Heisenberg doubles for Snyder type models
aa r X i v : . [ h e p - t h ] J a n Heisenberg doubles for Snyder type models
Stjepan Meljanac ∗ and Anna Pacho l † Division of Theoretical Physics, Rudjer Boˇskovi´c Institute,Bijeniˇcka c.54, HR-10002 Zagreb, Croatia Queen Mary, University of London, Mile End Rd., London E1 4NS, UK.
A Snyder model generated by the noncommutative coordinates and Lorentz gen-erators which close a Lie algebra can be equipped in a Hopf algebra structure. TheHeisenberg double is described for the dual pair of generators: noncommutative Sny-der coordinates and momenta (with non-coassociative coproducts). The phase spaceof the Snyder model is obtained as a result. Further, the extended Snyder algebra isconstructed by using the Lorentz algebra, in one dimension higher. The dual pair ofextended Snyder algebra and extended Snyder group is then formulated. Two fur-ther Heisenberg doubles are considered, one with the conjugate tensorial momentaand another with the Lorentz matrices. Explicit formulae for all Heisenberg doublesare given.
I. INTRODUCTION
Noncommutative coordinates and noncommutative spacetimes lead to a modification of theircorresponding relativistic symmetries, which are described by quantum groups (Hopf algebras).Having a dual pair of Hopf algebras (one describing the quantum symmetry group A and its dualquantum Lie algebra A ∗ ), one can construct the so called Heisenberg double. The Heisenbergdouble, although interesting from mathematical point of view, can be also seen as a generalizationof the quantum mechanics phase space (Heisenberg algebra). Such constructions have been ofinterest especially for the κ -Minkowski noncommutative spacetime and its deformed relativisticsymmetry κ -Poincar´e quantum group [1], [2], [3], [4], [5] as well as for the θ -deformation [5] andfor the general Lie algebra type noncommutative spaces [6].In the present paper we want to focus on the Heisenberg double construction applied to aSnyder model. In the 1940s, Snyder proposed the model of Lorentz invariant discrete spacetime ∗ Electronic address: [email protected] † Electronic address: [email protected] [7] as the first example of the noncommutative spacetime.The Snyder model has been attracting quite a lot of attention in the literature [8]-[26]. Fieldtheory on this space was considered, for example in [8], [9], [10], [11], [12], its extension to acosmological [13] and curved [14], [15], [16] backgrounds was proposed, deformed Heisenberg un-certainty relations were investigated [17], different applications to quantum gravity phenomenol-ogy have been considered as well, see, e.g. [13], [18],[19]. Different Snyder phase spaces arisingwithin the projective geometry context were investigated in [20], [21] and the κ -Snyder spacewith non associative star product was proposed in [22].In the Snyder model the coordinates do not commute and their commutation relation is pro-portional to the Lorentz generators. For this reason noncommutative coordinates by themselvesdo not close a Lie algebra and cannot be equipped in the Hopf algebra structure. In this paperwe investigate various ways of extending the Snyder space so that we can define the Lie algebracontaining the Snyder coordinates. The Hopf algebra structure then arises naturally and theHeisenberg double construction can be performed.We start with the algebra generated by Snyder coordinates ˆ x i and Lorentz generators M jk , sothat it becomes a Lie algebra and then we equip it with the Hopf algebra structure, where the Liealgebra generators have primitive coproducts. For the Heisenberg double construction we needthe dual Hopf algebra. However, since we are interested in the phase space resulting from theHeisenberg double, we focus only on the commuting momenta p i as dual to Snyder coordinatesˆ x i . We can then follow the Heisenberg double construction to obtain the relations betweenthe Snyder coordinates and their dual momenta. The full set of cross commutation relationsobtained this way is described in detail and the Heisenberg double phase space corresponds tothe known versions of the Snyder phase space used in various applications, e.g. in [14], [16],[18], [19]. We obtain the Heisenberg double relations for both (non-coassociative) coproducts ofmomenta: in the so-called Snyder realization [7], [8] as well as for the general realization [8]. Thecoproduct of momenta in general realization is given up to the second order in β (deformationparameter) and we can recover the expression for Snyder phase space corresponding to Snyder[8], Maggiore [23] and Weyl realizations [8].In the following sections we analyse the extended version of the Snyder model, where Snydercoordinates are identified as ˆ x i ∼ ˆ x iN = M iN ∈ so (1 , N ) /so (1 , N −
