Associative realizations of the extended Snyder model
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Associative realizations of the Snyder modelS. Meljanac † Rudjer Boˇskovi´c Institute, Theoretical Physics DivisionBljeniˇcka c. 54, 10002 Zagreb, Croatiaand
S. Mignemi † Dipartimento di Matematica, Universit`a di Cagliarivia Ospedale 72, 09124 Cagliari, Italyand INFN, Sezione di Cagliari
Abstract
The star product usually associated to the Snyder model of noncommutative geometry is nonassociative,and this property prevents the construction of a proper Hopf algebra. It is however possible to introduce awell-defined Hopf algebra by including the Lorentz generators and their conjugate momenta into the algebra.In this paper, we study the realizations of this extended Snyder spacetime, and obtain the coproduct andtwist and the associative star product in a Weyl-ordered realization, to first order in the noncommutativityparameter. We then extend our results to the most general realizations of the extended Snyder spacetime,always up to first order. † e-mail:[email protected] † e-mail:[email protected] . Introduction Since the origin of quantum field theory there have been proposals to add a new scale of length to thetheory in order to solve the problems connected to ultraviolet divergences. Later, the necessity of introducinga fundamental length scale has also arisen in several attempts to build a theory of quantum gravity. In thesecases, the scale could be identified in a natural way with the Planck length L p = q ¯ hGc ∼ . · − m [1].A naive application of the idea of a minimal length, as for example a lattice field theory, would howeverbreak Lorentz invariance. A way to reconcile the discreteness of spacetime with Lorentz invariance wasoriginally proposed by Snyder [2] a long time ago. This was the first example of a noncommutative geometry:the length scale should enter the theory through the commutators of spacetime coordinates, see [3,4]. Inparticular, the position operators obey the commutation relations[ x µ , x ν ] = iβJ µν , (1)where J µν are the generators of the Lorentz transformations and β is a parameter of dimension length squarethat sets the scale of noncommutativity. In more recent times, using ideas coming from the development of noncommutative geometry [5], thecoproduct and star product structures induced by the position operators of the Snyder model have beencalculated [6,7]. However, in the Snyder model, the algebra of the position operators does not close, andhence the bialgebra resulting from the implementation of the coproduct is not strictly speaking an Hopfalgebra, as in other noncommuatative geometries. In particular, the coproduct is not coassociative [6]. Aclosed Lie algebra can however be obtained if one adds to the position generators the generators of the Lorentzalgebra [7]. In this way one can define a proper Hopf algebra, with coassociative coproduct. The price topay is the addition of the momenta conjugated to the Lorentz generators. Also, the physical interpretationof the new degrees of freedom is not evident, they may be viewed for example as coordinates parametrizingextra dimensions [7].In this paper, we construct new realizations of this algebra, perturbatively in the parameter β . Moreprecisely, we consider a Weyl realization of the algebra in terms of a generalized Heisenberg algebra, andthen extend it to the most general one compatible with Lorentz invariance at order β , including the oneobtained in [7], and compute the coproduct and the star product in the general case. We also calculate thetwist in the Weyl realization.We recall here some of the most relevant recent advances in Snyder theory: in [9] the Snyder algebrawas generalized in such a way to maintain the Lorentz invariance; in [6] the coproduct was calculated, in[7] the same problem was investigated from a geometrical point of view, using the fact that the momentumspace of Snyder can be identified with a coset space; the twist was investigated in [10,11]. The constructionof a field theory was first addressed in [6,7] and then examined in more detail in [12]. Different applicationsto phenomenology have been considered in [13]. Finally, the extension to a curved background was proposedin [14] and further investigated in [15]. Also the nonrelativistic limit of the theory was studied in a largenumber of papers, but we shall omit a discussion of this topic.The paper is organized as follows: in sect. 2 we introduce Snyder model with an associative star productand discuss its Weyl realization in terms of an extended Heisenberg algebra; in sect. 3 we compute thecoproduct and the star product in this realization; in sect. 4 also the twist is calculated. In sect. 5, genericrealizations up to order β are introduced and coproducts and star products are obtained. Finally, in sect. 6the relations of these realizations with that of ref. [7] and with well-known nonassociative ones are discussed.In sect. 7 some conclusion are drawn.
