dragon Institut fu¨rTheoretische Physik, LeibnizUniversita¨t Hannover, Appelstraße 2, 30167 Hannover, Germany Abstract Using Mackey’sclassificationofunitary representationsofthe Poincarégrouponmasslessstates of arbitraryhelicity we disprove the claim that states with helicity h 1 cannotcouple to a conservedcurrentby constructing sucha current. | |≥ 6 Keywords: 1 0Unitary representation, Poincarégroup, Helicity, Massless states, Currents, Supersymmetry, Position operator 22008 MSC: 58D30, 81R05, 83A05 n a J The constitutive feature of a relativistic quantum sys- ByMackey’stheoremsallunitaryrepresentationsofthe 8temistheexistenceofaunitaryrepresentation(ofthecon- Poincaré group are sums or integrals of irreducible rep- 2nectedpartofthecover)ofthePoincarégroup: aquantum resentations and each irreducible, unitary representation system is relativistic if its Hilbert space carries such a of the Poincaré group is uniquely specified by its mass ] H hrepresentation. It endows the states with physical prop- shell , the orbit under the proper Lorentz group G of a M terties, e.g. the representation U = eiP·a of translations momentump,andanirreducible,unitaryrepresentationR - a px x+aisgeneratedbycommuting,hermiteanoperators of its stabilizer H = h G:hp=p . eP7→=(P0,P1,P2,P3), P a=P0a0 P1a1 P2a2 P3a3. An orbit = p:{p=∈gp,g G }can be viewed as the h · − − − M { ∈ } Therefore, by definition, P is the four-momentum and its set G/H of cosets gH on which G acts by left multiplica- [ eigenstatesarestateswithdefiniteenergyandmomentum. tion. The projection of G to the mass shell G/H 1 The representation has to be unitary and its genera- v G G/H 5torshaveto be antihermiteanfor consistencywiththe ba- π : → (2) g p=gp 2sic postulate of quantum mechanics: if one measures the (cid:26) 7→ 8state Ψ with an ideal device A, which for simplicity has attributes to each Lorentz transformation g the momen- 7discrete,nondegenerateresults,thenthe probabilityw for tump=gpwhichisobtainedbyapplyinggtop. Thefiber 0 the result a to occur is . i overapointpconsistsofalltransformationsg,whichmap 1 ptop. ItisthecosetgH ofthestabilizerH ofp. Anytwo 0 w(i,A,Ψ)= Λ Ψ 2 . (1) 6 |h i| i| fibers are either disjoint or identical and G is the union of 1 all fibers, it is a fiber bundle. Here Λ are the states (they exist by the assumption : i A local sectionof G is a collectionof maps σ , enumer- vthat the device A is ideal) which yield a with certainty, α i Xiw(i,A,Λj)=δij. ated by Greek indices which we exempt from the summa- tion convention. In its domain the map σ is a right r No formulation of quantum theory can do without this inverse of the projection π. In oUthαer words: σαassigns to astatement of what the physical situation is in case that α each p a transformation σ (p) which maps p to p, mathematically the system is in a state Ψ. We remind ∈ Uα α σ (p) p=p, the reader of these well known foundations of quantum α mechanics only to stress the importance of the antiher- (cid:0) (cid:1) G/H G mHitwiceirteynooftthheermgeinteearant,otrhse−niiHts goefntehreatreedprfleoswenitnatHioinlb.erItf σα :(cid:26) Uα ⊂p →7→ σα(p) , πσα =idUα . (3) space e−iHt :Ψ(0) Ψ(t) would not conserve the sum of The domains are required to cover the base manifold probabilities, iw7→(i,A,Ψ)=1. α α = G/HU.αIf this can be done already with one do- ∪ U The construction of unitary representations g Ug of main, =G/H,thenthesectioniscalledaglobalsection P 7→ U the Poincaré group by Wigner [1] were shown by Mackey and the fiber bundle is trivial, i.e. equivalent to the pro- [2] to yield all unitary representations for which for all Φ duct of a fiber times the base manifold. and Ψ in the maps g ΦUgΨ are measurable. In their common domain two sections differ by their H 7→h | i transition function h : H, αβ α β U ∩U → Email address: [email protected](Norbert Dragon) σβ(p)=σα(p)hαβ(p) , h−αβ1(p)=hβα(p) . (4) Preprint submittedto Physics LettersB January 29, 2016 As groupmultiplicationisassociative,the transitionfunc- are equivalent if and only if their mass shells agree and tions are related by their inducing representations are equivalent. If the representation R is trivial, R = 1, then the wave hαβhβγ =hαγ (5) functions Ψ R are constant along each fiber gH and ∈ H naturally define functions of the mass shell G/H. in each intersection of three domains. Let C2s+1 be a sUpaαc∩e Uonβ ∩wUhγich a unitary, irreducible If R is