1) [9], [24]. We construct twofull Heisenberg doubles, firstly for the extended Snyder algebra generated by tensorial coordinatesˆ x µν with its dual Hopf algebra generated by tensorial (conjugate) momenta p ρσ . This way we findthe Heisenberg double for the extended Snyder space which may be considered as the extendedSnyder phase space. Secondly, we consider another Heisenberg double for the extended Snyderalgebra with its dual Hopf algebra of functions on a group Λ ρσ (Lorentz matrices). We alsopresent the Weyl realization for these Lorentz matrices in terms of tensorial momenta. We finishwith brief conclusions. II. HEISENBERG DOUBLE FOR THE SNYDER MODEL
Snyder space is defined by the position operators ˆ x i obeying the following commutationrelations: [ˆ x i , ˆ x j ] = iβM ij (1)where M ij are the generators of the Lorentz algebra so (1 , N −
1) and β is the Snyder parameterof length square dimension (usually assumed to be of order of Planck length L p ) that sets thescale of noncommutativity (we use natural units ~ = c = 1). Note that here i, j = 0 . . . , N − so (1 , N − M ij generators satisfy thestandard commutation relations[ M ij , M kl ] = i ( η ik M jl − η il M jk + η jl M ik − η jk M il ) . (2)We also have the cross commutation relations between Lorentz generators and Snyder coordi-nates: [ M ij , ˆ x k ] = i ( η ik ˆ x j − η jk ˆ x i ) . (3)Relations (1), (2), (3) constitute a Lie algebra, which we will denote as algebra A .To construct the Heisenberg double corresponding to the noncommutative Snyder space weneed to equip the above Lie algebra A , with the Hopf algebra structure. It is enough to imposethe following: ∆ (ˆ x i ) = ˆ x i ⊗ ⊗ ˆ x i , (4) ǫ (ˆ x i ) = 0 and S (ˆ x i ) = − ˆ x i . (5)∆ ( M ij ) = M ij ⊗ ⊗ M ij , (6) ǫ ( M ij ) = 0 and S ( M ij ) = − M ij . (7)Since we are interested in the phase space resulting from the Heisenberg double construction,as generators dual to the Snyder coordinates ˆ x i , we consider commuting momenta p i which areequipped in the non-coassociative coalgebra structure. We denote the Hopf algebra generatedthis way as ˜ A ∗ .The defining relations of ˜ A ∗ are: [ p i , p j ] = 0 (8)and the (non-coassociative) coalgebra structure is the following:∆ p i = 1 ⊗ p i + 11 − βp k ⊗ p k p i ⊗ − β p βp p i p j ⊗ p j + p βp ⊗ p i ! ,ǫ ( p i ) = 0 , S ( p i ) = − p i , (9)where p = η ij p i p j is Lorentz invariant and i, j = 0 . . . , N −
1. Here we chose the so-calledSnyder realization [7], [8] for the coproduct.The duality relations < , > : ˜ A ∗ × A → C are: < p i , ˆ x j > = − iη ij , (10)and < p i , M jk > = 0 . (11)One can explicitly check that the duality satisfies the compatibility conditions between algebras.We consider the left Hopf action ⊲ of ˜ A ∗ on A which is defined as p i ⊲ ˆ x j = < p i , ˆ x j (2) > ˆ x j (1) = − iη ij . (12)We can now construct the corresponding Heisenberg double (we refer the reader to the Appendixfor the details of the Heisenberg double construction). The resulting cross commutation relationsare: [ p i , ˆ x j ] = ˆ x j (1) < p i (1) , ˆ x j (2) > p i (2) − ˆ x j p i = − i ( η ij + βp i p j ) . (13)We note that relations (13) between momenta and Snyder coordinates, obtained here via theHeisenberg double, are agreeing with the commutation relations for the phase space of the Snydermodel usually considered in the literature, see, e.g., [14], [16], [18], [19], although not obtainedfrom the Heisenberg double construction therein. A. Heisenberg double corresponding to the general realization for coproducts ofmomenta
In the previous section we have used the coalgebra sector for momenta in the so-called Snyderrealization [7], [8], [25]. There exist more possible realizations for momenta’s (non-coassociative)coproducts, for example Maggiore [8], [23] and Weyl realizations [8], [25] can be considered aswell.However, there is a general way to write the (non-coassociative) coproduct for momentacorresponding to the Snyder model, i.e. the so-called general realization , but formula we recall The general realization for the coproduct of momenta corresponds to the general realization for the Snydercoordinates, see eq. (6), (7) in [8], where ˆ x i ⊲ x i , M ij ⊲ e ik i ˆ x i ⊲ e ik i x i , see eq. (12), (13) in [8]. below [8] is calculated only up to the