2. Snyder model and Weyl ralization
We consider the subalgebra of the N -dimensional Snyder algebra generated by the N position operatorsˆ x i and the N ( N − / x ij , with i = 0 . . . , N − x i , ˆ x j ] = iλβ ˆ x ij , [ˆ x ij , ˆ x k ] = iλ ( η ik ˆ x j − η jk ˆ x i ) , [ˆ x ij , ˆ x kl ] = iλ ( η ik ˆ x jl − η il ˆ x jk − η jk ˆ x il + η jl ˆ x ik ) , (2) Throughout this paper we adopt natural units ¯ h = c = 1. Generally, Lie deformed quantum Minkowski spaces admit both Hopf algebra and Hopf algebroid struc-ture [8]. 2here λ and β are real parameters. In particular, β can be identified with the Snyder parameter which isusually assumed to be of size L p , while λ is a dimensionless parameter. The parameter β can take bothpositive and negative values, leading to quite different physical models. However, from an algebraic pointof view both cases can be treated in an essentially unified way. For β = 0, the commutation relations (2)reduce to those of the standard Lorentz algebra acting on commutative coordinates.The algebra (2) can be realized in terms of a generalized Heisenberg algebra, which includes also theLorentz generators, [ x i , x j ] = [ p i , p j ] = [ x ij , x kl ] = [ p ij , p kl ] = 0 , [ x i , p j ] = iη ij , [ x ij , p kl ] = i ( η ik η jl − η il η jk ) , [ x i , x jk ] = [ x i , p jk ] = [ x ij , x k ] = [ x ij , p k ] = 0 , (3)where p i and p ij are momenta canonically conjugate to x i and x ij respectively, and p ij = − p ji . The momentacan be realized in a standard way as p i = − i ∂∂x i , p ij = − i ∂∂x ij . (4)Note that, including the momenta p i in the algebra (2), with commutation relations[ p i , p j ] = 0 , [ˆ x ij , p k ] = iλ ( η ik p j − η jk p i ) , [ˆ x i , p j ] = i ( η ij + λ βp i p j ) , (5)one recovers the full original Snyder algebra [2].To proceed with the computations, it is convenient to exploit the isomorphism between the Snyderalgebra and so (1 , N ), and write the previous formulas more compactly defining, for positive β , ˆ x iN ≡ √ β ˆ x i , x iN ≡ √ βx i , p iN ≡ p i / √ β , with η NN = 1, and µ = 0 , . . . N . The generalized Heisenberg algebra (3)becomes then [ x µν , x ρσ ] = [ p µν , p ρσ ] = 0 , [ x µν , p ρσ ] = i ( η µρ η νσ − η µσ η νρ ) , (6)while the Snyder algebra (2) takes the form[ˆ x µν , ˆ x ρσ ] = iλC µν,ρσ,αβ ˆ x αβ , (7)where C µν,ρσ,αβ are the structure constants of the so (1 , N ) algebra, C µν,ρσ,αβ = 12 h − η νρ ( η µα η σβ − η σα η µβ ) + η µσ ( η ρα η νβ − η να η ρβ )+ η µρ ( η να η σβ − η σα η µβ ) − η νσ ( η ρα η µβ − η µα η ρβ ) i . (8)that satisfy the symmetry properties C µν,ρσ,αβ = − C νµ,ρσ,αβ = − C µν,σρ,αβ = − C µν,ρσ,βα = − C ρσ,µν,αβ = − C µν,αβ,ρσ .In general, if the coordinates ˆ x µ generate a Lie algebra [ˆ x µ , ˆ x ν ] = iC µνλ ˆ x λ with structure constants C µνλ , then the universal realization of ˆ x µ corresponding to Weyl-symmetric ordering is given by [16]ˆ x µ = x α φ αµ ( p ) = x α (cid:18) C − e −C (cid:19) µα , (9)where C µν = C αµν p α . This realization enjoys the property e ik µ ˆ x µ ⊲ e ik µ x µ , k µ ∈ R , (10) When β < so (2 , N − β must be taken under the square root and η NN = −
