Kappa-deformed Snyder spacetime
aa r X i v : . [ h e p - t h ] D ec June 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
KAPPA-DEFORMED SNYDER SPACETIME
S.MELJANAC ∗ , D.MELJANAC † , A.SAMSAROV ‡ , M.STOJIC § Rudjer Boskovic Institute,Bijenicka c.54, HR-10002 Zagreb, Croatia
We present Lie-algebraic deformations of Minkowski space with undeformed Poincar´ealgebra. These deformations interpolate between Snyder and κ -Minkowski space. Wefind realizations of noncommutative coordinates in terms of commutative coordinates andderivatives. Deformed Leibniz rule, the coproduct structure and star product are found.Special cases, particularly Snyder and κ -Minkowski in Maggiore-type realizations arediscussed. Our construction leads to a new class of deformed special relativity theories. Keywords : noncommutative physics; snyder space; kappa-Minkowski space.PACS Nos.: 03.65.Fd, 11.10.Nx, 11.30.Cp.
1. Introduction
Noncommutative (NC) physics has became an integral part of present-day highenergy physics theories. It reflects a structure of space-time which is modified incomparison to space-time structure underlying the ordinary commutative physics.This modification of space-time structure is a natural consequence of the appearanceof a new fundamental length scale known as Planck length 1 ,
2. There are two mainphysical contexts within which a signal for the existence of a Planck length scaleappears. The first one lies within a loop quantum gravity framework in which thePlanck length plays a fundamental role. There, a presence of a new fundamentallength scale leads after quantization to discrete eigenvalues of the area and volumeoperators. It appears that in loop quantum gravity, the area and volume operatorshave discrete spectra, with minimal eigenvalue proportional to a square and cube ofthe Planck length, respectively. The second physical context where one can find asignal for the existence of a fundamental length scale comes from some observationsof ultra-high energy cosmic rays which seem to contradict the usual understandingof some astrophysical processes like, for example, electron-positron production incollisions of high energy photons. It turns out that deviations observed in theseprocesses can be explained by modifying dispersion relation in such a way as to ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] 1 une 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla S.Meljanac et al. incorporate the fundamental length scale 3. NC space-time has also been revivedin the paper by Seiberg and Witten 4 where NC manifold emerged in a certain lowenergy limit of open strings moving in the background of a two form gauge field.As a new fundamental, observer-independent quantity, Planck length is incor-porated in kinematical theory within the framework of the so called doubly specialrelativity theory (DSR) 5 ,
6. The idea that lies behind DSR is that there exist twoobserver-independent scales, one of velocity, identified with the speed of light, andthe other of mass, which is expected to be of the order of Planck mass. It can alsobe considered as a semi-classical, flat space limit of quantum gravity in a similarway special relativity is a limit of general relativity and, as such, reveals a structureof space-time when the gravitational field is switched off.Following the same line of reasoning, the symmetry algebra for doubly specialrelativity can be obtained by deforming the ordinary Poincare algebra to get somekind of a quantum (Hopf) algebra, known as κ -Poincar´e algebra 7 ,
8, so that κ -Poincar´e algebra is in the same relation to DSR theory as the standard Poincar´ealgebra is related to special relativity. κ -Poincar´e algebra is an algebra that describes in a direct way only the energy-momentun sector of the DSR theory. Although this sector alone is insufficient toset up physical theory, the Hopf algebra structure makes it possible to extend theenergy-(angular)momentum algebra to the algebra of space-time. It is shown in 9that different representations (bases) of κ -Poincar´e algebra correspond to differentDSR theory. However, the resulting space-time algebra, obtained by the exten-sion of energy-momentum sector, is independent of the representation, i.e. energy-momentum algebra chosen 9 , κ -Minkowskispace-time into space-time with noncommutative structure described by the algebrafirst introduced by Snyder 11. In 10, the use of Snyder algebra provided NC space-time structure of Minkowski space with undeformed Lorentz symmetry. In the samepaper it is argued that the algebra introduced by Snyder provides a structure ofconfiguration space for DSR and thus can be used to construct the second orderparticle Lagrangian, what would make it possible to define physical four-momentadetermined by the particle dynamics. This would be significant step forward becausethe theoretical development in this area has been mainly kinematical so far. Onesuch dynamical picture has been given recently in 12 ,