second order in the parameter β :∆ p i = 1 ⊗ p i + p i ⊗ β (cid:18)(cid:18) c − (cid:19) p i ⊗ p + (cid:18) c − (cid:19) p i p k ⊗ p k + c (cid:0) p ⊗ p i + 2 p k ⊗ p k p i (cid:1)(cid:19) + O ( β ) , (14) ǫ ( p i ) = 0 , S ( p i ) = − p i . (15)This formula describes (non-coassociative) coproduct for momenta in any general realization.The choice of parameter c encodes: • Snyder realization if c = (and this case, in its finite form, was considered in the previoussection); • Maggiore realization if c = 0; • Weyl realization if c = .We can now construct the Heisenberg double corresponding to the general realization of themomenta coproduct for the Snyder model (the duality relations and the left Hopf action remainunchanged (10), (11), (12)). It results in the following cross commutation relations betweenmomenta and Snyder coordinates:[ p i , ˆ x k ] = ˆ x k (1) < p i (1) , ˆ x k (2) > p i (2) − ˆ x k p i = − iη ik (cid:18) β (cid:18) c − (cid:19) p (cid:19) − icβp k p i + O ( β ) . (16)This reduces to (for the specific, above mentioned, choices of c parameter): • Snyder realization [ p i , ˆ x k ] = − i ( η ik + βp i p k ) (17)cf. (13); • Maggiore realization [ p i , ˆ x k ] = − iη ik (cid:18) − β p (cid:19) + O ( β ) (18) • Weyl realization [ p i , ˆ x k ] = − iη ik (cid:18) − β p (cid:19) − i βp k p i + O ( β ) . (19)The relations (16) offer the general quantum phase space for the Snyder model and interestingphysical applications could be based on this description. It is worth to note that in the limit of β → p i = 1 ⊗ p j + p i ⊗ β = 0 (classical case) there exists full duality between algebra A generators x i , M ij andgroup elements p i , Λ ij (where Λ ij are matrix elements of Lorentz matrices, see e.g. [1], [2], [3],[4], [5]).In this section we cannot define the dual elements to the Lorentz generators M ij for β = 0(cf. footnote 1). And the momenta dual to Snyder coordinates do not allow for expanding theLorentz algebra (2) to the Poincar´e algebra. The only way to construct the full dual Hopf algebraand the full Heisenberg double is described in Sections III and IV and requires introducing theextended noncommutative coordinates ˆ x ij . III. UNIFIED NOTATION FOR THE SNYDER ALGEBRA: EXTENDED SNYDERMODEL
In the previous section, to be able to consider a Hopf algebra related with the Snyder model,we have expanded the commutation relations between Snyder coordinates (1) by the Lorentzalgebra (2) and included the cross commutation relations (3) which allowed us to define a Hopfalgebra related to the Snyder model (1) - (7). After defining the dual momenta (with non-coassociative coproducts) we were able to calculate the cross commutation relations comingfrom the Heisenberg double construction compatible with the Snyder model (1) and obtain theSnyder phase space as a result. However, in that framework it was not possible to define the dualelements to the Lorentz generators, hence we were not able to obtain the full Heisenberg doublefor the Snyder model. The full Heisenberg double for the Snyder space requires an introductionof extended noncommutative coordinates which we discuss in this section.Another way to obtain a Lie algebra from the Snyder model (1) is to extend it by identifyingthe Snyder coordinates as ˆ x i ∼ ˆ x iN = M iN ∈ so (1 , N ) /so (1 , N −
1) [9], [24]. This way one candefine a Lie algebra corresponding to the extended Snyder space and further one can also definethe (coassociative) Hopf algebra structure [24]. In this approach the Snyder coordinates are seenas generators of the Lorentz algebra, but the Lorentz algebra considered now has one dimensionhigher (i.e. so (1 , N ) instead of so (1 , N −
1) from Sec. II). Thanks to this unified extended version of the Snyder model [9], [24] the Heisenberg double construction completely mimics theconstruction of the undeformed Heisenberg double for the Lorentz algebra so (1 , N ) with its dualalgebra of functions on a group SO (1 , N ) (Lorentz matrices), however the noncommutativityparameter β related with the Snyder space (1) will be now implicitly included in the cross Many authors use the word ” generalized ” for the version of Snyder space in a different meaning, see e.g. [21]or [26],therefore, following [24], we shall call the version used here as ” extended ” instead of generalized - sinceit is unified with the Lorentz generators. commutation relations. This will be considered in Section IV B. In Section IV C we will alsopresent the realizations for the Lorentz matrices in the so-called Weyl