1. All results areidentical, with the appropriate choice of the sign of β .3here the action ⊲ is given by x µ ⊲ f ( x α ) = x µ f ( x α ) , p µ ⊲ f ( x a ) = − i ∂f ( x a ) ∂x µ , (11)or, in our case, x µν ⊲ f ( x αβ ) = x µν f ( x αβ ) , p µν ⊲ f ( x αβ ) = − i ∂f ( x αβ ) ∂x µν = [ p µν , f ( x αβ )] , (12)Hence, the corresponding Weyl realization of ˆ x µν in terms of the generalized Heisenberg algebra (6) reads[16] ˆ x µν = x αβ (cid:18) λ C − e − λ C (cid:19) µν,αβ = x µν + λ x αβ C µν,αβ + λ x αβ (cid:0) C (cid:1) µν,αβ + O ( λ ) . (13)where C µν,αβ = 12 C ρσ,µν,αβ p ρσ = 12 ( η µα p νβ − η µβ p να + η νβ p µα − η να p µβ ) , (cid:0) C (cid:1) µν,αβ = 12 X k =0 (cid:18) k (cid:19) (cid:16) ( p k ) µα ( p − k ) νβ − ( p − k ) µβ ( p k ) να (cid:17) , (14)and p µν is written in matricial notation.Inserting C in (13), we find up to order λ ,ˆ x µν = x µν + λ x µα p να − x να p µα ) − λ
12 ( x µα p νβ p αβ − x να p µβ p αβ − x αβ p µα p νβ ) . (15)One has then[ˆ x µν , p ρσ ] = i ( η µρ η νσ − η µσ η νρ ) + iλ η µρ p νσ − η νρ p µσ + η νσ p µρ − η µσ p νρ ) − iλ
12 ( η µρ p να p σα − η µσ p να p ρα − η νρ p µα p σα + η νσ p µα p ρα + 2 p µρ p νσ − p νρ p µσ ) . (16)One can rewrite eq. (15) in terms of its components asˆ x i = x i + λ (cid:0) x k p ik − βx ik p k (cid:1) − λ (cid:0) x κ p kl p il + β ( − x k p k p i + x i p k − x ik p l p kl − x kl p k p il ) (cid:1) , ˆ x ij = x ij + λ (cid:0) x i p j + x ik p jk − ( i ↔ j ) (cid:1) − λ (cid:0) x ik p jl p kl − x kl p ik p jl − x i p k p jk + 2 x k p i p jk + βx ik p k p j − ( i ↔ j ) (cid:1) . (17)In the limit λβ = L p , λ = 0, the algebra (2) becomes the DFR (Moyal) algebra [3] and the realization(15) takes the form ˆ x i = x i − L p x ik p k , ˆ x ij = x ij . (18)The corresponding Lorentz generators are M ij = x i p j − x j p i + x ik p jk − x jk p ik . (19)4 . Coproduct and star product in Weyl realization In order to compute the coproduct of the Hopf algebra, we use the formalism introduced in [17]. Wedefine a function P µν ( tk αβ ) that satisfies the differential equation d P µν dt = i p µν , k ρσ ˆ x ρσ ] (cid:12)(cid:12) p →P ( tk ) = k ρσ Φ µν,ρσ ( P αβ ) , (20)with initial condition P µν (0) = q µν . The function Φ µν,ρσ ( p αβ ) is defined from (15) as ˆ x µν = x ρσ Φ ρσ,µν . Inour case, equation (20) takes the form d P µν dt = k µν − λ k µα P να − k να P µα ) − λ