nontrivial, then by composition with a local section σ one obtains a one-to-one correspondence of each representationR of the stabilizerH ofsome momentum p α function Ψ to a section of the vector bundle over acts. Consider the space HR of functions G/H consist∈inHg Rof the collection of functions Ψα, Ψ:G C2s+1 (6) → Ψα =Ψ σα : α C2s+1 , (14) ◦ U → withafixeddependencealongeachfiber(forallg Gand h H) ∈ which by (7) are related in Uα∩Uβ by ∈ Ψ(gh)=R(h−1)Ψ(g) . (7) Ψ =Ψ(σ h )=R h−1 Ψ =R(h )Ψ . (15) β α αβ αβ α βα α As R is unitary, the scalar product of the values of two A transformation Λ G(cid:0) ma(cid:1)ps σ (p) by left multiplica- functions Φ,Ψ R at g ∈ α ∈H tion to Λσ (p) in the fiber of (Λp) for some β which α β ∈U Φ,Ψ = Φ(g),Ψ(g) = R(h−1)Φ(g),R(h−1)Ψ(g) by (4) is related to σβ(Λp) by an H-transformation, the g Wigner rotation W (Λ,p), βα (cid:0) (cid:1) (cid:0)= Φ(gh),(cid:1)Ψ(g(cid:0)h) = Φ,Ψ (cid:1)(8) gh Λσ (p)=σ (Λp)W (Λ,p) . (16) (cid:0) (cid:1) (cid:0) (cid:1) α β βα is constant along each fiber p = gH and therefore a func- tionof =G/H. Thisallowstodefinethescalarproduct Inserted into (12) one obtains the transformation of the M in HR as wave function Ψβ Φ Ψ = d˜p Φ,Ψ . (9) h | i p U Ψ (Λp)= U Ψ (σ (Λp))= ZM Λ β Λ β (cid:0) (cid:1) We choose the integration measure Lorentz invariant and = UΛΨ ((cid:0)Λσα((cid:1)p)Wβ−α1)=(cid:0)R(W(cid:1)βα) UΛΨ (Λσα(p))= with the conventionalnormalization factors (cid:0) =(cid:1)R(Wβα)Ψ(σα(p))=R(W(cid:0)βα)Ψ(cid:1)α(p) , d3p d˜p= . (10) U Ψ (Λp)=R W (Λ,p) Ψ (p) . (17) (2π)32 m2+~p2 Λ β βα α The transf(cid:0)orma(cid:1)tion U rep(cid:0)resents the(cid:1)group G because Then becomes a Hilbertpspace if in addition we re- Λ HR R represents the stabilizer H and the Wigner rotations quire its elements, the wave functions, to be measurable and square integrable with respect to d˜p. R carries a satisfy for p∈Uα, Λ1p∈Uβ and Λ2Λ1p∈Uγ by (16) H unitary representationofthe Poincarégroup: translations W (Λ Λ ,p)=W (Λ ,Λ p)W (Λ ,p) . (18) γα 2 1 γβ 2 1 βα 1 are represented on the wave functions Ψ multiplicatively For p0 = m2+p~2 we spell out the transformations of U Ψ (g)=eip·aΨ(g) , p=gp . (11) a massive m>0 and massless m=0 states. p Lorentz tran(cid:0)sform(cid:1)ations Λ of the wave functions are de- Themassshellofaparticlewithmassm>0istheorbit of p=(m,0,0,0), fined to act by inverse left multiplication of the argument U Ψ (g)=Ψ(Λ−1g) . (12) = (p0,p~):p0 = m2+~p2 , p~ R3 . (19) Λ m M { ∈ } Consistent with ((cid:0)7) the(cid:1)y map unitarily to itself, The stabilizer of p is the gproup of rotations, H = SO(3). R H Eachofits irreducible, unitary representationsR is deter- U Φ U Ψ = d˜(Λp) U ΦU Ψ = mined by its dimension 2s+1, where s, the spin of the h Λ | Λ i ZM Λ | Λ Λp (13) particle, is nonnegative and half integer or integer. = d˜(Λp) ΦΨ = d˜p(cid:0)ΦΨ = (cid:1)Φ Ψ . The manifold SO(1,3)↑ is the product SO(3)× R3 as | p | p h | i each LorentztransformationΛ=L O can be uniquely de- ZM ZM p (cid:0) (cid:1) (cid:0) (cid:1) composedinto a rotationO and a boost L [3, (6.32)]. So p The map Λ U is a representation of G in and is 7→ Λ HR there exists a global section σ : m SO(1,3)↑ sparing said to be induced by the representation R of the stabi- M → us to write an index α or β for local sections. lizer H. By Mackey’s theorems the representation of the Poincaré group is irreducible if it is induced by an irre- σ(p)=L , L p=p (20) p p ducibleunitaryrepresentationR. Viceversa,eachunitary, irreducible representationof the Poincarégroupis unitar- Composedwiththissection,thefunctionsΨ become R ∈H ily equivalent to an induced representation. Two of them wave functions of . m M 2 Each Lorentz transformation in the neighbourhood of The componentfunctions ofthe differentialoperatoron the unit element is the exponential Λ = eω of an element the right side of (22) are the negative of the infinitesimal of its Lie algebra. As the group multiplication is ana- motion δp of the points p. This motion occurs on the left lytic (and if the local section is analytic) (17) maps ana- sideof(17). Becauseofthissignonehastodistinguishthe lytic states to each other because the parameters ω of the variation of a point p from the change of the coordinate transformationswhichdonotleaveapointintheorbitin- functions hi :p pi which change by δhi = δpi. 7→ − variantcanbechosenascoordinatesinitsneighbourhood. We choose the basis matrices l