realization [24], [27], [29].However, in analogy to obtaining the phase space from the Heisenberg double (as it was donein Sec. II) we can also consider a dual Hopf algebra of momenta and find the correspondingextended Snyder phase space. In the remaining part of the paper, we want to focus on findingthe explicit formulae for such Heisenberg doubles corresponding to the extended Snyder model.To that aim, we first need to define the Hopf algebra related with the extended Snyder modeland also define the dual Hopf algebra of objects which would play the role of momenta. We startwith embedding the Snyder algebra relations (1), (2), (3) in an algebra which is generated bythe N position operators denoted by ˆ x i and N ( N − / x ij ,transforming as Lorentz generators [9], [24]. This larger algebra has the following commutationrelations: [ˆ x i , ˆ x j ] = iλβ ˆ x ij , [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 ) , (20)[ˆ x ij , ˆ x k ] = iλ ( η ik ˆ x j − η jk ˆ x i ) , (21)where λ and β are real parameters. We can easily notice that these commutation relationsreduce to those of the standard Lorentz algebra acting on commutative coordinates in the limitof β → λ →
1, and to the Lie algebra from Sec. II ((1), (2), (3)) in the limit λ → so (1 , N ), and write the previous formulas (20), (21)more compactly defining, for positive β , ˆ x i = p β ˆ x iN . (22)The extended Snyder algebra then takes the form [9], [24] of the Lorentz algebra so (1 , N ) (to beprecise it is U so (1 ,N ) [[ λ ]]), given by one set of commutation relations as[ˆ x µν , ˆ x ρσ ] = iλ ( η µρ ˆ x νσ − η νρ ˆ x µσ + η νσ ˆ x µρ − η µσ ˆ x νρ ) , (23)with η NN = 1 and η kN = 0, here µ = 0 , , . . . N (Greek indices are running from 0 up to N ,whereas the Latin indices are i, j = 0 , , . . . , N − • to the Snyder noncommutative spacetime relations (20)[ˆ x jN , ˆ x iN ] = [ 1 √ β ˆ x j , √ β ˆ x i ] = iλ ( η ji ˆ x NN − η Ni ˆ x jN + η NN ˆ x ji − η jN ˆ x Ni ) = iλ ˆ x ji (note that ˆ x NN = 0 due to antisymmetricity), • to the commutation relations for Lorentz generators (20)[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 ) , • and to cross commutation relations of Lorentz generators acting on coordinates (21)[ˆ x jk , ˆ x iN ] = [ˆ x jk , √ β ˆ x i ] = iλ ( η ji ˆ x kN − η ki ˆ x jN + η kN ˆ x ji − η jN ˆ x ki ) == iλ √ β ( η ji ˆ x k − η ki ˆ x j ) . In turn, these all reduce to (1), (2), (3) from Sec. II, respectively for λ → A. Generalized Heisenberg algebra
To discuss the extended phase space associated with this extended Snyder model (23) as aresult of the Heisenberg double construction, we need to first recall few facts about the general-ized Heisenberg algebra and Weyl realization of the Lorentz algebra based on results presentedin [27].The generalized Heisenberg algebra can be introduced as an unital, associative algebra gener-ated by (commutative) x µν and p µν (both antisymmetric), satisfying the following commutationrelations: [ x µν , x αβ ] = 0 , (24)[ p µν , p αβ ] = 0 , (25)[ p µν , x ρσ ] = − i ( η µρ η νσ − η µσ η νρ ) . (26)Here we consider the elements p µν as canonically conjugate to x µν which can be realized instandard way as p µν = − i ∂∂x µν .Commutative coordinates x µν can be viewed as the classical limit (when λ →
0) of ˆ x µν generators of so (1 , N ) used in the extended version for the Snyder algebra (23). In other words,the Lie algebra so (1 , N ), more specifically its universal enveloping algebra U so (1 ,N ) [[ λ ]], generatedby ˆ x µν can be seen as a deformation of the underlying commutative space x µν with λ as thedeformation parameter.Now, since we are interested in the Snyder model (in its extended version) and the cor-responding deformed phase space, first let us notice that we can make use of the analogousrelation to (22) for the commutative coordinates, i.e. take x i = √ βx iN and similarly for theconjugate canonical momenta p µν we can introduce: p i = p iN √ β . This allows us to reduce theabove generalized Heisenberg algebra (24) - (26) to: • the usual Heisenberg algebra sector, i.e. quantum mechanical phase space correspondingto the commutative (classical) space-time:[ x i , x j ] = 0 , [ p i , p j ] = 0 (27)[ p j , x i ] = [ p iN , x jN ] = − i ( η ij η NN − η iN η Nj ) = − iη ij , (28) • and ”the remaining part”, consisting of commutation relations between x ij - tensorialcoordinates and p ij - their corresponding canonical momenta:[ x ij , x kl ] = [ x ij , x k ] = 0 , (29)[ p ij , p kl ] = [ p ij , p k ] = 0 , (30)[ p ij , x kl ] = − i ( η ik η jl − η il η jk ) , (31)[ p i , x kl ] = [ p ij , x k ] = 0 . (32)Therefore, relations (24)-(26) indeed describe the generalization of the quantum mechanicalphase space as it contains as subalgebra the relations (27)-(28).The Weyl realization of the Lorentz algebra in terms of this generalized Heisenberg algebra(24)-(26) (as formal power series) have been discussed in detail in [27]. IV. EXTENDED SNYDER SPACE AND ITS HEISENBERG DOUBLES
