Continuous Spin Representation from Contraction of the Conformal Algebra
aa r X i v : . [ h e p - t h ] F e b Continuous Spin Representation from Contraction of theConformal Algebra
Abu Mohammad Khan ∗ Department of Computer Science & EngineeringBRAC University46 Mohakhali, Dhaka -1212, BangladeshFebruary 08, 2021
Abstract
In this paper, we discuss the In¨on¨u-Winger contraction of the conformal algebra.We start with the light-cone form of the Poincar´e algebra and extend it to writedown the conformal algebra in d dimensions. To contract the conformal algebra, wechoose five dimensions for simplicity and compactify the third transverse directionin to a circle of radius R following Kaluza-Klein dimensional reduction method. Weidentify the inverse radius, 1 /R , as the contraction parameter. After the contraction,the resulting representation is found to be the continuous spin representation in fourdimensions. Even though the scaling symmetry survives the contraction, but thespecial conformal translation vector changes and behaves like the four-momentumvector. We also discussed the generalization to d dimensions. PACS: 20-a, 11.25Hf, 11.25-w, 11.30Cp
Recently the interest in continuous spin representation (CSR) of the Poincar´e group hasgrown tremendously. The eigenvalues of the Casumir operators for these repesentationsin d dimensions are the momentum squared which is zero and the non-zero values of thesquares of the Pauli-Lubansky forms[1]. In four dimensions, these are the four-momentumsquared and the square of the Pauli-Luba´nski vector. These representations are alsoknown as the Wigner’s infinite spin representation in the literature. The CSRs did notattract much attention until recently mainly because no known particles seem to obey thisrepresentation, and also because of the ‘No-Go Theorem’ by Weinberg and Witten[2].However, in 2002 Brunetti et.al. [3] showed the possibility of localization of the CSRfields in spacelike cones. This observation renewed the interest in CSR of the Poincar´egroup. In 2005, Mourad[4] showed that the CSRs may arise in the context of String theory.Metsaev[5, 6, 7, 8] argued their existence in different contexts. In 2015, Longo et.al. [9] andSchroer[10] showed that the CSRs may have only non-point like localization as in StringTheory. A beautiful discussion on incompatibility of CSRs with point-like localizationis presented by K¨ohler[11]. Rehren[12] also constructed a string-localized infinite spinquantum stress-energy tensor even without a classical action.On the other hand, Schuster and Toro[13, 14] proposed the action principle for theparticles with integer CSRs, and later Bekaert et.al. [15] presented the action for particleswith half-odd integer CSRs. Very recently Najafizadeh[16] presented the supersymmetrytransformation to construct the supermultiplets associated with the CSRs. Buchbinder ∗ Email: [email protected], [email protected] t.al. [17, 18] developed the BRST method to construct the Lagrangian for bosonic fields.They also obtained the field realization of the infinite spin N = 1 supersymmetry. UsingKirillov method, Gracia-Bond´ıa et.al. [19] quantized the fields. These developments andother approaches has been concisely summarized by Bekaert and Skvortsov in a recentarticle[20].The rapid progress in the area of CSR after the work of Brunetti et.al. is very en-couraging for us and it motivates us to think how do these infinite spin representationsarise from a theory that has conformal symmetry or AdS symmetry, like String Theory.It is very well known for a long time that the Conformal group contains the Poincar´egroup as a sub group, and it only admits regular massless representations of the Poincar´egroup[21, 22, 23]. This clearly states that the CSR can not be obtained by any non-singulartransformations. Hence, only other possibility is to consider a singular transformation, likegroup contraction, to find out if the conformal group give rise to the CSR in lower dimen-sions.In an earlier paper[22], we discussed how the CSR can be obtained in lower dimensionsby contracting the Poincar´e group in higher dimensions. In that paper, we used the In¨on¨u-Winger group contraction method[24] by considering the inverse radius of the Kaluza-Klein(KK) compactification[25, 26] as the contraction parameter. It is also very well-known that the contraction of AdS groups[27] give rise to the Poincar´e group with masslessrepresentation in lower dimensions. However, the contraction of the conformal group hasnot yet been discussed. Naively it is not surprising to expect that the contraction ofthe conformal group would only yield regular massless