12 ( k µα P αβ P νβ − k να P αβ P µβ − k αβ P µα P νβ ) , (21)and with the given initial condition has solution P µν = q µν + tk µν − λt (cid:16) k µα q να − k να q µα (cid:17) − λ (cid:16)(cid:0) k µα q αβ q νβ − k να q αβ q µβ − k αβ q µα q νβ (cid:1) t + (cid:0) k µα k αβ q νβ − k να k αβ q µβ − k µα k νβ q αβ (cid:1) t (cid:17) . (22)We can now define P µν ( k µν , q µν ) ≡ P µν ( t = 1), so that P µν ( k µν , q µν ) = k µν + q µν − λ (cid:16) k µα q να − k να q µα (cid:17) − λ (cid:16) k µα q αβ q νβ − k να q αβ q µβ − k αβ q µα q νβ + k µα k αβ q νβ − k να k αβ q µβ − k µα k νβ q αβ (cid:17) . (23)Defining then K µν ( k µν ) ≡ P µν ( q µν = 0), one has K µν = k µν , and therefore also its inverse function K − µν ( k µν ) = k µν .It can be shown that the generalized momentum addition law is given by [17] k µν ⊕ q µν ≡ D µν ( k αβ , q αβ ) = P µν ( K − αβ , q αβ ) , (24)and hence in our case D µν ( k αβ , q αβ ) = P µν ( k αβ , q αβ ). This yields the coproduct∆ p µν = D µν ( p µν ⊗ , ⊗ p µν ) = ∆ p µν − λ (cid:16) p µα ⊗ p να − p να ⊗ p µα (cid:17) − λ (cid:16) p µα ⊗ p αβ p νβ − p να ⊗ p αβ p µβ − p αβ ⊗ p µα p νβ + p µα p αβ ⊗ p νβ − p να p αβ ⊗ p µβ − p µα p νβ ⊗ p αβ (cid:17) , (25)with ∆ p µν = p µν ⊗ ⊗ p µν . It is straightforward to explicitly check the coassociativity of this coproduct.It is also easy to see that the antipode is trivial, S ( p µν ) = − p µν .Recalling our definitions ˆ x i = √ β ˆ x iN and p i = p iN / √ β , we can write the functions D αβ in terms oftheir components, namely D i ( k, q ) = k i + q i − λ h k j q ij − k ij q j i + λ h β ( k i k j q j − k j q i ) − k j k jk q ik + 2 k ik k j q jk + k ij k jk q k + β ( k j q j q i − k i q j ) + k ij q jk q k − k jk q k q ij − k j q jk q ik i , (26) D ij ( k, q ) = k ij + q ij − λ h k ik q jk + βk i q j − ( i ↔ j ) i + λ h k ik k jl q kl − k ik k kl q jl + β ( k i k k q jk − k ik k k q j − k i k jk q k ) + k kl q ik q jl − k ik q kl q jl − β ( k ik q k q j − k i q k q jk − k k q i q jk ) − ( i ↔ j ) i . (27)5he functions D ( q, k ) satisfy the symmetry properties D i ( q, k ) (cid:12)(cid:12) λ = D i ( k, q ) (cid:12)(cid:12) − λ , D ij ( q, k ) (cid:12)(cid:12) λ = D ij ( k, q ) (cid:12)(cid:12) − λ . (28)It also holds e i k µν ˆ x µν e i q ρσ ˆ x ρσ = e i D µν ( k,q )ˆ x µν , (29)and e i k µν x µν ⋆ e i q ρσ x ρσ = e i k µν ˆ x µν e i q ρσ ˆ x ρσ ⊲ e i D µν ( k,q )ˆ x µν ⊲ e i D µν ( k,q ) x µν . (30)Moreover, we can write e i k µν ˆ x µν = e ik i ˆ x i + i k ij ˆ x ij ,e ik i x i + i k ij x ij ⋆ e iq k x k + i q kl x kl = e i D i x i + i D ij x ij . (31)In particular, from (26) and (27) one can obtain the star product for plane waves. Notice that the starproduct of two translations clearly