We are now ready to discuss how to construct two full Heisenberg doubles corresponding tothe extended Snyder space (23), in its unified so (1 , N )-like version (with the generators ˆ x µν ).For the purpose of this section let us denote the extended Snyder algebra, defined by relations(23) as an algebra B . To construct the Heisenberg double for the extended Snyder algebra B we first need to equip it in the Hopf algebra structure (which is straightforward as we can usethe primitive coproducts for ˆ x µν ) and then we need to define the dual Hopf algebra.We equip the Snyder algebra B (as so (1 , N )), defined by relations (23), with the Hopf algebrastructure as follows: ∆ (ˆ x µν ) = ∆ (ˆ x µν ) , (33) ǫ (ˆ x µν ) = 0 and S (ˆ x µν ) = − ˆ x µν . (34)With the above relations algebra B , using (22), leads to the extended Snyder Hopf algebra. A. Extended Snyder phase space from the Heisenberg double construction
To discuss the phase space corresponding to the extended Snyder space, we consider the Hopfalgebra generated by p µν (satisfying (25)), as a dual Hopf algebra B ∗ , which is equipped with the0Hopf algebra structure introduced in [24][see, eq.(25) therein] where the coproducts, calculatedup to the third order, have the following form:∆ p µν = ∆ p µν − λ p µα ⊗ p να − p να ⊗ p µα ) + − λ
12 ( 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 ) + O ( λ ) . (35)Counits are ǫ ( p µν ) = 0 and antipodes are S ( p µν ) = − p µν . This defines the coassociative Hopfalgebra B ∗ as the dual to the extended Snyder Hopf algebra B . The above coproducts formomenta are corresponding to the so-called Weyl realization for the ˆ x µν . One could use thecoproducts for generic realization but they depend on 5 free parameters and are calculated upto the second order in λ [24].The duality relation < , > : B ∗ × B → C is as follows: < p µν , ˆ x ρσ > = − i ( η ρµ η σν − η σµ η ρν ) . (36)We take the left Hopf action ⊲ of B ∗ on B , which is defined by p ρσ ⊲ ˆ x µν = < p ρσ , ˆ x µν (2) > ˆ x µν (1) = − i ( η ρµ η σν − η σµ η ρν ) . (37)We can now construct the corresponding Heisenberg double resulting in the following crosscommutation relations:[ p µν , ˆ x ρσ ] = ˆ x ρσ (1) < p µν (1) , ˆ x ρσ (2) > p µν (2) − ˆ x ρσ p µν == − i ( η ρµ η σν − η σµ η ρν ) + iλ η ρµ p νσ − η σµ p νρ − η ρν p µσ + η σν p µρ ) ++ iλ
12 [ η ρµ p σβ p νβ − η σµ p ρβ p νβ − η ρν p σβ p µβ + η σν p ρβ p µβ − p µρ p νσ + 2 p µσ p νρ ] + O (cid:0) λ (cid:1) . (38)For description of the phase space corresponding to the extended Snyder model (in Snydercoordinates ˆ x i , ˆ x ij ), we make use of the isomorphism (22) ˆ x i = √ β ˆ x iN and p i = p iN √ β . The aboveduality (36) then becomes: < p j , ˆ x i > = − iη ij , (39) < p k , ˆ x ij > = 0 , (40) < p kl , ˆ x i > = 0 , (41) < p kl , ˆ x ij > = − i ( η ik η jl − η jk η il ) . (42) In general, if noncommutative coordinates close a Lie algebra, as it is the case for ˆ x µν , then the correspondingcoproducts of momenta are coassociative [28], [29]. The Weyl realization for the extended Snyder space is defined as e ik i ˆ x i + i k ij ˆ x ij ⊲ e ik i x i + i k ij x ij whereˆ x i ⊲ x i , ˆ x ij ⊲ x ij , see eq. (17) in [24]. Note that the action differs from the one described in footnote1. • commutation relations between Snyder coordinates and their coupled momenta:[ p k , ˆ x i ] = [ p kN , ˆ x iN ] = − iη ik (1 − βλ p l p l ) − iβλ p k p i + iλ p ki + iλ p il p kl + O (cid:0) λ (cid:1) , (43) • commutation relations between tensorial coordinates and their coupled momenta:[ p kl , ˆ x ij ] = − i ( η ik η jl − η jk η il ) + i λ η ik p lj − η jk p li − η il p kj + η jl p ki ) + iλ
12 [( η ik p jm p lm − η jk p im p lm ) − ( η il p jm p km − η jl p im p km ) − p ki p lj + 2 p kj p li ] + O (cid:0) λ (cid:1) , (44) • and mixed relations:[ p kl , ˆ x i ] = h p kl , p β ˆ x iN i = i λ β ( η ik p l − η il p k ) − i λ β [ η ik p m p lm − η il p m p km +2 p ki p l − p k p li ]+ O (cid:0) λ (cid:1) , (45)[ p k , ˆ x ij ] = (cid:20) √ β p kN , ˆ x ij (cid:21) = − i λ η ik p j − η jk p i ) − i λ