which generate mn On analytic states and only on them the transformation LorentztransformationsinMinkowskispaceR1,3withma- U is given by a series in the transformationparameter ω trix elements (η =diag(1, 1, 1, 1)) Λ − − − times differential and multiplicative antihermitean generators iM , U =exp iωmnM . (lmn)rs =δmrηns δnrηms (23) − mn Λ −2 mn − TheCauchycompletionoftheanalyticstatesdefinesan invariant Hilbert space o(cid:0)f the irreduc(cid:1)ible representation. consistent with ωrs = 21ωmnlmnrs. Their commutators satisfy the Lorentz algebra Therefore the Cauchy completion coincides with . R H Wave functions Ψ which differ only in a set of vanish- [l ,l ]= η l +η l +η l η l . (24) kl mn km ln kn lm lm kn ln km ing measure correspond to the same vector in because − − H ΨΨ = 0 if and only if Ψ = 0, so the value of the wave So lij rotates the basis vectors ei to ej (i,j 1,2,3 . h | i ∈ { } functioninapointisirrelevant. However,continuouswave The infinitesimal boost l maps e to e , moves points 0i 0 i − functions cannot be changed in a point to some other p by (l p)j = p0δj , p0 = m2+p~2, and changes wave 0i i − equivalent continuous function as each equivalence class functions by p0∂pi. p of functions which coincide nearly everywhere contains at The calculation of the Wigner rotation W(L ,p) of a q most one and then unique continuous function. States on boost Lq in terms of the lengths of ~q and p~ and their in- which the generators of an analytic transformation group cluded angle is a ‘Herculean task’ [4] or requires ‘tedious actbydifferentialoperatorsareevenmorerestrictedtobe manipulations’ [5]. The result is stated in [4] or e.g [6]. analytic functions ofthe orbitandarealreadydetermined We derive it comparing the products of matrices la and w in each orbit by their behavior in a neighbourhoud of one whichrepresentinSL(2,C)[3,chap. 6]boostsL withra- p point. pidity a = artanh(p~/p0) in direction ~n and rotations W | | The expansion of (17) gives UΨ (p)+(ωp)i∂ (UΨ)(p) around an axis ~n by an angle δ (a′ =a/2 , δ′ =δ/2) i to first order and in the latter term we can replace UΨ by Ψ yielding Ψ(p) iωmnM (cid:0) Ψ(p(cid:1))+(ωp)i∂ Ψ(p)1. la =cha′ sha′~na ~σ , w=cosδ′ i sinδ′~n ~σ . (25) − 2 mn i − · − · A rotation D = eω coincides with its Wigner rotation The products of two boosts l l and of a boost l with a b a −c W(D,p), DL =L D. Let its unitary representationbe p Dp rotation w yield givenbyR(D)=exp1ωijΓ withantihermiteanmatrices, 2 ij Γij = −Γ†ij = −Γji, which represent the Lie algebra of chb′−shb′~nb·~σ cha′−sha′~na·~σ = rotations (i,j,k,l 1,2,3 ) (chb′cha′+shb′sha′~n ~n )1 ∈{ } (cid:0) (cid:1)(cid:0) a· b − (cid:1) (cha′ shb′~n +chb′ sha′~n ) ~σ [Γij,Γkl]=δikΓjl δilΓjk δjkΓil+δklΓik . (21) − b a · − − − i shb′ sha′(~n ~n ) ~σ , a b So the right side of (17) is Ψ + 12ωijΓijΨ up to higher chc′+−~nc ~σshc′ cosδ×′ isi·nδ′~n ~σ = (26) orders and we obtain · − · (chc′cosδ′ ishc′ sinδ′~n ~n)1+ (cid:0) (cid:1)(cid:0) c (cid:1) − · iM Ψ (p)= pi∂ pj∂ Ψ(p)+Γ Ψ(p) . (22) +shc′(cosδ′~n +sinδ′~n ~n) ~σ − ij − pj − pi ij c c× · − i chc′sinδ′~n ~σ . (cid:0)As the o(cid:1)perators(cid:0)(J1,J2,J3) =(cid:1) (M ,M ,M ) gen- − · 23 31 12 erate rotations, they are the components of the angular Both products are equal, l l = l w (16), if the com- b a −c momentum J~. They consistoforbitalangularmomentum plex coefficients of 1 and of the σ-matrices match as the L~ = ip ∂p and spin S~ contributed by the matrices iΓ, matrices are linearly independent. − × J~ = L~ +S~. Orbital angular momentum commutes with The coefficients of 1 agree only if ~n ~n = 0. The axis c · spin, [Li,Sj]=0. ~n of the Wigner rotation is orthogonalto the direction~n c of the resulting boost and~n lies in the plane spanned by c ~n and ~n because l and l are in the subgroup SO(1,2) a b b a 1Thesummationindexienumeratescoordinatesofthemassshell, of boosts and rotations in this plane. e.g. the spatialcomponents ofthemomentum. Thoughthecompo- With ~n ~n = cosϕ and ~n ~n =~n sinϕ the compa- nopenetrsataonrdisthineddeepreivnadteivnetsodfecpoeonrddionnattehse.cIotorisditnhaeterse,stthriectdioiffnerteonttihael rison of thbe·coaefficients of 1 aand×ofb~n ~σ yields Lmoarsesnsthzefllloowftohfefovuerc-tmorofimeeldntPum3m.