will have a component also in the direction of rotations, e ik i x i ⋆ e iq j x j = e i [ k i + q i − λ β ( q j k i − k j q j q i + k j q i − k j q j k i ) ] x i − i λβk i q j x ij ,e i k ij x ij ⋆ e i q kl x kl = e i [ k ij + q ij − λk ik q jk − λ ( k ik q kl q jl − k kl q ik q jl + k ik k kl q jl − k ik k jl q kl ) ] x ij ,e ik k x k ⋆ e i q ij x ij = e i [ k i − λ k j q ij − λ k j q jk q ik ] x i + i [ q ij + λ βk i k k q jk ] x ij ,e i k ij x ij ⋆ e i q k x k = e i [ q i + λ k ij q j + λ k ij k jk q k ] x i + i [ k ij − λ βk ik q k q j ] x ij . (32)This star product is associative. One can also check that the star products of the coordinates x i and x ij satisfy the Snyder algebra. In fact, according to [7], denoting k the vector k i , l the tensor l ij and so on, anddefining e k, l = e k i x i + l jk x jk , the star product of the coordinates can be evaluated as follows: x i ⋆ x j = Z dk dq d l d r δ ( k ) δ ( q ) δ ( l ) δ ( r ) ∂ k i ∂ q j ( e k, l ⋆ e q, r ) = ˆ x i ⊲ x j = x i x j + i λβ x ij ,x ij ⋆ x kl = Z dk dq d l d r δ ( k ) δ ( q ) δ ( l ) δ ( r ) ∂ l ij ∂ r kl ( e k, l ⋆ e q, r ) = ˆ x ij ⊲ x kl = x ij x kl + i λ (cid:16) η ik x jl − η jk x il − η il x jk + η jl x ik (cid:17) ,x k ⋆ x ij = Z dk dq d l d r δ ( k ) δ ( q ) δ ( l ) δ ( r ) ∂ k k ∂ r ij ( e k, l ⋆ e q, r ) = ˆ x k ⊲ x ij = x k x ij − i λ (cid:16) η ik x j − η jk x i (cid:17) ,x ij ⋆ x k = Z dk dq d l d r δ ( k ) δ ( q ) δ ( l ) δ ( r ) ∂ l ij ∂ q k ( e k, l ⋆ e q, r ) = ˆ x ij ⊲ x k = x ij x k + i λ (cid:16) η ik x j − η jk x i (cid:17) . (33)Therefore, [ x i , x j ] ⋆ = iλβx ij , [ x ij , x k ] ⋆ = iλ ( η ik x j − η jk x i ) , [ x ij , x kl ] ⋆ = iλ ( η ik x jl − η jk x il − η il x jk + η jl x ik ) , (34)which is isomorphic to the algebra (2).
4. The twist for the Weyl realization
In this section, we construct the twist operator at second order in λ , using a perturbative approach.The twist is defined as a bilinear operator such that ∆ h = F ∆ h F − for each h ∈ so (1 , N ).The twist in a Hopf algebroid sense can be computed by means of the formula [10,18] F − ≡ e F = e − i p µν ⊗ x µν e i p ρσ ⊗ ˆ x ρσ . (35)6y the CBH formula e A e B = e A + B + [ A,B ]+ ... , one gets F = i p µν ⊗ (ˆ x µν − x µν ) − p µν p ρσ ⊗ [ x µν , ˆ x ρσ ] + . . . (36)where we can safely ignore further terms because it can be explicitly checked that they give contributions oforder λ .Substituting (15) in (36), one obtains F = iλ p αγ ⊗ x αβ p γβ − iλ (cid:0) p αγ ⊗ x αβ p βδ p γδ − p γδ ⊗ x αβ p αγ p βδ − p αγ p βδ ⊗ x αβ p γδ + p αγ p δγ ⊗ x αβ p δβ (cid:1) . (37)Using the Hadamard formula e A Be − A = B + [ A, B ] + [ A, [ A, B ]] + . . . , it is easy to check that F ∆ p µν F − = ∆ p µν , (38)with ∆ p µν given in (25), as expected.
