On Light-like Deformations of the Poincaré Algebra
aa r X i v : . [ h e p - t h ] J a n On Light-like Deformations of the Poincar´e Algebra
Zhanna Kuznetsova ∗ and Francesco Toppan † January 4, 2019 ∗ UFABC, Av. dos Estados 5001, Bangu,cep 09210-580, Santo Andr´e (SP), Brazil. † CBPF, Rua Dr. Xavier Sigaud 150, Urca,cep 22290-180, Rio de Janeiro (RJ), Brazil.
Abstract
We investigate the observational consequences of the light-like deformations ofthe Poincar´e algebra induced by the jordanian and the extended jordanian classesof Drinfel’d twists. Twist-deformed generators belonging to a Universal EnvelopingAlgebra close nonlinear algebras. In some cases the nonlinear algebra is responsiblefor the existence of bounded domains of the deformed generators. The Hopf algebracoproduct implies associative nonlinear additivity of the multi-particle states. Asubalgebra of twist-deformed observables is recovered whenever the twist-deformedgenerators are either hermitian or pseudo-hermitian with respect to a common in-vertible hermitian operator.
CBPF-NF-002/18 ∗ e-mail: [email protected] † e-mail: [email protected] Introduction
This paper addresses the problem of the observational consequences of twist-deformingthe Poincar´e algebra. We work within a quantization scheme which has been previouslyapplied to Drinfel’d twist deformations of quantum theories in a non-relativistic setting .Open questions are investigated. We mention, in particular, the nature of the observables:which of them are consistently maintained in the deformed theory either as hermitian orpseudo-hermitian operators? To be specific, in the large class of deformed Poincar´e theo-ries (which include, e.g., κ -Poincar´e theories [1]-[3], Deformed Special Relativity theories[4, 5], light-like noncommutativity in Very Special Relativity [6, 7] and many other exam-ples [8]-[10]) we focus on the Drinfel’d twist [11, 12] deformations of a light-like direction.Due to this reason the deformations we consider here are based on the jordanian [13]-[15]and on the extended jordanian [16] twist (for physical applications of the Jordanian twistsee [17, 18] and, for the extended Jordanian twist, [19]-[24]).The deformations (see Appendix B for a more detailed discussion) can be encodedin twist-deformed generators which, essentially, correspond to twist-covariant generaliza-tions of the Bopp-shift [25]. The operators which in the undeformed case are associatedwith generators of the Lie algebra are, under a twist, mapped into given elements of theUniversal Enveloping Algebra.Observables in connection with deformed generators were addressed in [26] for twistdeformations of the Poincar´e-Weyl algebra with the dilatation operator entering the twist.In our work the twist is defined in terms of Poincar´e generators only. The framework weare using in this paper is based on [27]-[30]. Its presentation is quickly summarized inAppendix B .Different deformed theories are obtained from the original twist and its flipped version(obtained by a permutation of the tensor space).It is worth mentioning that quantum deformations of the Poincar´e algebra were classi-fied in [31] in terms of classical r -matrices for Poincar´e Poisson structures. The connectionwith twist-deformations was presented in [32]. The extended Jordanian twist discussedbelow is recovered from the second case of Table 1 given in [32] by setting equal to zerotwo of the three deformation parameters.A common feature of the deformed theories is that the deformed generators define aclosed nonlinear algebra (antisymmetry and Jacobi identities are respected, but the righthand side is a nonlinear combination of the generators). Furthermore, the deformationmodifies the domain of the physical parameters. For instance, in the simplest non-trivialcase, the jordanian deformation implies the introduction of a maximal momentum alonga light-cone direction. Induced by the coproduct (see [30]), nonlinear addition formulasare obtained for multi-particle states. The nonlinear addition formulas satisfy the asso-ciativity condition. Their main raison d’ˆetre is that they allow to respect the domainof validity of the physical quantities (in the example above, the composite momentumalong the light-cone direction is bounded by the maximal value). We postpone to theConclusions a more detailed discussion of the implications of our results.The scheme of the paper is as follows. The jordanian and the extended jordaniantwists are recalled in Section . In Section twist-deformed generators are introduced.In Section the (pseudo)-hermiticity property of twist-deformed generators is discussed.2he arising of a nonlinear algebra is investigated in Section . The bounded domains ofdeformed physical observables and nonlinear additive formulas are discussed in Section .For completeness in the Appendix A the (undeformed) Poincar´e algebra in the light-conebasis is presented and in Appendix B the framework of twist-deformation of Hopf algebraused in this paper is succinctly explained. We recall, see [33, 34] for details, that a Drinfel’d twist deformation of a Hopf Algebra A is induced by an invertible element F ∈ A ⊗ A which satisfies the cocycle condition( ⊗ F )( id ⊗ ∆) F = ( F ⊗ )(∆ ⊗ id ) F . (1)In Sweedler’s notation [35] F can be expanded according to F = f β ⊗ f β , F − = f β ⊗ f β . (2)For A = U ( G ), the Universal Enveloping Algebra of a Lie algebra G , the elements of F are taken from an even-dimensional subalgebra of G . Therefore the simplest cases of twistare found for two-dimensional subalgebras.There are (over C ) two inequivalent two-dimensional Lie algebras (we denote thegenerators as a, b ): i ) the abelian algebra [ a, b ] = 0 and ii ) the non-abelian algebra [ a, b ] = ib .The case i ), the abelian twist, leads to constant non-commutativity (see, e.g., [36, 27]).In this paper we focus on the second case. The non-abelian algebra ii ) is, for example,the Borel subalgebra of sl (2). The sl (2) generators can be presented as D, H, K , satisfyingthe commutation relations[
D, H ] = iH, [ D, K ] = − iK, [ K, H ] = 2 iD. (3)We can identify D ≡ a and H ≡ b . This leads, see [29], to non-commutativity of Snydertype.We can also regard ii ) as the subalgebra of a d -dimensional Poincar´e algebra P ( d ).From its generators P , P , M , whose commutators are[ P , P ] = 0 , [ M , P ] = − iP , [ M , P ] = iP , (4)one can identify the ii ) subalgebra from the positions a ≡ − M , b ≡ P + = P + P .The abelian algebra i ) induces the abelian twist F = exp( − iαa ⊗ b ) , (5)where α is the (dimensional) deformation parameter.The non-abelian algebra ii ) induces the non-abelian (jordanian) twist [13]-[15] F = exp( − ia ⊗ ln(1 + αb )) , (6)3here α is the (dimensional) deformation parameter.Under the transposition operator τ ( τ ( v ⊗ w ) = w ⊗ v ), the transposed twist F τ := exp( − i ln(1 + αb ) ⊗ a ) , (7)still satisfies the cocycle condition (1). We are using both F and F τ in our paper.The jordanian and the extended jordanian twist of the d -dimensional Poincar´e algebracan be expressed, in terms of light-cone coordinates (see the Appendix for the d = 4 case)and Einstein’s convention, through the position (see, e.g., [20]) F = exp (cid:18) iM ⊗ ln(1 + αP + ) + iǫM + j ⊗ ln(1 + αP + ) P j P + (cid:19) . (8)The jordanian case is recovered for ǫ = 0; the extended jordanian case is recovered for ǫ = 1. In two dimensions the two twists coincide since there are no transverse directions.Under transposition, the F τ twist is given by F τ = exp (cid:18) i ln(1 + αP + ) ⊗ M + iǫ ln(1 + αP + ) P j P + ⊗ M + j (cid:19) . (9)The following four kinds of twist-deformations can be considered: the jordanian defor-mations ( ǫ = 0) based on F (case I ) and F τ (case II ) and the extended jordaniandeformations ( ǫ = 1) based on F (case III ) and F τ (case IV ). A twist deformation can be expressed