=T0h(ωepd)emri∂vmatiwvehsic∂hmis,tmang=en0t,1t,o2t,h3e, (chc′)(cosδ′)=(chb′)(cha′)+(sh·b′)(sha′)(cosϕ) , hthoewyevaerre,ncoanfunnocttiboensapopflRie1d,3.separately to the wave functions Ψ as (chc′)(sinδ′)=(shb′)(sha′)(sinϕ) . (27) 3 Theratioofbothequationsdeterminesthelookedforangle ThestabilizerH ofpisgeneratedbyinfinitesimaltrans- formations ω, ηω = (ηω)T with ωm +ωm = 0 which δ sinϕ a b − 0 z tan = , k=(coth )(coth ) . (28) consequently are of the form 2 k+cosϕ 2 2 aT It has the same sign as ϕ, i.e. W(b,a) rotates in the di- ω(a,ωˆ)= a ωˆ a . (34) rection from~n to ~n . a b  −  aT The derivative of δ with respect to b at b=0 is   dδ a p~ HereaisaD 2-vectorandωˆ isa(D 2) (D 2)matrix, =(sinϕ) tanh =(sinϕ) | | . (29) − − × − db|b=0 2 p0+m whichgeneratesarotationw ∈SO(D−2). Becauseω(a,0) and ω(b,0) commute they generate translations in D 2 As the infinitesimal Wigner rotation of a boost from p to − dimensions. They arerotatedby w. So the stabilizerH is e rotates from p to e in the same sense as from e i i i theEuclideangroupE(D 2)oftranslationsandrotations − − to p it is represented by Γ pj/(p0+m). Combining both − ij in D 2 dimensions. infinitesimal changes, we obtain the generators of boosts − In D = 4 the inducing representation R of an irre- of wave functions ducible, massless unitary representation of the Poincaré pj group is an irreducible, unitary representation of E(2). iM Ψ (p)=p0∂ Ψ(p)+Γ Ψ(p) . (30) − 0i pi ijp0+m Such a representation is characterized by the SO(2) orbit of some point q R2 and a unitary representation Rˆ (cid:0) (cid:1) ∈ The generators (22, 30) are antihermitean with respect of its stabilizer Hˆ SO(2) [2]. If q = 0, the orbit is a to the scalar product (9) with the measure d˜p and rep- ⊂ 6 circle and the stabilizer is trivial. E(2) is represented on resent the Lorentz algebra (24). This is simple to check wave functions of a circle. Contrary to its denomination for [ iM , iM ] and manifest for [ iM , iM ], be- ij kl ij 0k ‘continuous spin’ such a representationleads to integer or − − − − cause pi,∂ and Γ transform as vectors or products of pi ij halfinteger,thoughinfinitely many,helicities (the Fourier vectors under the joint rotations of momentum and spin; modes of the wave functions). These representations are [ iM , iM ]=iM holds as if by miracle, 0i 0j ij phenomenologically excluded as infinitely many massless − − statespergivenmomentummakethe specificheatofeach pk pl [p0∂ +Γ ,p0∂ +Γ ] cavity infinite. pi ikp0+m pj jlp0+m If q = 0 R2, then the orbit is the point q = 0 and = p0pi∂ + 1 (p0(p0+m)δ l pipl)Γ the translati∈onsin E(2)are representedtrivially. The sta- p0 pj (p0+m)2 i − jl − bilizer is H =SO(2). Each unitary, irreducible represen- (cid:0) pj 1 (cid:1) tation R of its cover or each ray representation is one di- p0 ∂ + (p0(p0+m)δ l pjpl)Γ + − p0 pi (p0+m)2 j − il mensionalandrepresentstherotationDδ bytheangleδby multiplicationwithR(D )=e−ihδ wherehisanarbitrary, (cid:0) pkpl (cid:1) δ + δ Γ δ Γ δ Γ +δ Γ real number, which characterizes R. (p0+m)2 ij kl− il kj − kj il kl ij InD =4thereisnoglobalsection2 withwhichtorelate =− −(pi∂pj(cid:0)−pj∂pi)+Γij . (cid:1) (31) wave functions in HR (7) to wave functions of M0 (14) Such a section cannot exist as S3 = S2 S1 is the Hopf For d(cid:0)ifferent masses the rep(cid:1)resentations U of transla- 6 × a bundleofcircleswhereeachcirclewindsaroundeachother tions are inequivalent as the spectra of P differ. The rep- circle [8]. resentations U of the Lorentz group on states with mass Λ SoweusetwolocalsectionsN andS whicharedefined m>0,however,areunitarilyequivalentbythescaletrans- p p in the north outside = (p ,0,0, p ) and the formation U to the representation on states Ψ with unit UN A− { | z| −| z| } m south outside = (p ,0,0, p ) mass p0 = 1+p~2, US A+ { | z| | z| } p (UmΨ)(mp)=mΨ(p) . (32) UN ={p∈M0 :|p~|+pz >0} , (35) = p : p~ p >0 . S 0 z U { ∈M | |− } Themassshell ofamasslessparticle,theorbitunder 0 M the properLorentzgroupofalightlikemomentumsuchas The section N =D B boosts (1,0,0,1)in 3-direction p p p p=(1,0,0,...1), is the manifold SD−2 R, toB p=(p~,0,0, p~)andthenrotatesbythesmallestan- × p | | | | gle namely by θ around the axis ( sinϕ,cosϕ,0) along a 0 = (p0,p~):p0 = p~ >0 , p~ SD−2 R , (33) great circle to p= p~(1,sinθcosϕ−,sinθsinϕ,cosθ). With M { | | ∈ × } | | θ′ = θ/2, tanθ′ = (p~ p )/(p~ +p ) and p′ = p~ as p~ = 0 is specified by its direction and its nonvanishing | |− z | | z | | 6 modulus p~ =eλ >0. The tip of the lightcone p=0 does p p | | not belong to the orbit. Already the differential operators 2Only in the 4 cases D−1 = 1,2,4,8 does the group manifold |p~|∂pi, which generate the flow of boosts, are not analytic SO(1,D−1)∼SO(D−1)×RD−1 factorizeintoM0×E(D−2)∼ there. SD−2×SO(D−2)×RD−1 [7]. 