12 [( η ik p jl p l − η jk p il p l ) − p ki p j +2 p kj p i ]+ O (cid:0) λ (cid:1) . (46)One can notice that the commutator between momenta generators and the Snyder coordinates(43) obtained from (38) actually resembles the Snyder model phase space for the Weyl realization(19) obtained in Sec.II. The first three terms agree up to the factor λ but the remaining termsinclude the tensorial momenta.We have now obtained the full extended Snyder phase space (43)-(46) resulting from theHeisenberg double construction.It is worth to mention that some authors, see e.g. [15], [20], also consider another version ofthe Snyder phase spaces, where momenta do not commute, but this has not been in our focusin this work and we have not considered this type of phase spaces here, the momenta sector isalways commutative, for both the Snyder model considered in Sec. II and the extended Snydermodel in Sec. IV A. B. Another Heisenberg double for the extended Snyder algebra
To construct another Heisenberg double for the extended Snyder algebra B written in anunified so (1 , N )-like form (23), it is quite straightforward to mimic the Heisenberg double con-struction for the Lorentz algebra.We take the extended Snyder Hopf algebra B (23) equipped with the Hopf algebra structure(33),(34), as before. We define the dual Hopf algebra algebra D (as the dual to the extended2Snyder Hopf algebra B ) as an algebra of functions on a group SO (1 , N ) which is generated byLorentz matrices Λ αβ , i.e.: D = F ( SO (1 , N )) = { Λ αβ : [Λ αβ , Λ µν ] = 0 : Λ T η Λ = η } , (47)∆ (Λ ρσ ) = Λ ρα ⊗ Λ ασ ; ǫ (Λ ρσ ) = δ ρσ ; S (Λ ρσ ) = (Λ − ) ρσ = Λ σρ . (48)Note that the Greek indices are running up to N , i.e. α, β = 0 , , . . . , N . The duality relation is < , > : D × B → C is given by: < Λ ρσ , ˆ x µν > = − iλ ( η ρµ η σν − η ρν η σµ ) . (49)We consider the left Hopf action ⊲ of D on B , which is defined byΛ ρσ ⊲ ˆ x µν = < Λ ρσ , ˆ x µν (2) > ˆ x µν (1) = η ρσ ˆ x µν − iλ ( η ρµ η σν − η ρν η σµ ) . (50)And we calculate the cross commutation relations defining the Heisenberg double as:[Λ ρσ , ˆ x µν ] = ˆ x µν (1) < Λ ρσ (1) , ˆ x µν (2) > Λ ρσ (2) − ˆ x µν Λ ρσ (51)= − iλ ( η ρµ Λ νσ − η ρν Λ µσ ) . (52)For the description of the Heisenberg double for the extended Snyder model (in Snyder co-ordinates ˆ x i , ˆ x ij ), we again make use of the isomorphism (22), and the above formulae leadto: • duality for the Snyder coordinates with Lorentz matrices: < Λ jk , ˆ x i > = p β < Λ jk , ˆ x iN > = − iλ p β ( η ji η kN − η jN η ki ) = 0 , (53) < Λ jN , ˆ x i > = p β < Λ jN , ˆ x iN > = − iλ p βη ji , (54) < Λ Nk , ˆ x i > = p β < Λ Nk , ˆ x iN > = iλ p βη ki , (55) < Λ NN , ˆ x i > = p β < Λ NN , ˆ x iN > = 0 , (56) • duality of the Lorentz generators with their dual Lorentz matrices: < Λ jk , ˆ x ip > = − iλ ( η ji η kp − η jp η ki ) , (57) < Λ jN , ˆ x ip > = 0 = < Λ Nk , ˆ x ip >, (58) < Λ NN , ˆ x ip > = 0 . (59)Similarly the cross commutation relations from the Heisenberg double construction (52) thenbecome as follows:3 • cross commutation relations between the Snyder coordinates and Lorentz matrices:[Λ jk , ˆ x i ] = p β [Λ jk , ˆ x iN ] = − iλ p βη ji Λ Nk , (60)[Λ jN , ˆ x i ] = p β [Λ jN , ˆ x iN ] = − iλ p βη ji Λ NN , (61)[Λ Nk , ˆ x i ] = p β [Λ Nk , ˆ x iN ] = iλ p β Λ ik , (62)[Λ NN , ˆ x i ] = p β [Λ NN , ˆ x iN ] = iλ p β Λ iN , (63) • and the cross commutation relations between the Lorentz generators (of so (1 , N − jk , ˆ x ip ] = − iλ ( η ji Λ pk − η jp Λ ik ) , (64)[Λ jN , ˆ x ip ] = − iλ ( η ji Λ pN − η jp Λ iN ) , (65)[Λ Nk , ˆ x ip ] = − iλ ( η Ni Λ pk − η Np Λ ik ) = 0 , (66)[Λ NN , ˆ x ip ] = 0 . (67)The primitive coproduct for ˆ x µν reduces to the primitive coproduct for ˆ x ip (Lorentz generatorsof so (1 , N − x i (Snyder coordinates), as in (6), (4) in Sec.II, respectively.We also note that [Λ µν , p ρσ ] = 0 . (68) C. Realizations for Lorentz matrices