in terms of the twist-deformed generators (see[36, 37, 27]). To make the paper self-consistent, the framework we are using is succinctlydescribed in Appendix B . Under deformation, a Lie algebra generator g ∈ G is mappedinto the Universal Enveloping Algebra element g F ∈ U ( G ), given by g F = f β ( g ) f β , ( F − = f β ⊗ f β ) . (10)In this paper we consider the Hopf algebras U ( G ) and U ( P ). They are defined over theUniversal Enveloping Algebras of, respectively, the Lie algebra G = { P µ , M µν , x µ , ~ } andits subalgebra P = { P µ , M µν } , which is the d -dimensional Poincar´e algebra (see Appendix A for the explicit presentation of G and P for the ordinary d = 4 spacetime).We point out that ~ is a central element of G and is a primitive element of theHopf algebra structure defined on the Universal Enveloping Algebra U ( G ) (for details andmotivations of the construction see, e.g., [30]).We present the twist deformations for the previous Section cases I , II and IV , whosetwist deformed generators can be presented in closed form.4n the case I ( F twist with ǫ = 0) we have P F + = P +
11 + αP + ,P F− = P − (1 + αP + ) ,P F j = P j ,M F = M,M F + j = M + j
11 + αP + ,M F− j = M − j (1 + αP + ) ,N F = N (11)and ~ F = ~ ,x F + = x +
11 + αP + ,x F− = x − (1 + αP + ) ,x F j = x j . (12)The undeformed generators can be expressed in terms of the deformed generators on thebasis of inverse formulas. In particular we have P + = P F + − αP F + ,
11 + αP + = 1 − αP F + , αP + = 11 − αP F + . (13)The twist-deformed generators for the transposed twists F τ , with ǫ = 0 ,
1, are P F• = P • ,M F = 1 + 2 αP + αP + M + ǫ (cid:18) αP j αP + − ln(1 + αP + ) P j P + (cid:19) M,M F + j = M + j + ǫ ln(1 + αP + ) M + j ,M F− j = M − j + 2 αP j αP + M + ǫ (cid:18) ln(1 + αP + ) P j P + δ jk + 2 αP j P k (1 + αP + ) P + − αP + ) P j P k P +2 (cid:19) M + k ,N F = N − ǫǫ jk ln(1 + αP + ) P k P + M + j (14)and ~ F = ~ ,x F + = x + ,x F− = x − + 2 α ~ αP + M + 2 ǫ ~ (cid:18) α αP + − ln(1 + αP + ) P + (cid:19) P j P + M + j ,x F j = x j − ǫǫ jk ~ ln(1 + αP + ) 1 P + M + k . (15)5he bullet • denotes ± , j (all translation generators are undeformed). The summationover repeated indices is understood. For an operator Ω the hermiticity condition is Ω † = Ω.The pseudohermiticity condition [38] isΩ † = η Ω η − , (16)for some invertible hermitian operator η = η † .For the jordanian twist the pseudo-hermiticity properties of the deformed generatorsare the following.In case I (jordanian F twist with ǫ = 0): P F• † = η λ P F• η − λ , ∀ λ ∈ R ,M F † = M F , i.e. λ = 0 ,M F + j † = η λ M F + j η − λ , ∀ λ ∈ R ,M F− j † = ηM F− j η − , i.e. λ = 1 ,N F † = η λ N F η − λ , ∀ λ ∈ R , (17)together with x F + † = η λ x F + η − λ , ∀ λ ∈ R ,x F−† = ηx F− η − , i.e. λ = 1 ,x F j † = η λ x F j η − λ , ∀ λ ∈ R , (18)for the hermitian operator η = 1 + αP + .One can observe that, in this case, the subset of hermitian ( λ = 0) deformed opera-tors is given by { P F± , P F j , M F , N F , M F + j , x F + , x F j , ~ F } , since the operators M F− j , x F− are nothermitian.In the case II (jordanian F τ twist with ǫ = 0) we have P F• † = η λ P F• η − λ , ∀ λ ∈ R ,M F † = ηM F η − , i.e. λ = 1 ,M F + j † = η λ M F + j η − λ , ∀ λ ∈ R ,N F † = η λ N F η − λ , ∀ λ ∈ R , (19)6ogether with x F + † = η λ x F + η − λ , ∀ λ ∈ R ,x F j † = η λ x F j η − λ , ∀ λ ∈ R , (20)for the hermitian operator η = αP + αP + .Unlike the other deformed generators, M F− j , x F− do not satisfy the pseudo-hermiticitycondition for any choice of η .It is worth pointing out that, by taking the choice λ = 1, we formally obtain the sameset of P F± , P F j , M F , N F , M F + j , x F + , x F j , ~ F deformed generators as in the previous case. Theyare now, of course, different operators which satisfy a pseudo-hermiticity condition. Aswe shall see, they close a nonlinear algebra.For the extended jordanian twist, cases III and IV , one can verify