4 the preimage of N in SL(2,C) is As R(eiδ)=e−ihδ, the representation ihp /(p~ +p ) p y z − | | of the infinitesimal Wigner rotation accompanies e.g. the cosθ′ p′sinθ′e−iϕ infinitesimal rotation in the 3-1-plane. np = sinp′θp′′eiϕ − p′cosθ′ ! AnalogouslyonedeterminestheWignerangleofaboost 1 √|p~|+pz px−ipy (36) tanδ = −(nxpy−nypx) , = √2|p~|p√x|+|p~p~i||p+ypz −√√|p~|p~|+|+ppzz . 2 d(δ|p~|+p=z)−(c(ontxhpβ2y)−+nnyzp|xp~)|+. ~n·p~ (42)   dβ p~ +p One easily checks that n transforms pˆ = 1 σ3 to |β=0 | | z p − nppˆ(np)† = |p~|−~p·~σ = pˆ [3, (6.63)] In UN the section and the generatorsof boosts, bearing in mind that −iM0i np is analytic in p. It is discontinuous on A−, as the limit boosts from e0 to −ei, θ π depends on ϕ. → iM Ψ (p)= p ∂ p ∂ +ih Ψ (p) , The section − 12 N − x py − y px N p sp = copss′θinp′′eθ−′ iϕ p−′cpo′ssiθn′eθiϕ′ ! (cid:0)(cid:0)−−iiMM3321ΨΨ(cid:1)(cid:1)NN((pp))==−−(cid:0)(cid:0)ppzz∂∂ppyx −−ppyx∂∂ppzz −+iihh(cid:1)|p~p~|p++xyppz(cid:1)ΨΨNN((pp)),, = √12|p~√|p√x|p~−||p~−i|p−pyzpz −p√x+|p~i|p−ypz (37) (cid:0)−iM01Ψ(cid:1)N(p)= |p~(cid:0)|∂px −ih|p~|p+ypz Ψ|N|(p) ,z(cid:1)  |p~| √|p~|−pz  (cid:0)−iM02Ψ(cid:1)N(p)=(cid:0)|p~|∂py +ih p~ p+xpz(cid:1)ΨN(p) , is defined and analytic in . The corresponding south- | | US (cid:0) iM Ψ(cid:1) (p)=(cid:0)p~∂ Ψ (p) . (cid:1) (43) ern section Sp rotates Bpp in the 1-3-plane by π to − 03 N | | pz N |p~|(1,0,0,−1)andthenalongagreatcirclebythesmallest I(cid:0)n particu(cid:1)lar the helicity, the angular momentum p~ J~/p~ angletop,namelybyπ θ around~n′ =(sinϕ, cosϕ,0). · | | − − in the direction of the momentum ~p, is a multiple of 1, In theircommondomaintwo sectionsdiffer by the mul- namely the real number h, tiplicationfromtheright(4)withamomentumdependent matrix in the SL(2,C) preimage of H =E(2) (p M +p M +p M )Ψ (p)=h p~ Ψ (p) . (44) x 23 y 31 z 12 N | | N e−iδ/2 0 (cid:0)Outside thedifferentialan(cid:1)dmultiplicativeoperators w = a eiδ/2 . (38) iM satAis−fytheLiealgebra(24)ofLorentztransforma- (cid:18) (cid:19) − mn tions for each value of the real number h. s differs in from n by a rotation around the p UN ∩US p Butthe functions(MmnΨ)N arenotdefinedon − and 3-axis by 2ϕ, the area ofthe zone (spherical biangle)with A seem to contradict Mackey’s results that analytic states vertices~e and ~e and sides through~e and p~/p~, z − z x | | in R are transformed to each other (12). One cannot H require the wave functions Ψ to vanish on because e−iϕ N A− sp =np eiϕ , rotations map them to functions which vanish elsewhere. (cid:18) (cid:19) One also cannot turn a blind eye to the singularity of p +ip p +ip (39) eiϕ = x y = x y . (43) taking unwarranted comfort in the misleading argu- p2 +p2 p~ pz p~ +pz ment [5, 9] that one can change a wave function in a set x y | |− | | of vanishing measure. The problem is not a set of mea- q p p The angle δ of the Wigner rotation W(Λ,p), which ac- sure zero but the factor py/(p~ +pz) which near to − | | A companies a Lorentz transformation, ΛNp = NΛpW(Λ,p) grows as 2|pz|py/(p2x + p2y) proportional to pz/r where (16) canbe readoff fromthe 22-elementofthe product of r =(p2+p2)1/2istheaxialdistance. FordifferentiableΨ x y N the SL(2,C) preimage λ with np, the derivative terms of M13Ψ stay bounded and the inte- grandof M Ψ M Ψ (9)divergesas4h2 Ψ 2(p /r)2. 13 13 N z λn =n w(Λ,p) , (40) h | i | | p Λp Cutting out a tube of radius ε around on and inte- − grating in polar coordinates d3p = dp (r dAr)dϕ the in- as δ/2 is the phase of (λn ) = (n w) eiδ/2 (36, 38). z p 22 | Λp 22| tegral over r ε diverges if ε goes to zero, h2 1dr/r = If Λ is arotationby α aroundthe axis~n then λ isgiven ≥ ε h2lnε . by (25) and the Wigner angle δ and its infinitesimal value − →∞ R If Ψ does not vanish on then for M Ψ to ex- are − 13 A k k ist Ψ must not be differentiable there. The derivative N tan2δ = (|p~|+pz)(ncoz|tp~α2|+)+~nn·xp~py−nypx , (41) steinrgmula−rpitzy∂pwxhhicahs itsoaclsaonclienlenareairnApz−. tThehismcualtnipcelilclaattiiovne ddαδ|α=0 = nz||p~p~||++p~nz·p~ . (cid:0)∂∂ppyx+−2i2hihpxp/y(/p(2xp2x++p2y)p2yf)(cid:1)=f 0=de0teramndin,efofr(pMx,2p3yΨ)=toe−e2xiihsϕt,. 