We can actually relate the dual momenta discussed in Sec. IV A with the dual Lorentzmatrices discussed in Sec. IV B.This can be done by introducing the realizations of the elements of the dual algebra D , i.e.functions on a group SO (1 , N ) - Lorentz matrices Λ αβ . These realizations can be expressed asa formal power series of the tensorial momenta introduced in Sec. III A. For more details werefer the reader to [27] where the Lorentz algebra extension by its dual counterpart has beendiscussed in detail. The formulas presented below base on the Theorem III.1 from [27]. TheWeyl realization [27], [29] for the generators of the algebra D , satisfying (47), can be written ina very compact form as follows: Λ ρσ = (cid:0) e λp (cid:1) ρσ . (69)If we want to calculate the explicit formulae for the realization of the elements dual to theLorentz sector in the extended Snyder algebra written in the form of tensorial coordinates ˆ x ij kl = ∞ X n =0 λ n n ! ( p n ) kl , Λ Nl = ∞ X n =1 λ n n ! p Nk (cid:0) p n − (cid:1) kl = p Nk (cid:18) Λ − ηp (cid:19) kl , Λ kN = ∞ X n =0 λ n n ! ( p n ) kN = ∞ X n =1 λ n n ! (cid:0) p n − (cid:1) kl p lN = (cid:18) Λ − ηp (cid:19) kl p lN = (cid:0) Λ − (cid:1) Nk , Λ NN = ∞ X n =0 λ n n ! ( p n ) NN = η NN + ∞ X n =2 λ n n ! p Nk (cid:0) p n − (cid:1) kl p lN = η NN + p Nk (cid:18) Λ − η − λpp (cid:19) kl p lN , where we also used the following notation ( p ) αβ = η αβ and (Λ ) ρσ = η ρσ . V. CONCLUSIONS
In this paper we have investigated three Heisenberg doubles related with the two types ofnoncommutative Snyder models. The Heisenberg double construction was widely investigatedfor other noncommutative spacetimes [1], [2], [3], [4], [5], but not yet (up to our knowledge) forthe Snyder space. Therefore this work offers the first study on Heisenberg doubles for the Snydermodel as well as for the extended Snyder model.In Sec. II we use the Heisenberg duality between Snyder coordinates and momenta with the(non-coassociative) coproducts related to the so-called Snyder realization (9). As a result of crosscommutation relations, we obtain the usual Snyder phase space (13) considered in the literature.Then we use the momenta with the (non-coassociative) coproducts in the general realizationwhich provides more general version of the Snyder phase space and reduces to all known casesfor certain choices of the parameter c (which parametrises the non-coassociative coproduct formomenta generators). However, we point out that the construction of the full Heisenberg doublefor Snyder space requires introduction of extended noncommutative coordinates.Next, we use the fact that the Snyder model can be embedded in a larger algebra: U so (1 ,N ) [[ λ ]],for which the dual algebra admits the coassociative coalgebra structure. We then construct theHeisenberg double for this extended Snyder model in two ways. Firstly, by introducing thedual tensorial momentum space. Secondly, by using the Lorentz matrices, i.e. functions on theLorentz group. The Heisenberg double of the extended Snyder algebra with their dual momentacan be interpreted as the extended Snyder phase space. The formulation of Heisenberg doubles asextended Snyder phase spaces, proposed in this work, can be used in many further applications.Additional advantage is that, since the noncommutative coordinates generate the Lie algebrathen the corresponding coproducts of momenta are coassociative, and the related star productsbetween coordinates are associative [24], which opens a way to a great number of applications5where the associative star product is required. The only drawback of this approach is theunclear physical interpretation of tensorial coordinates and corresponding conjugated momentaappearing in this picture.Nevertheless, by using the algebraic scheme of Heisenberg doubles, one can introduce thecovariant Snyder phase spaces related to two types of Snyder spacetimes and further investigatethe applications where the consistent definition of a phase space is crucial, for example suchdeformed (extended) Snyder phase space may lead to deformed Heisenberg uncertainty relations[17], or may be considered in the context of quantum gravity phenomenology [13], [18],[19], orwhen investigating cosmological [13] and curved [14], [15], [16] backgrounds coupled to Snyderspacetime. We consider our work presented here as a first step towards such applications. Acknowledgements
AP would like to acknowledge the contribution of the COST Action CA18108.
Appendix 1: Heisenberg double construction.
Let A and A ∗ be dual Hopf algebras. To construct the Heisenberg double A ⋊ A ∗ we startwith the left Hopf action ⊲ of A ∗ on A defined as: a ∗ ⊲ a = < a ∗ , a (2) > a (1) , (70)where we use the Sweedler notation ∆ ( a ) = a (1) ⊗ a (2) for the coproduct and a ∗ ∈ A ∗ , a ∈ A .Duality needs to satisfy the following compatibility conditions between algebras: < a ∗ (1) , a >< a ∗ (2) , a ′ > = < a ∗ , a · a ′ >, (71) < a ∗ , a (1) >< a ∗′ , a (2) > = < a ∗ · a ∗′ , a > . (72)The Heisenberg double corresponding to these data can be then constructed as the crossedproduct algebra (aka “smash product”) A ⋊ A ∗ . The (left) product in the crossed productalgebra (Heisenberg double) becomes:( a ⊗ a ∗ ) ⋊ ( a ′ ⊗ a ∗′ ) = a (cid:0) a ∗ (1) ⊲ a ′ (cid:1) ⊗ a ∗ (2) a ∗′ = < a ∗ (1) , a ′ (2) > aa ′ (1) ⊗ a ∗ (2) h ′ , (73)which leads to the following (left) products: a ∗ ◦ a = (1 ⊗ a ∗ ) ⋊ ( a ⊗