by explicit com-putation through a Taylor expansion in the deformation parameter α , that most of thedeformed generators are neither hermitian nor pseudo-hermitian. In the case I (the jordanian F twist with ǫ = 0) the deformed generators induce nonlinear(at most quadratic) algebras recovered from their commutators (see the comments inAppendix B ). The basis of deformed generators defining the G dfr nonlinear algebra isgiven by G dfr : { ~ F , x F± , x F j , P F± , P F j , M F , N F , M F± j } . (21)The deformed generators are obtained from the G = { P µ , M µν , x µ , ~ } set of Lie algebragenerators after applying the (10) mapping. Some of its relevant subalgebras are thedeformed Poincar´e subalgebra P dfr , with basis of deformed generators given by P drf : { P F± , P F j , M F , N F , M F± j } , (22)as well as the subalgebra of mutually consistent observables (generators with the samehermiticity/pseudo-hermiticity property). The G obs and the P obs subalgebras of observ-ables are respectively given by G obs : { ~ F , x F + , x F j , P F± , P F j , M F , N F , M F + j } (23)and P obs : { P F± , P F j , M F , N F , M F + j } . (24)7he non-vanishing commutation relations of the G dfr nonlinear algebra are explicitly givenby [ P F± , M F ] = ± iP F± ∓ iαP F + P F± , [ P F− , M F + j ] = 2 iP F j , [ P F− , M F− j ] = 2 iαP F− P F j , [ P F j , M F± k ] = iδ jk P F± , [ M F , M F± j ] = ∓ iM F± j ± iαM F± j P F + , [ N F , M F± j ] = iǫ jk M F± k , [ M F + j , M F− k ] = − iδ jk M F − iǫ jk N F − iαM F + j P F k , [ M F− j , M F− k ] = 2 iα ( M F− j P F k − M F− k P F j ) , [ P F + , x F− ] = 2 i ~ F (1 − αP F + ) , [ P F− , x F + ] = 2 i ~ F , [ P F− , x F− ] = 2 iα ~ F P F− , [ P F j , x F k ] = − iδ jk ~ F , [ M F , x F± ] = ∓ ix F± ± iαx F± P F + , [ N F , x F j ] = iǫ jk x F k , [ M F + j , x F− ] = − ix F j − iα ~ F M F + j , [ M F− j , x F + ] = − ix F j + 2 iαx F + P F j , [ M F− j , x F− ] = 2 iα ( ~ F M F− j − x F− P F j ) , [ M F± j , x F k ] = − iδ jk x F± , [ x F + , x F− ] = − iα ~ F x F + . (25)From the above formulas one can check that P dfr , G obs and P obs close as nonlinear subal-gebras.It follows from the last formula of the (25) algebra that the associated non-commutativespace-time belongs to the class of Lie-deformed space-times, see e.g. [39], with light-likedeformation.In case II (the jordanian F τ twist) the algebra of the deformed observables, givenby the operators ~ F , x F + , x F j , P F± , P F j , M F , N F , M F + j obtained from formulas (14) and (15)with ǫ = 0, also closes as a nonlinear algebra. In this class of operators the only one whichdiffers from its undeformed counterpart is M F , given by M F = ZM , Z = 1 + 2 αP + αP + = 1 + 2 αP F + αP F + . (26)It turns out that the commutators with nonlinear right hand side are the following ones:[ M F , P F± ] = ∓ iZP F± , [ M F , M F + j ] = − iZM F + j , [ M F , x F + ] = − iZx F + . (27)8ll the remaining commutators are linear and coincide with the ones (see the Appendix A ) for the undeformed generators. The second order (mass-term) Casimir of the Poincar´e algebra C = P + P − − ( P j ) (28)remains undeformed under the twist-deformations I, II, IV introduced in Section . Weget C = P F + P F− − ( P F j ) . (29)All ten twisted generators (collectively denoted as g F I , I = 1 , . . . ,
10) entering their re-spective deformed Poincar´e algebras, commute with C :[ g F I , C ] = 0 , ∀ g F I . (30)To analyze the physical consequences of the deformation we consider here the simplestsetting, namely the case I (the F jordanian twist with ǫ = 0). We will discuss theproperties of the deformed momenta and their nonlinear addition formulas.Let’s set, for simplicity, the transverse momenta P j ≡