5 (cid:0) (cid:1) IfΨ issquareintegrabelandanalyticin ,thenM Ψ R(eiδ) = e−ihδ of rotations around the 3-axis. However, N R mn U is square integrable and analytic if and only if in to be induced by an irreducible representationon sections N S U ∩U the function Ψ = e−2ihϕΨ is the product of the gauge over an orbit does not make the representation of SO(3) N S transformation e−2ihϕ times a function Ψ , which is ana- irreducible. S lytic in , To determine its invariant subspaces we recall that S U Ψ (p)=e2ihϕΨ (p) . (45) eachangularmomentum multiplet with total angularmo- S N mentum j contains a state Λ which is an eigenvector of Multiplying (43) with the transition function e2ihϕ and J = M with eigenvalue j and which is annihilated by using (45) one obtains 3 12 J =M +iM + 23 31 iM Ψ (p)= p ∂ p ∂ ih Ψ (p) , − 12 S − x py − y px − S (M j)Λ=0, (M +iM )Λ=0 . (48) (cid:0) iM Ψ(cid:1) (p)= (cid:0)p ∂ p ∂ +ih(cid:1) py Ψ (p), 12− 23 31 − 31 S − z px − x pz p~ p S (cid:0)−iM32Ψ(cid:1)S(p)=−(cid:0)pz∂py −py∂pz −ih |p~|p−xpz(cid:1)ΨS(p), Bcoym(e43ea)stihlyesseolavraebldeiffifewreentcioanlseidqueratΛioNnsasfoarfΛunNc.tiTonheoyftbhee- | |− z complex stereographic coordinates (cid:0) (cid:1) (cid:0) py (cid:1) iM Ψ (p)= p~∂ +ih Ψ (p) , − 01 S | | px p~ p S p +ip p ip | |− z w= x y , w¯ = x− y , (49) (cid:0)−iM02Ψ(cid:1)S(p)=(cid:0)|p~|∂py −ih p~ pxp (cid:1)ΨS(p) , |p~|+pz |p~|+pz z (cid:0) iM Ψ(cid:1) (p)=(cid:0)p~∂ Ψ (p) .| |− (cid:1) (46) which map the northern domain to C. Then the differen- − 03 S | | pz S tial equations read T(cid:0)hesameg(cid:1)eneratorsensueif,alongthelinesofthederiva- (w∂ w¯∂ +h j)Λ =0, (w2∂ +∂ +hw)Λ =0 . tion of (43), one reads the Wigner angle δ from the phase w w¯ N w w¯ N − − of the 12-element (λs ) = (s w) eiδ/2. (50) p 12 Λp 12 The transitionfunction e2−ih|ϕ is defin|ed andanalyticin Recollecting that w∂w measures the homogeneity in w, onlyif2h Zisinteger. Thisiswhythehelicityof w∂w(w)r = r(w)r, the first equation is solved by ΛN = UN∩US ∈ wj−hg(w2) and the second equation implies g to be ho- amasslessparticleisintegerorhalfinteger. Foreachother | | mogeneous in 1+ w2 of degree j, value iM are not antihermitean operators in Hilbert mn | | − − space. (j h+(cid:0)h)g+((cid:1)w2+1)g′ wj−h+1 =0 , For helicity h = 0 each continuous momentum wave − | | 6 functionΨhastovanishalongsomeline,aDiracstringin (cid:0) wj−(cid:1)h (51) Λ (w,w¯)= . momentumspace,fromp=0toinfinity. Namely,ifonere- N (1+ w2)j | | movesthe set , whereΨ vanishes,fromthe domains N and thenthNeremainingsets ˆ and ˆ cannotbothUbe ThestateΛisanalyticonlyifj hisanonnegativeinteger. simpUlySconnected. Because in ÛNthe phUaSse of Ψ is con- It is square integrable with res−pect to the measure N N U tinuous and its winding number along a closed path does not change under deformations of the path in ˆN as this dΩ=dcosθ dϕ=dwdw¯ 2i (52) U (1+ww¯)2 number is integerandcontinuous. Fora contractible path this windingnumber vanishes,becausethe phasebecomes ifalsoj+hisnonnegativewhichisjusttherestrictiontobe constant upon the contraction. If ˆ is simply connected UN analytic also in southern stereographic coordinates. They thenitcontainsacontractiblepatharound whichalso lies in ˆ . On this path the phase of Ψ Ah+as vanishing are related to the northern coordinates in their common US N domain by inversionat the unit circle, winding number but the phase of Ψ = e2ihϕΨ winds S N 2h-fold. So the path cannot be contractible in UˆS. w′ = px+ipy = 1 , w¯′ = px−ipy = 1 , (53) Restrictedtothesubgroupofrotationstheinducedrep- p~ p w¯ p~ p w z z resentation is a direct integral of representations. In co- | |− | |− ordinates E = p~ and coordinates u for the sphere S2 of and Λ is analytic only if j+h is a nonnegative integer S | | directions(withrotationinvariantsurfaceelementdΩ)the scalar product Λ (w′,w¯′)=( w 2hΛ = w′j+h . (54) S w N (1+ w′ 2)j 1 ∞ | | | | ΦΨ = dE E dΩ Φ∗(E,u)Ψ(E,u) (47) (cid:1) h | i 2(2π)3 Z0 ZS2 So j ≥|h|: there is no round photon with j =0. The representation R(eiδ) = e−ihδ of SO(2) induces in revealsthe Hilbert space as tensor product (R) (S2) the space of sections over the sphere no SO(3) multiplet where (R), the space of wave functionsHof the⊗eHnergy with j < h and one multiplet for j = h, h +1,..., E, is leHft pointwise invariant under rotations and (S2) | | | | | | H is the space of states on the unit sphere. In (S2) the 0 if j < h representationofrotationsisinducedbythereprHesentation nh(j)= 1 if j =|h|, h +1,... . (55) (cid:26) | | | | 6 The multiplicity n (j) exemplifies the Frobenius