5. Generic realizations
We consider now the most general realization of the commutation relations (2) in terms of the elements ofthe generalized Heisenberg algebra (3), up to second order in λ . Of course, this will deform the commutationrelations between coordinates and momenta.The generic form of the Lorentz-covariant combinations of the generators of the algebra (3), linear in x i , x ij , up to order λ is given by ˆ x i = x i + λ (cid:0) βc x ik p k + c x k p ik (cid:1) + λ (cid:0) β ( c x i p k + c x k p k p i + c x ik p kl p l + c x kl p k p il ) + c x k p kl p il (cid:1) , ˆ x ij = x ij + λ (cid:0) d x ik p jk + d x i p j − ( i ↔ j ) (cid:1) + λ (cid:0) βd x ik p k p j + d x ik p kl p jl + d x kl p ik p jl + d x i p k p jk + d x k p ik p j − ( i ↔ j ) (cid:1) . (39)In order to satisfy (2) to first order in λ one must have c = − , d = 12 , c + d = 1 . (40)Hence, at this order one has one free parameter. In particular, in the Weyl realization (17), d = c = .To second order in λ , one has ten new parameters c , . . . , c , d , . . . , d that must satisfy the six inde-pendent relations c − c + c = d , c c + c = 12 , d − d = − ,c − d = 14 , c c − d = 0 , c − c d + c + d = 0 . (41)Hence up to second order one has five free parameters. For example, one may choose as free parameters c , c , c , d and d , so that d = 1 − c and c = 1 − c c , c = 12 − c − c c = c − c − d ,d = 14 − c − c , d = −
14 + 2 d , d = c − c − d . (42) In principle, one may add further terms to (39), namely the terms x i p kl p kl and x kl p kl p i to ˆ x i , and x ij p k p k , x ij p kl p kl , x kl p kl p ij , x k p k p ij to ˆ x ij . However these terms must vanish if one requires that theSnyder algebra is satisfied. 7t is easy to verify that the coefficients of the Weyl realization (17) satisfy the above relations with c = , c = − c = − d = − d = − .Note that setting β = 0 in (39) one obtains realizations of the Poincar´e algebra. For example, the Weylrealization for the operators ˆ x i and ˆ x ij of the Poincar´e algebra becomesˆ x i = x i + λ x k p ik − λ x k p kl p il , ˆ x ij = x ij + (cid:20) λ (cid:0) x i p j + x ik p jk (cid:1) − λ (cid:0) x ik p jl p kl − x kl p ik p jl − x i p k p jk + 2 x k p i p jk (cid:1) − ( i ↔ j ) (cid:21) . (43)Through the same procedure as in the previous section, one can determine the coproduct for the genericrealization (39). The differential equations for P i ( tk ) and P ij ( tk ) are d P i dt = i h p i , k k ˆ x k + 12 k kl ˆ x kl i(cid:12)(cid:12)(cid:12)(cid:12) p →P ( tk ) ,d P ij dt = i h p ij , k k ˆ x k + 12 k kl ˆ x kl i(cid:12)(cid:12)(cid:12)(cid:12) p →P ( tk ) , (44)with initial conditions P i (0) = q i and P ij (0) = q ij . After some calculations, one can write down the functions D i ( k, q ) and D ij ( k, q ) that appear in the star product of plane waves, D i ( k, q ) = k i + q i + λ ( − c k j q ij + d k ij q j ) + λ h β ( c c + c ) k j q i + β ( − c c + 2 c + c ) k i k j q j + (cid:0) c − c d − c d + c + d (cid:1) k j k jk q ik + ( c d + c d + c − d ) k ik k j q jk + (cid:0) d + d − d (cid:1) k ij k jk q k + 2 βc k i q j + 2 βc k j q j q i + 2 d k ij q jk q k + 2 d k jk q j q ik + 2 c k j q jk q ik i , (45)and D ij ( k, q ) = k ij + q ij + λ ( − d k ik q jk + βc k i q j − ( i ↔ j )) + λ h β ( − c c + c − c ) k i k k q jk + ( − d + d + 2 d ) k ik k kl q jl + (cid:0) d + d (cid:1) k ik k jl q kl + β ( c d + c + d ) k ik k k q j + β ( c d + c d + c − d ) k i k jk q k + 2 βd k ik q k q j + 2 d k ik q kl q jl + 2 d k kl q ik q jl + 2 βc k i q k q jk + 2 βc k k q ik q j − ( i ↔ j ) i . (46)From these functions one can easily obtain the star product and the coproduct in the general case, see (25)and (31). In particular, for c = − and k ij = q ij = 0, one has e ik i x i ⋆ e iq j x j = e i [ k i + q i + λ β ( c q j k i +2 c k j q j q i +( c − c ) k j q i +(2 c + c + c ) k j q j k i )] x i − i λβk i q j x ij , (47)which for c = , c = − c = − reduces to the first relation in (32).