0. For convenience we introducenew variables, defined as r = P + , l = P − ,e = P , p = P . (31)Since P ± = P ± P , we have e = 12 ( r + l ) , p = 12 ( r − l ) , (32)with e representing the energy of the system.For a massive representation we get, on shell, rl = m , e − p = m . (33)Let’s set, without loss of generality, m = 1. Therefore rl = 1 ,e − p = 1 ,l = 1 r ,e = 12 ( r + 1 r ) ,p = 12 ( r − r ) . (34)9n order to have a positive energy e , r should be non-negative. The observables aretherefore bounded in the domains r ∈ ]0 , + ∞ ] ,l ∈ ]0 , + ∞ ] ,e ∈ [1 , + ∞ ] ,p ∈ [ −∞ , + ∞ ] . (35)The rest condition for the P momentum corresponds to p = 0 obtained at r = 1. Forthis value the energy is minimal ( e = 1).The twist-deformed variables will be denoted with a bar, the deformation parameterbeing α . We have r = r αr , l = l (1 + αr ) . (36)The condition α ≥ l = 1 r (1 + αr ) ,e = 12 ( r + l ) = 12 (cid:18) r αr + 1 r (1 + αr ) (cid:19) ,p = 12 ( r − l ) = 12 (cid:18) r αr − r (1 + αr ) (cid:19) . (38)The rest condition p = 0 for the deformed P momentum is obtained for r = 11 − α . (39)Since r should be positive, one finds the restriction α <
1. Therefore, the range of thedeformation parameter α is 0 ≤ α < . (40)As a consequence, the domains of the deformed operators are modified with respect totheir undeformed counterparts. We get r ∈ ]0 , α [ ,l ∈ ] α, + ∞ ] ,e ∈ [1 , + ∞ ] ,p ∈ [ −∞ ,
12 ( α − α )[ . (41)10ollowing [30] (see also the explanations in Appendix B ) the additive formulas of thedeformed momenta are induced by their deformed coproducts. In particular, the 2-particleaddition formula for the deformed P + momenta r , r of, respectively, the first and thesecond particle reads, in terms of the undeformed momenta, as r = r + r α ( r + r ) . (42)Closely expressed in terms of the deformed momenta it is given by r = r + r − αr r − α r r . (43)The binary operation defined by Eq. (43) is a group operation with r = 0 as identityelement and inverse element given by r − = − r − αr . The associativity is satisfied due tothe relation r (1+2)+3 = r = r + r + r − α ( r r + r r + r r ) + 3 α r r r − α ( r r + r r + r r ) + 2 α r r r . (44)It should be noted, however, that the physical requirement of r belonging to the domainin Eq. (41) excludes the identity and the inverse element from the physical values. Thus,on physical states, the addition formula (43) only satisfies the properties of a semigroupoperation.It is useful to compare the formula (43) with the nonlinear addition of velocities inspecial relativity. Let us change variables once more and set r , = v , , α = c .In special relativity we have v ,s.r. = v + v c v v . (45)In the above jordanian deformation we get v ,jd. = v + v − c v v − c v v . (46)Both nonlinear addition formulas are symmetric in the v ↔ v exchange and associa-tive.They can also be defined in the interval 0 ≤ v , ≤ c , so that the nonlinear additivevelocities belong to the [0 , c ] range (in both cases if v = 0, then v = v and, if v = c , v = c ).The main difference is that in special relativity the formula can be nicely extended tonegative velocities belonging to the − c ≤ v , ≤ c interval. This is not the case for thenonlinear addition formula based on the jordanian deformation.It is worth to point out that an important feature of the nonlinear additivity consistsin respecting the physical domain of the variables defining the multi-particle state.For completeness and as an example we present the coproducts for the twist-deformedtranslation generators P F± , P F j induced by the case I twist. The nonlinear addition for-mulas are recovered as a consequence. In principle one should use the twist-deformed11oproduct ∆ F . However, as recalled in Appendix B , the application of the ordinarycoproduct ∆ (whose computations are easier) produces unitarily equivalent results.We set, for convenience, r = P F + ⊗ , r = ⊗ P F + , s = P F− ⊗ , s = ⊗ P F− , t j , = P F j ⊗ , t j , = ⊗ P F j . The coproducts are ∆( P F + ) = ∆( P + ) · + α ∆( P + ) , (47)which implies the nonlinear additive formula (43),∆( P F− ) = P F− ⊗ + ⊗ P F− + αP F− ( − αP F + ) ⊗ P F + ( − αP F + ) + αP F + ( − αP F + ) ⊗ P F− ( − αP F + ) , (48)which implies the nonlinear additive formula s = s + s + αs r − αr − αr + αr s − αr − αr , (49)∆( P F j ) = P F j ⊗ + ⊗ P F j , (50)which gives, since the transverse momenta are undeformed, t j , = t j , + t j , . (51) In this paper we investigated the observational consequences of the light-like deformationsof the Poincar´e algebra induced by the jordanian and the extended jordanian twists (whichboth belong to the class of Drinfel’d twist deformations). We used the framework of [27]-[30] where, in particular, the deformed quantum theory is recovered from deformed twist-generators belonging to a Universal Enveloping Algebra. The Hopf algebra structure of thetwist-deformation controls the physical properties of the theory. The deformed generatorsinduce nonlinear algebras, while the coproduct implies associative nonlinear additivity ofthe multi-particle states. In the simplest setting of the jordanian deformation along alight-cone direction, these nonlinearities consistently imply the existence of a maximallight-cone momentum. The undeformed theory is recovered by allowing the maximallight-cone momentum to go to infinity. This situation finds a parallelism, as we noted,with the light velocity as the maximal speed in special relativity.A question that we raised regards the status, as observables of a quantum theory,of the deformed generators. It is rewarding that for the jordanian deformation a largesubset of the deformed Poincar´e generators are pseudo-hermitian for the same choice of an12nvertible hermitian η operator, see e.g. formula (19). Therefore, they are observable withrespect to the η -modified inner product [38]. This subset of observables close a nonlinearalgebra as a consistent deformation of a Poincar´e subalgebra.The picture is different for the extended jordanian twist. One can explicitly check, byTaylor-expanding in power series of the deformation parameter, that most of the Poincar´edeformed generators are neither hermitian nor pseudo-hermitian. It is still an open ques-tion the status as observables of the extended jordanian twist-deformed generators.We point out that, so far, no systematic study has been conducted to investigate thephysical consequences of all 21 classes of Poincar´e twist deformations given in [32]. Thecase we are addressing here is particularly relevant because, according to [32], each twistcan be obtained as the product of factorized twists. The three-parameter twist enteringthe second case of Table 1 is the product of three factorized twists, two of them beingabelian, while the third one is our extended jordanian twist. Setting to zero the two extradeformation parameters as done in this paper corresponds to analyze the “pure” extendedjordanian twist. The two extra abelian deformations are easier to be invesitigated. Appendix A: Poincar´e algebra in the light-cone ba-sis.
We work with the metric η µν = diag ( − , , ,
1) ( µ, ν = 0 , , , P µ , P ν ] = 0 , [ M µν , P ρ ] = iη µρ P ν − iη νρ P µ , [ M µν , M ρσ ] = iη µρ M νσ − iη µσ M νρ − iη νρ M µσ + iη νσ M µρ . (52)In terms of the spacetime coordinates x µ ( x µ ), with µ = 0 , j and j = 1 , ,
3, the Poincar´egenerators can be realized through the positions P = i ~ ∂∂x ,P j = − i ~ ∂∂x j ,M j = − ix ∂∂x j − ix j ∂∂x ,M jk = − ix j ∂∂x k + ix k ∂∂x j . (53) P µ , M µν , together with the coordinates x µ and the central charge ~ close a Lie algebra13 = { P µ , M µν , x µ , ~ } whose remaining non-vanishing commutators are given by[ P , x ] = i ~ , [ P j , x k ] = − iδ jk ~ , [ M j , x ] = − ix j , [ M j , x k ] = − ix δ jk , [ M jk , x r ] = − ix j δ kr + ix k δ jr . (54)The light-cone coordinates mix the time-direction x and the space direction x ; theremaining space coordinates are the transverse directions. The light-cone generators arelabeled by the indices ± and j = 1 ,