representations. However, in thispaper, we have found that it gives rise to the CSR as well.In this paper, we first consider the conformal algebra in five dimensions in light-coneform, apply the Kaluza-Klein(KK) dimensional reduction, followed by the IW contractionto get the generators in four dimensions. We show that after the contraction, the confor-mal symmetry is lost. In fact, the special conformal translation vectors become a linearcombination of the four-momentum vectors and the light-cone translation vectors. Theresulting algebra and the symmetry properties are those of the continuous spin represen-tations of the Poincar´e group in four dimensions. We also discuss the generalization to d dimensions.We organize the paper as in the following ways: in Section-2, we review the light-coneformulation of the Poincar´e algebra and its conformal extension, in Section-3, we developthe contraction parameter by KK compactification and use it to contract the conformalalgebra to obtain the CSR, and finally in Section-4, we discuss the result and make theconcluding remarks. In this section, we briefly present the conformal algebra in the light cone form. We beginwith the Poincar´e algebra in d dimensions and then extend it by adding generators ofscaling and special conformal transformations to complete the conformal algebra in d dimensions. Since the conformal symmetry requires the particles to be massless, onlythe massless representation of the Poincar´e group can be embedded in the conformalrepresentation. In this subsection, we present a short review on the light-cone form of the Poincar´e algebraintroduced by Dirac[28] and its isomorphic form in a infinite momentum frame shown byBacry and Chang[29]. 2n d -dimensions, the Poincar´e group generators, P µ and M µν , satisfy the followingcommutation relations,[ P µ , P ν ] = 0 , (1)[ M µν , P α ] = i ( η µα P ν − η να P µ ) , (2)and h M µν , M αβ i = i (cid:16) η µα M νβ + η αν M βµ + η νβ M µα + η βµ M αν (cid:17) , (3)where µ, ν, α, β = 0 , , , · · · , ( d − η µν = − + + · · · +. Therepresentations are characterized by the eigenvalues of the Casimirs: P ≡ P µ P µ and thesquares of the Pauli-Luba´nski k -forms which are given by[1] W ( k ) = ǫ µ µ ··· µ k µ k +1 µ k +2 µ k +2 ··· µ d − µ d P µ d M µ k +1 ν k +2 · · · M µ d − µ d − q k !2 ( d − k +1) / (( d − k − / d − k − / , where k = 1 , , · · · , ( d −
3) if d is even and k = 0 , , · · · , ( d −
3) if d is odd. In momentumspace the canonical representation of the rotation and boost generators of the Poincar´egroup are given by M ij = − i (cid:16) p i ∂ jp − p j ∂ ip (cid:17) + S ij , (4)and M i = − ip ∂ ip + (cid:18) − p + m (cid:19) p j S ij , (5)respectively, where i, j = 1 , · · · , ( d −
1) and ∂ ip = ∂∂p i . The generators S ij satisfy the following commutation relation h S ij , S mn i = i (cid:16) η im S jn + η mj S ni + η jn S im + η ni S mj (cid:17) . For simplicity, we restrict ourselves to d = 4 dimensions. Following Dirac, we introducethe light-cone coordinates x ± = 1 √ (cid:16) x ± x (cid:17) and p ± = 1 √ (cid:16) p ± p (cid:17) . The commutation relations now read[ x − , p + ] = − i and [ x a , p b ] = iη ab , where a, b = 1 , P + = p + , P a = p a and P − = p a p a + m p + . (6)Here P − is called the light-cone Hamiltonian. The six spacetime generators of the Poincar´egroup now split up as M µν = (cid:16) M ij , M i (cid:17) → (cid:16) M + − , M ab , M + a , M − b (cid:17) , where i, j = 1 , ,
3. Using the canonical representations in Eqs.(4,5), we now compute thelight-cone forms of the generators. The M + − generator becomes, M + − = − M , = ip + ∂ + − √ p a S a p + + p − − √ m , (7)3here we used ∂∂p = √ p + p + + p − ∂∂p + , (8)because of the mass-shell condition in Eq.(6). The light-cone Hamiltonian, M − a , becomes M − a = 1 √ (cid:16) M a − M a (cid:17) , = 1 √ (cid:20) − ip ∂ ap + (cid:18) − p + m (cid:19) p j S aj + i (cid:16) p ∂ ap − p a ∂ p (cid:17) − S a (cid:21) , = − ip − ∂ ap − i √ p a ∂ p − √ − p + p + m − p + m ! S a + 1 √ p b S ab − p + m , = − ip − ∂ ap − i p a p + p + + p − ∂ + p − √ p − − mp + + p − − √ m ! S a − p b S ab p + + p − − √ m . (9)The generators, M + a , also change to the following forms, M + a = − ip + ∂ ap + i p a p + p + + p − ∂ + p + √ p + − mp + + p − − √ m ! S a − p b S ab p + + p − − √ m . (10)The transverse rotation generators, M ab , do not change. We now apply a boost alongthe third direction. The generators are mapped to the boosted frame and scaled in thefollowing ways p ± → b p ± = e iφM p ± e − iφM = e ± φ p ± , (11) p a → b p a = e iφM p a e − iφM = p a , (12) M ± a → c M ± a = e iφM M ± a e − iφM = e ± φ M ± a , (13) M ab → c M ab = e iφM M ab e − iφM = M ab , (14)and M + − → c M + − = e iφM M + − e − iφM = M + − , (15)where the rapidity parameter φ is related to the speed v of the boost by the followingrelation, e φ = s v − v . To go to the infinite momentum frame, we take the limit φ → ∞ . In this limit, thecanonical representations of the Poincar´e algebra in light-come form in Eqs.