reci- The generatorsof the Poincarétransformationof the con- h procity [2] for G=SO(3) and H=SO(2): The representa- tinuum basis are the transposed of the generators on the tion h of the subgroupH induces eachrepresentationj of wave function, i.e. their derivative part has the reversed thegroupGwiththemultiplicitym (h)withwhichthere- signfrompartialintegrationandthemultiplicativepartis j strictionofjtothesubgroupH containsh,m (h)=n (j). transposed. This is compatible with the Lorentz algebra j h AcompleteSO(3)multipletwithspinsatfixedmomen- as U is linear and therefore commutes with derivatives Λ tum consists of helicities h= s, s+1,...s and induces and matrices which multiply the continuum basis. − − SO(3) representations j with multiplicity Admittedly, the transformation of the continuum basis (60)ismoreintuitivethan(17)andformanypurposesitis s 2j+1 if j s sufficienttoworkwithonecoordinatepatche.g. in and N N(j)= nh(j)= 2s+1 if j ≤s . (56) to forget the analyticity requirements (45) for theUstates h=−s (cid:26) ≥ X on which the generators (43) act. However, employing the continuum basis one is prone to subtle mistakes, e.g. This fits to the multiplicity N (j) with which total an- s⊗l to overlook that helicity states have angular momentum gular momentum j is contained in the product of spin s j which is at least h (55) or to miss the Dirac string with orbital angular momentum l, | | in momentum space on which continuous wave functions 1 if j s+l,s+l 1,... s l have to vanish. Ns⊗l(j)= 0 else∈. { − | − |} (57) UsingacontinuumbasisWeinbergandWitten[10]con- (cid:26) clude that each matrix element Φ jm(0)Ψ vanishes for h | i Aslrangesover0,1,2,... oneobtainsN(j)= N (j). states with h > 1/2 if jm(x) = eiPxjm(0)e−iPx is a con- l s⊗l | | Ifeachhelicitystatewouldinduceonemultipletforeachj, served current, 0 = ∂ jm(x) = i[P ,jm(x)] which trans- m m thenforintegerspinthestatesofthespinmultPipletwould forms as a field U jm(0)U−1 =Λ−1m jn(0). Λ Λ n induce 2s+1 singlets j =0 rather than N(0)=1. The conclusion follows from the vanishing of Throughoutquantumfieldtheoryoneusesacontinuum Γ jm(0)Γ in momentum eigenstates. Under rota- −p~ p~ h | i basis of momentum eigenstates Γ to map test functions tionsD aroundp~theytransformbyU Γ =e−ihδΓ and p δ D p~ p~ of to states Ψ in , U Γ =eihδΓ as each state has angular momentum h 0 D −p~ −p~ M H in direction of its momentum, so Ψ= d˜p Ψ(p)TΓ , Ψ(p)= Γ Ψ . (58) p h p| i U Γ U jm(0)U−1U Γ ZM D −p~ D D D p~ (62) Strictly speaking the eigenstatesΓp are distributions with =(cid:10) e−2ihδD(cid:12)(cid:12) −1mn hΓ−p~|jn(0)Γp~(cid:11)i scalar product and the matrix elements constitute an eigenvector of D with eigenvalue e−2ihδ. But a rotationD of a four vector Γ ΓT =(2π)32 m2+~p2δ3(~q ~p)1 . (59) δ q p − has only eigenvalues 1 and e±iδ. So, if 2h >1, then | | (cid:10) (cid:12) (cid:11) p and only i(cid:12)ntegrals of the continuum basis with test func- Γ jm(0)Γ =0 . (63) tions arevectorsΨin with the scalarproduct(9). This h −p~| p~i H is why we work directly with the wave functions. More- The catch in the argument is that there is no basis of overthestatesofmasslessparticlesarenofunctionsof 0 statesΓ forall momenta. Thedomain doesnotcover M p~ N but sections of a nontrivial bundle. To describe them in U the south pole. Rotations around some axis ~n are not a continuum basis a proper notation has to use Γ or p,N defined by (17) in for momenta which lie in the south Γ in their appropriate domains with Γ = e2ihϕΓ UN p,S p,N p,S pole’sorbitundertheserotations. Inparticular,thestates in andwiththerestrictiononanalyticstatesthat N S Γ and Γ are separatedby this orbitand do not lie in a U ∩U p~ −p~ their wave functions be analytic in and and be re- N S common coordinate patch in which U acts smoothly. U U D lated by (45). We show that the conclusion of [10] is wrong and con- Thecontinuumbasistransformscontragradientlytothe struct a conserved current for arbitrary helicity h. For wave functions Ψ (17) this purpose we enlarge the Hilbert space by an auxiliary multiplet with helicity h+1/2 with states Ψ such that U Γ =RT(W(Λ,p))Γ (60) + Λ p Λp states are two component column vectors (Ψ ,Ψ ). On + h these states we define in northern coordinates generators (if p and Λp are covered by the same coordinate patch) Q ,Q ,Q¯ = (Q )† and Q¯ = (Q )† to multiply the wave suchthatU Ψcorrespondsto the transformedwavefunc- 1 2 1˙ 1 2˙ 2 Λ functions Ψ(p) with tion, p ip d˜p Ψ(p)TUΛΓp = d˜p Ψ(p)TRT(W(Λ,p))ΓΛp Q1(p)=− xp~−+pyz a† , Q2(p)= |p~|+pza† , Z Z (61) | | p (64) p +ip = d˜p (UΛΨ)T(Λp) ΓΛp . Q¯1˙(p)=−pxp~ +py a , Q¯2˙(p)= |p~|+pza . z Z | | p 7 p where a† and a are the matrices 3 With help of the supersymmetry generators we specify a current jm of charge q 0 1 0 0 a† = , a= . (65) 0 0 1 0 (cid:18) (cid:19) (cid:18) (cid:19) Φjm(0)Ψ = d˜pd˜p′ Φ∗(p)jm(p,p′) Ψ(p′) , Q and Q¯ commute with Pm and satisfy the supersym- h | i α α˙ Z (74) metry algebra jm(p,p′)=σ¯mβ˙α Q (p)Q¯ (p′)+Q¯ (p)Q (p′) f α β˙ β˙ α {Qα,Qβ}=0={Q¯α˙,Q¯β˙}, {Qα,Q¯β˙}=σαmβ˙Pm1, (66) (cid:16) (cid:17) wheref denotesafunctionof(p p′)2 withf(0)=q. The and the Weyl equation (σ¯0 =σ0,σ¯i = σi) current is conserved, 0 = p′ jm(−p′,p) = p jm(p′,p) (67). − m m The integralof the density j0(x)=eiPxj0(0)e−iPx yields Q¯ (p)σ¯mβ˙αp =0 , p σ¯mβ˙αQ (p)=0 . (67) β˙ m m α the charge q, They areanalyticin as(p +p )±1/2 isanalyticthere. N z On the negative 3-aUxis |th|eir phase is discontinuous d˜pd˜p′d3x Φ∗(p)ei(p−p′)xj0(p,p′)Ψ(p′) − A (px+ipy)/ p~ +pz =eiϕ p~ pz (39). Z (75) As (Q1,Qp2|) i|s proportiopna|l t|o−the second column of np = d˜pj0(p,p)Φ∗(p)Ψ(p) , (36) Q satisfy (40) 2p0 α Z DαβQβ(p)=Qα(Λp)eiδ(Λ,p)/2 (68) because the supersymmetry algebra Q ,Q¯ = P σm { α β˙} m αβ˙ whereD(Λ)istheSL(2,C)representationofΛandδ(Λ,p) implies jm(p,p)=2qpm. the Wigner angle. This property,the induced transforma- The multiplet with h +1/2 is auxiliary, because only tion of helicity states the h-h-components of the 2 2 matrices Q Q¯ are × α β˙ needed for the current. Explicitly the 4 components of (UΛΨ)(Λp)=Ψ(p)e−ihδ , (69) jm(p,p′)/f((p p′)2) are − and that Q is a multiplicative operator allow to simplify α p +ip p′ ip′ Dα−1β(QβUΛΨ)(Λp)=Dα−1βQβ(Λp)(UΛΨ)(Λp) xp~ +py xp~−′ +yp′ + |p~|+pz |p~′|+p′z , =Qα(p)e−iδ/2Ψ(p)e−ihδ (70) ppx| +| ipyz p|p~′| +pz′ + pp~ +p pp′x−ip′y , and show (reading (69) for helicity h+1/2 backwards) |p~|+pz p| | z p| | z |p~′|+p′z (76) Dα−1β(QβUΛΨ)(Λp)=(UΛQαΨ)(Λp) . (71) −ippxp~++ippy |p~′|+p′z +i |p~|+pzpp′xp~−′ +ip′yp′ , ScoonQjuαgaatreertehperecsoemntpaotinoenn)t,s of a SL(2,C) spinor (D¯ is the ppx|+| ipyz pp′x−ip′y + pp~ +p pp|~′ |+p′z. − p~ +p p~′ +p′ | | z | | z U Q U−1 =D−1βQ , U Q¯ U−1 =D¯−1β˙Q¯ , (72) | | z | | z p p Λ α Λ α β Λ α˙ Λ α˙ β˙ p p allowing to check explicitly jm(p,p)=2qpm and the con- and carry helicity 1/2. servation 0 = p jm(p,p′) = p′ jm(p,p′). Neither the su- m m The supersymmetry generators map analytic sections persymmetrygeneratorsnorthecurrentjm dependonthe with helicities h+1/2 and h to each other as they pro- valuehofthehelicity. Thereisnoobstructiontoconserved vide the relative phase e±iϕ, or a factor p~ +p which z currentswithnonvanishingmatrixelements instateswith | | vanishes on , to make ei(2h+1)ϕ(QΨ) and e2ihϕ(QΨ) A− +p h arbitrary helicity. analytic in . US Massless states do not allow hermitean spatial position In southern coordinates the generators of supersymme- operators Xi which generate commuting translations of try are related to (64) by the spatial momentum [11, 12]. By the Stone von Neu- Q (p)=eiϕQ (p) , Q¯ (p)=e−iϕQ¯ (p) , mann theorem [13] they act analytically on the subspace αS α β˙S β˙ ofsquareintegrableanalyticfunctionsofR3. Thisspaceis p +ip Q (p)= p~ p a† , Q (p)= x y a† , notmappedtoitselfbyLorentztransformations. Theyact 1S z 2S − | |− p~ pz analytically on analytic sections over S2 R. So there is p | |− × Q¯ (p)= p~ p a , Q¯ (p)= ppx−ipy a . nocommondenseandinvariantsubspaceofanalyticstates 1˙S − | |− z 2˙S p~ p on which Lorentz transformationsand translations of mo- z p | |− (73) mentumcanbeexpandedandonwhichthegeneratorscan p be applied repeatedly. 3With a matrix representation of several fermionic creation and For h 6= 0, the partial derivatives of of ΨN and ΨS are annihilation operators a†i and aj, {ai,aj} = 0, {a†i,aj} = δij, one not in the Hilbert space of square integrable functions as obtains thegenerators Qαi,Q¯α¯j ofextended supersymmetry. thederivationof(45)shows. Welldefinedarethecovariant 8 derivatives D =∂+A and D =e2ihϕD e−2ihϕ, N N S N ∂ p px ih y D = ∂ p , N  py− p~(p~ +p )− x ∂pz N | | | | z 0 (77)     ∂ p px ih y D = ∂ + p . S  py p~(p~ p )− x ∂pz S | | | |− z 0     But their commutator yields the field strength of a momentum space monopole of charge h at p=0, pk [D ,D ]=F =∂ A ∂ A =ihε . 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