2. The latter is used for the transverse coordinates.In the light-cone basis the generators are expressed through the positions x ± = x ± x , x j = x j +1 (55)and P ± = P ± P , P j = P j +1 ,M := M + − = M ,M ± j = M ,j +1 ± M ,j +1 ,N = M . (56)In the above formulas relating the two presentations of space-time coordinates and Poincar´egenerators, for reason of clarity, the Poincar´e generators and space-time coordinates ex-pressed in the light-cone variables are overlined. Since however in the main text we onlywork with light-cone quantities and no confusion will arise, for simplicity, these quantitieswill not be overlined.In the light-cone basis the non-vanishing commutators of the Poincar´e algebra read asfollows [ M, P ± ] = ∓ iP ± , [ M, M ± j ] = ∓ iM ± j , [ M ± j , P ∓ ] = − iP j , [ M ± j , P k ] = − iδ jk P ± , [ M + j , M − k ] = − iδ jk M − iǫ jk N, [ M ± j , N ] = − iǫ jk M ± k , [ N, P j ] = iǫ jk P k (57)(the constant antisymmetric tensor is ǫ = − ǫ = 1).The non-vanishing commutators of the Poincar´e generators with the light-cone coor-14inates x ± , x j are [ P ± , x ∓ ] = 2 i ~ , [ P j , x k ] = − iδ jk ~ , [ M, x ± ] = ∓ ix ± , [ M ± j , x ∓ ] = − ix j , [ M ± j , x k ] = − ix ± δ jk , [ N, x j ] = iǫ jk x k . (58) Appendix B: comments on the Drinfel’d twist de-formation of Hopf algebras.
In this paper we investigated Drinfel’d twist deformations of Hopf algebras. We sum-marize here, for self-consistency, the main features of our approach. Our framework hasbeen detailed before, e.g. in reference [28]. It is based on the construction discussed in[37].We point out at first that the cocycles (8) and (9) are counitary 2-cocycles satisfyingi) ( ⊗ F )( id ⊗ ∆) F = ( F ⊗ )(∆ ⊗ id ) F ,ii) ( ε ⊗ id ) F = ( id ⊗ ε ) F ,iii) F = ⊗ + O ( α ).Any Universal Enveloping Lie Algebra U ( G Lie ) = A over, let’s say, the complex field C , isa Hopf algebra endowed with multiplication m , unit η , coproduct ∆, counit ε , antipode S . A twist-deformation satisfying properties i,ii,iii), see [37, 28], maps the original Hopfalgebra into a new Hopf algebra with structures and costructures given by [11]( A, m, η, ∆ , ε, S ) → ( A, m, η, ∆ F , ε, S F ) , (59)where, ∀ a ∈ A , ∆ F = F ∆( a ) F − ,S F = χSχ − (60)for χ = m (( id ⊗ S ) F ).For simplicity the new deformed Hopf algebra is denoted as A F . As a vector space A F = A is identified with the original algebra A . As a vector space A = U ( G Lie ) can be spannedboth by the original Lie algebra generators g i ∈ G Lie or by its twist-deformed generatorsobtained from (10), namely g F i = f β ( g i ) f β , ( F − = f β ⊗ f β ) . (61)This formula was derived in [37] on the basis of the [40] construction.In this paper, following the above prescriptions, we constructed the twist-deformedalgebras U F ( G ), U F ( P ), respectively obtained from the Lie algebra G = { P µ , M µν , x µ , ~ } and its P = { P µ , M µν } Poincar´e subalgebra. They are both Hopf algebras.15ollowing the construction of [27, 28, 30], once the twist-deformed generators are expressedas elements of the Universal Enveloping algebra, relevant information about the twist-deformed Hopf algebra is encoded in their commutators.To obtain the nonlinear addition formulas of Section one needs to use a co-structure,given by the twisted coproduct ∆ F ( g F ). There is an important remark; for practical pur-poses it is more convenient to compute just ∆( g F ), see the explanation given in ref. [4].Indeed, when applied to a Hilbert space, F ≡ F becomes a unitary operator, so that \ ∆ F ( g F ) = F \ ∆( g F ) F − . Therefore, the operators \ ∆ F ( g F ) and \ ∆( g F ) are unitarily equiv-alent. The non-linearity of the addition formulas is nevertheless guaranteed by the factthat the deformed generators from equation (61) are not unitarily equivalent to the un-deformed generators. Acknowledgments
The work received support from CNPq under PQ Grant No. 308095/2017-0.
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