(6,7,9,10) andthe transverse generator reduce to the following simple forms respectively (expressed inthe usual position space), b P + = b p + , b P a = b p a , b P − = ( b p a ) + m b p + , (16) c M + − = − b x − b p + , (17) c M − a = b x − b p a − { b x a , b P − } + 1 b p + ( b T a − b p b S ab ) , (18) c M + a = − b x a b p + , (19) c M ab = b x a b p b − b x b b p a + S ab . (20)Here the light-cone translation vector is defined as b T a ≡ mS a . The generalization to d dimensions can be obtained by extending the transverse direction upto ( d − i.e. , b = 1 , , · · · , ( d − S ab and the light-cone vectors b T a formthe transverse little group SO ( d −
2) and satisfy the following algebra h S ab , S cd i = i (cid:16) η ac S bd + η cb S da + η bd S ac + η da S cb (cid:17) , h S ab , b T c i = i (cid:16) η ac b T b − η bc b T a (cid:17) , and h b T a , b T b i = im S ab . For massless particles, we put m = 0 and b T a ’s become commuting vectors. The states canbe labelled by two ways: • b T a = 0. This is the regular massless representation. All known massless particlesbelong to this representation. The states satisfy b T a | ψ > = 0, and are labelled by the SO ( d −
2) little group, in addition to b p + and b p i . • b T a = 0. This is the CSR. The states satisfy b T a | ψ > = ξ a | ψ > , and are labelled bythe short little group SO ( d −
3) that leaves b T a invariant, b p + , b p a and the length ofthe light-cone vector, ξ a .In the following, we add the scaling operator and the special conformal translation operatorto the Poincar´e algebra, and complete the conformal algebra relations in d dimensions inlight-cone forms. We also drop the ‘ hat ’ symbols to represent the generators in the infinitemomentum frame for simplicity. The conformal symmetry is the symmetry for massless particles. Hence it is very naturalto assume that the massless representations of the Poincar´e group also have conformalsymmetry. Since the Conformal group is larger than the Poincar´e group, the massless rep-resentations of the Poincar´e group can reside inside the conformal group as an embedding.The scaling operator D , also known as the Dilatation operator, and the special con-formal translation vector K µ are given by D = 12 ( x · p + p · x ) , and K µ = 2 x ν M µν + x p µ = 2 x µ D + 2 x ν S µν − x p µ , where µ, ν = 0 , , , · · · , ( d − d dimensions, these two generators satisfy the following commutation relations[30],[ M µν , D ] = 0 , (21)[ D , p µ ] = ip µ , (22)[ D , K µ ] = − iK µ , (23)[ p µ , K ν ] = − i ( η µν − M µν ) , (24)and [ M µν , K α ] = i ( η µα K ν − η να K µ ) . (25)The commutation relations in Eqs.(1-3) and together with the commutation relations inEqs.(21-25) complete the algebra of the Conformal group in d dimensions.To obtain the light-cone form, we first express the generators, D and K µ , in light-conecoordinates, and apply a boost by setting x + = 0. We use the expressions from Eqs.(6-18)to write down the light-cone forms of these generators and get, D = 12 (cid:0) − (cid:8) x − , p + (cid:9) + { x a , p a } (cid:1) , (26)5 + = − x a x a p + , (27) K − = 2 x − D − x a x a P − + 2 p + (cid:16) x a T a − x a p b S ab (cid:17) , (28)and K a = 2 x a D − x c x c p a + 2 x b S ab , (29)where a, b = 1 ,
2. Now we compute the following commutators,[ p a , K − ] = − i (cid:18) x − p a − (cid:8) x a , P − (cid:9) + 1 p + (cid:16) T a − p b S ab (cid:17)(cid:19) , and [ P − , K a ] = 2 i (cid:18) x − p a − (cid:8) x a , P − (cid:9) − p + (cid:16) p b S ab (cid:17)(cid:19) . To satisfy the commutation relation in Eq.(24), the right-hand sides of the above twoequations must be negative of each other. This is possible only if T a = 0 in Eq.(18), andhence in Eq.(28). This clearly indicates that only the regular massless representation ofthe Poincar´e algebra has conformal extension. The light-cone boost now becomes, M − a = x − p a − { x a , P − } − p + p b S ab . The above results are, of course, not new. We present here for clarity and completenessof our work in this paper.In the following section-3, we will contract the conformal representation. Even thoughonly the regular massless representation has conformal extension, the contraction mayprovide different massless representation, namely the CSR.