1) = (cid:0) a ∗ (1) ⊲ a (cid:1) ⊗ a ∗ (2) = < a ∗ (1) , a (2) > a (1) ⊗ a ∗ (2) , (74) a ◦ a ∗ = ( a ⊗ ⋊ (1 ⊗ a ∗ ) = aa ∗ . (75)6So the cross commutation relation becomes:[ a ∗ , a ] = a (1) < a ∗ (1) , a (2) > a ∗ (2) − aa ∗ . (76)There are some special cases worth considering:1. If the coproduct on A is primitive then the above formula reduces to:[ a ∗ , a ] = < a ∗ (1) , a > a ∗ (2) . (77)2. If the coproduct on the algebra A ∗ is opposite, i.e. ∆ a ∗ = a ∗ (2) ⊗ a ∗ (1) then the commutatorbecomes: [ a ∗ , a ] = a (1) < a ∗ (2) , a (2) > a ∗ (1) − aa ∗ . (78)And if, in addition A has the primitive coproduct then we get:[ a ∗ , a ] = < a ∗ (2) , a > a ∗ (1) . (79) References [1] P. Kosinski and P. Maslanka, The duality between κ - Poincare algebra and κ -Poincare group,(1994) [arXiv:hep-th/9411033];[2] S. Giller, C. Gonera, P. Kosinski, and P. Maslanka, A Note on Geometry of κ -Minkowski Space,Acta Phys. Pol. B, 27, 2171-2177 (1996);[3] J. Lukierski and A. Nowicki, Heisenberg double description of κ -Poincar´e algebra and κ -deformedphase space, Proc. 21st Intl. Colloquium on Group Theoretical Methods in Physics (Group 24:Physical and mathematical aspects of symmetries, V. K. Dobrev and H. D. Doebner, eds.), HeronPress, Sofia , 186-192 (1997) [arXiv:qalg/9702003];[4] G. Amelino-Camelia, J. Lukierski, and A. Nowicki, κ -Deformed Covariant Phase Spaceand Quantum-Gravity Uncertainty Relations, Phys. Atomic Nuclei, 61, 1811-1815 (1998)[arXiv:hep-th/9706031];[5] A. Borowiec, A. Pacho l, Heisenberg doubles of quantized Poincar´e algebras, Theor. Math. Phys.169 (2), 1620-1628 (2011);[6] Z. ˇSkoda, Heisenberg double versus deformed derivatives, Int.J. of Modern Physics A 26, Nos. 27& 28, 4845-4854 (2011) [arXiv:0909.3769];[7] H. S. Snyder, Quantized Space-Time, Phys. Rev. 71, 38, 0031-899 (1947); [8] M.V. Battisti and S. Meljanac, Scalar Field Theory on Non-commutative Snyder Space-Time,Phys. Rev. D 82, 024028 (2010) [arXiv:1003.2108];[9] F. Girelli and E. Livine, Scalar field theory in Snyder space-time: alternatives, JHEP 1103, 132(2011) [arXiv:1004.0621];[10] S. Meljanac, D. Meljanac, S. Mignemi and R. Strajn, Quantum field theory in generalised Snyderspaces, Phys. Lett. B 768, 321-325 (2017)[arXiv:1701.05862];[11] S. Meljanac, S. Mignemi, J. Trampetic and J. You, Nonassociative Snyder φ Quantum FieldTheory, Phys. Rev. D 96, 045021 (2017) [arXiv:1703.10851];[12] S. Meljanac, S. Mignemi, J. Trampetic and J. You, UV-IR mixing in nonassociative Snyder φ theory, Phys. Rev. D 97, 055041 (2018) [arXiv:1711.09639];[13] M. V. Battisti, Loop and braneworlds cosmologies from a deformed Heisenberg algebra, Phys. Rev.D 79, 083506 (2009) [arXiv:0805.1178];[14] S. Mignemi, Classical and quantum mechanics of the nonrelativistic Snyder model in curved space,Class. Quant. Grav. 29, 215019 (2012) [arXiv:1110.0201];[15] B. Ivetic, S. Meljanac, S. Mignemi, Classical dynamics on curved Snyder space, Class. QuantumGrav. 31 105010 (2014) [arXiv:1307.7076];[16] S. Mignemi and R. Strajn, Snyder dynamics in a Schwarzschild spacetime, Phys. Rev. D 90, 044019(2014) [arXiv:1404.6396];[17] M.V. Battisti and S. Meljanac, Modification of Heisenberg uncertainty relations in non-commutative Snyder space-time geometry, Phys. Rev. D 79, 067505 (2009) [arXiv:0812.3755];[18] S. Mignemi and A. Samsarov, Relative-locality effects in Snyder spacetime, Phys. Lett. A 381,1655 (2017) [1610.09692];[19] S. Mignemi and G. Rosati, Relative-locality phenomenology on Snyder spacetime, Class. QuantumGrav. 35, 145006 (2018) [1803.02134];[20] A. Ballesteros, G. Gubitosi, and F. J. Herranz, Lorentzian Snyder spacetimes and theirGalilei and Carroll limits from projective geometry, Class. Quantum Grav. 37, 195021 (2020)[arXiv:1912.12878];[21] G. Gubitosi, A. Ballesteros, and F. J. Herranz, Generalized noncommutative Snyder spaces andprojective geometry, PoS (CORFU2019) 376, 190 (2020) [arXiv:2007.09653];[22] S. Meljanac, D. Meljanac, A. Samsarov and M. Stojic, Mod. Phys. Lett. A25, 579 (2010);S. Meljanac, D. Meljanac, A. Samsarov and M. Stojic, Phys. Rev. D 83, 065009 (2011);[23] M. Maggiore, A Generalized Uncertainty Principle in Quantum Gravity, Phys. Lett. B 304, 65(1993) [arXiv:hep-th/9301067];M. Maggiore, Quantum Groups, Gravity, and the Generalized Uncertainty Principle, Phys. Rev.8