6. Comparison with the Girelli-Livine approach
The authors of [7] studied our model in 3D Euclidean space using geometric methods, with a verydifferent parametrization, adapted to the coset space nature of the Snyder momentum space. In our notations,their star product for plane waves takes the form, at second order in λ , e ik i x i ⋆ e iq j x j = exp (cid:20) i (cid:18) k i + q i + λ β k j q j k i + k j q i + 2 k j q j q i ) (cid:19) x i − i λβ k i q j x ij (cid:21) . (48)This expression corresponds to a realization (39) with c = − , d = and c = c = 0. It follows from(42) that c = 1, but the other coefficients are not determined and depend on three free parameters. If one8lso requires d = 0, this may be called a generalized Snyder realization, since it obeys all the commutationrelations of the original Snyder model [2], given by (2) and (5). Note that the momenta p ij do not appear inthese relations. Of course, additional commutation relations are obeyed by the momenta p ij , but they arenot of interest for our considerations.One may consider more general realizations belonging to the previous class, with c = − , d = , c = 0 and three free parameters. For example, c = − , implies c = 0 and gives rise to a realization that,for d = 0, reproduces at order β the commutation relations of the Maggiore realization introduced in [9].More generally, these representations generalize those introduced in [10], with arbitrary c and c =1 + 2 c . In particular, one can choose the free parameters such thatˆ x i = x i + λ β h ( c − x i p k + 2 c x k p k p i i − λβ m ik p k ˆ x ij = ˆ m ij + λ ( x i p j − x j p i ) , (49)where the ˆ m ij generate the Lorentz algebra so (1 , N −
1) and[ ˆ m ij , x k ] = [ ˆ m ij , p k ] = 0 . (50)For example, in the Weyl realization of ˆ m ij , d = − d = − , leaving c as a free parameter. In the limit β = 0, ˆ x i reduces to x i .
7. Conclusions
The coalgebra usually associated to the Snyder model is noncoassociative, and this fact prevents thedefinition of a proper Hopf algebra, whose coproduct is by definition coassociative. The reason is that thealgebra of the position operators of the Snyder model does not close. However this can be remedied byincluding the Lorentz generators in the defining algebra [7]. In this way a standard coassociative Hopfalgebra can be defined.In this paper we have studied the realizations of this algebra in terms of the deformations of an extendedHeisenberg algebra, which contains tensorial elements that in the deformation assume the role of Lorentzgenerators. We have obtained the coproduct, the star product and the twist in the case of a Weyl realization.We have also considered the most general realization of the algebra up to second order in the expansionparameter λ (or equivalently at first order in the Snyder parameter β ) and calculated the correspondingcoproduct and star product.Although this approach may be considered more rigorous than the standard one from a mathematicalpoint of view, the physical interpretation of the new degrees of freedom, related to the Lorentz generators andtheir momenta, is still an issue. For example, in ref. [7] they were interpreted in a Kaluza-Klein perspective,as coordinates of extra dimensions, but a more compelling view might be figured out.One may also consider the construction of a field theory based on this formalism, along the lines ofref. [6]. Even if the shortcomings due to the nonassociativity of the star product [12] are absent in thepresent framework, different problems arise because of the intertwining between the position and the extradegrees of freedom [7].To conclude, we observe that also the standard commutative theory, as well as DFR spacetime [3], canbe formulated in this extended framework, as we have observed several times in the text. The investigationof these elementary cases could be a good starting point to better understand the physical implications ofthe present formalism, in particular in relation with quantum field theory. References [1] L.J. Garay, Int. J. Mod. Phys.
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