In this section, we contract the conformal algebra following the procedure introduced byIn¨on¨u and Wigner[24]. Here we proceed by compactifying a direction on to a circle ofradius R , known as the KK dimensional reduction method[25, 26]. The inverse radius,1 /R , is considered to be the contraction parameter. When the contraction limit, R → ∞ or equivalently 1 /R → In their paper[24], In¨on¨u and Winger obtained the inhomogeneous Euclidean group E bycontracting the homogeneous SO (3) group. The rotation generators of the SO (3) groupare L ij for i, j = 1 , ,
3. Defining L i = ǫ ijk L jk , the generators satisfy the relation,[ L i , L j ] = iǫ ijk L k L + L + L = l ( l + 1) ,L = m , when acts on a state. The representation space is l ( l + 1) dimensional. Now we define, b L i ≡ ǫL i , where ǫ is an arbitrary contraction parameter. For a fixed m , we have, b L + b L + b L = ǫ l ( l + 1) . Now if we take the limits, ǫ → l → ∞ such that ǫl ≡ Ξ = fixed, we get, b L + b L = Ξ , b L = ǫL = ǫ m → ǫ → m . The vectors b L a with a = 1 ,
2, satisfy the following commutation relations[ b L , b L ] = 0 , [ L , b L ] = i b L and [ L , b L ] = − i b L , which is the algebra of the inhomogeneous Euclidean group E . The contraction parameteris arbitrary and has no obvious physical meaning. In the following, we apply IW contrac-tion by identifying ǫ as the inverse KK-radius to obtain the CSR form the conformalalgebra. We now apply the IW contraction to the generators of the Conformal group. For simplicity,we choose to work in five dimensions. So the light-cone little group is SO (3). In fivedimensions, there are three transverse directions. We compactify the third transversedirection, x , on to a circle of radius R , x = x + 2 πR , and hence the momentum along the third direction become p = nR , where n is the mode number, called the KK-mode. The light-cone Hamiltonian in Eq.(6)nowreads, P − = 12 p + p a p a + n R ! , a = 1 , . The term M n ≡ n /R is the mass term in four dimensions after compactification. Wenow evaluate all generators in light-cone form at x = 2 πR . The Poincar´e generatorsbecome[22] M +3 = − πR p + ,M a = 2 πR p i − nx a R + S a ,M − = nx − R − πR p − − p + (cid:16) p a S a (cid:17) , and M − a = x − p a − { x a , P − } + 1 p + (cid:18) nR S a − p b S ab (cid:19) . The scaling and special conformal transformation generators reduce to the following forms D = 12 (cid:0) −{ x − , p + } + { x a , p a } (cid:1) + 2 nπ ,K + = − x ⊥ p + − π R p + ,K − = 2 x − D + 4 nπx − − x ⊥ p − − π R p − − p + (cid:18) x a p b S ab + 2 πR p a S a − x a (cid:16) nR S a (cid:17)(cid:19) ,K a = 2 x a D − x ⊥ p a − π R p a + 2 x b S ab − πR S a , and K = 4 πRD − nR x ⊥ − π nR + 2 x a S a , where a, b = 1 ,
2, and we used x ⊥ = x a x a and also p ⊥ = p a p a . The remaining generatorsdo not change.We now define the light-cone translation vector, T a = (cid:18) nR (cid:19) S a . (30)7learly T a ’s and the generator S ab satisfy[ S ab , T c . ] = i ( η ac T b − η bc T a ) , [ T a , T b ] = n R S ab ≡ M n S ab . The commutation relations are the SO (3) algebra in four dimensions with mass M n ,which is known as the regular massive representation. Wigner called these regular massiverepresentations as the ‘ ’ faithful representations and denoted these by D ( j ) toclassify all irreducible representations of the Poincar´e group[31].To contract the algebra, we identify the inverse KK radius, 1 /R , as the contractionparameter. In the limit, 1 /R →
0, or equivalently R → ∞ , the KK mass term vanish, andhence the representation becomes massless. In the contraction limit, the various generatorsreduce to the following forms, c M − ≡ lim R →∞ M − R = − np + p a T a , c M a ≡ lim R →∞ M a R = 2 πp a − n T a , c M +3 ≡ lim R →∞ M +3 R = − πp + , c M − a ≡ lim R →∞ M − a = x − p a − { x a , P − } + 1 p + (cid:16) T a − p b S ab (cid:17) , c K + ≡ lim R →∞ K + R = − π p + , c K − ≡ lim R →∞ K − R = − π p − − πnp + ( p a T a ) , c K a ≡ lim R →∞ K a R = − π p a − πn T a , c K ≡ lim R →∞ K R = 0 . The remaining generators that are completely transverse to the compactified direction,such as M ab = x a p b − x b p a + S ab , do not change. From the above contraction of thealgebra, it is clear that, the special conformal translation vector reduces to the linearcombination of four-momentum vector p µ and the light-cone translation vector T a , andhence can be absorbed by relabelling or shifting the Poincar´e generators. The dilatationoperator picks up a constant term which can be absorbed easily. Since the representationbecomes massless after the contraction limits has been applied, it is expected that theDilatation operator would not vanish. The little group of the contracted representation isgenerated by the following algebra,[ S ab , T c . ] = i ( η ac T b − η bc T a ) , (31)[ T a , T b ] = 0 . (32)The above relation is the algebra of E group, and the Casimir eigenvalue is, T a T a = n R (cid:16) ( S ) + ( S ) + ( S ) (cid:17) − n R ( S ) , = n j ( j − R − n m z R , where m z is the eigenvalue of S . We now take the limits, R → ∞ , j → ∞ such that jR ≡ Ξ8emains constant. Hence, we obtain, in the contraction limit, T a T a = Ξ . (33)The states are now labelled by the values of m z , known as the helicity, and the length ofthe light-cone vector Ξ, and of course by p + and p a . This is known as the CSR, becausea finite boost on a state generates an infinite tower of equally spaced helicity states.We also obtain an extra term p a T a due to contraction. This gives the projection of thelight-cone translation vector along the transverse momentum direction. It is interestingto note that p a transform as a vector under the orbital rotation L ab , and T a transform asa vector under the internal rotation S ab with opposite results,[ L ab , p c T c ] = − [ S ab , p c T c ] = − i ( p b T a − p a T b ) . Hence, the term p a T a commute with M ab ≡ L ab + S ab and this means that the orthogonalpart of T a to the momentum vector p a is physically relevant.From the above analysis, the generalization to any higher dimension is trivial. Theindices of transverse directions now run as a, b, · · · = 1 , , , · · · , ( d − SO ( d − In this article we analyzed the IW-contraction of the conformal algebra in d dimensions,and discussed the conditions to obtain the CSR. Since the conformal group contains onlythe regular massless representations, we intuitively expected that the contraction of thealgebra would yield only regular massless representations. Instead, we obtain the CSRin the double limit as both R, j → ∞ such that j/R = Ξ is fixed. Even though theresult looks simple, but it is not obvious, because the IW contraction could be appliedby compactifying more than one direction. In that case, the contraction may or may notprovide the CSR.Here we have only considered to contract one direction. But from a higher dimension,there are different routes to contract to lower dimensions, and it is not obvious whichrepresentation will occur under contraction. It is still an open question. Of particularinterest is the AdS/CFT duality. It would very interesting to see the consequence ofgroup contraction on AdS/CFT duality.We also did not consider the contraction of the algebra generated by the SO (2 , d )or SO (1 , d + 1) in d dimensions. The generators of these groups also satisfy the samecommutation relations of the conformal algebra. It is yet not known if the contractionwould yield any CSR or not. I thank Pierre Ramond for introducing me into this work, and helping me to understandand insisting me to explore further.
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