13 where it was shown thatreparametrisation symmetry of the proposed Lagrangian, together with the appro-priate change of variables and conveniently chosen gauge fixing conditions, leads toan algebra which is a combination of κ -Minkowski and Snyder algebra. This general-ized type of algebra describing noncommutative structure of Minkowski space-timeis shown to be consistent with the Magueijo-Smolin dispersion relation. This typeof NC space is also considered in 14. It has to be stressed that NC spaces in neitherof these papers are of Lie-algebra type.In order to fill this gap, in the present paper we unify κ -Minkowski and Sny-une 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla Kappa-deformed Snyder spacetime der space in a more general NC space which is of a Lie-algebra type, with Lorentzgenerators and NC space-time coordinates closing the Lie algebra. In addition, itis characterized by the undeformed Poincar´e algebra and deformed coalgebra. Inother words, we shall consider a type of NC space which interpolates between κ -Minkowski space and Snyder space in a Lie-algebraic way and has all deformationscontained in the coalgebraic sector. Particularly, in this paper we shall be interestedin finding a coproduct for translation generators, which corresponds to a general-ized momentum addition rule. First example of NC space with undeformed Poincar´ealgebra, but with deformed coalgebra is given by Snyder 11. Some other types ofNC spaces are also considered within the approach in which the Poincar´e algebrais undeformed and coalgebra deformed, in particular the type of NC space with κ -deformation 9 , , , ,
17. Here we present a broad class of Lie-algebra type de-formations with the same property of having deformed coalgebra, but undeformedalgebra. The investigations carried out in this paper are along the track of devel-oping general techiques of calculations, applicable for a widest possible class of NCspaces and as such are a continuation of the work done in a series of previous papers15 , , , , , , ,
22. The methods used in these investigations were taken overfrom the Fock space analysis carried out in 23 , , , , , κ -type of deformations and deformations of the Snyder type.In section 3 we analyze realizations of NC space in terms of operators belonging tothe undeformed Heisenberg-Weyl algebra. Section 4 is devoted to an analysis of theeffects which deformations we are considering have on the coalgebraic structure ofthe symmetry algebra. In the same section we specialize the general results obtainedto some interesting special cases, such as Snyder space and κ -Minkowski space.
2. Noncommutative coordinates and Poincar´e algebra
We are considering a Lie algebra type of noncommutative (NC) space generated bythe coordinates ˆ x , ˆ x , . . . , ˆ x n − , satisfying the commutation relations[ˆ x µ , ˆ x ν ] = i ( a µ ˆ x ν − a ν ˆ x µ ) + sM µν , (1)where indices µ, ν = 0 , . . . , n − a , a , . . . , a n − are componenets of a four-vector a in Minkowski space whose metric signature is η µν = diag ( − , , ··· , . Thequantities a µ and s are deformation parameters which measure a degree of devia-tion from standard commutativity. They are related to length scale characteristicfor distances where it is supposed that noncommutative character of space-time be-comes important. When parameter s is set to zero, noncommutativity (1) reducesto covariant version of κ -deformation, while in the case that all components of afour-vector a are set to 0 , we get the type of NC space considered for the first timeby Snyder 11. The NC space of this type has been annalyzed in the literature fromdifferent points of view 29 , , , , S.Meljanac et al.
The symmetry of the deformed space (1) is assumed to be described by anundeformed Poincar´e algebra, which is generated by generators M µν of the Lorentzalgebra and generators D µ of translations. This means that generators M µν , M µν = − M νµ , satisfy the standard, undeformed commutation relations,[ M µν , M λρ ] = η νλ M µρ − η µλ M νρ − η νρ M µλ + η µρ M νλ , (2)with the identical statement as well being true for the generators D µ , [ D µ , D ν ] = 0 , (3)[ M µν , D λ ] = η νλ D µ − η µλ D ν . (4)The undeformed Poincar´e algebra, Eqs.(2),(3) and (4) define the energy-momentumsector of the theory considered. However, full description requires space-time sectoras well. Thus, it is of interest to extend the algebra (2),(3) and (4) so as to includeNC coordinates ˆ x , ˆ x , . . . , ˆ x n − , and to consider the action of Poincar´e generatorson NC space (1),[ M µν , ˆ x λ ] = ˆ x µ η νλ − ˆ x ν η µλ − i ( a µ M νλ − a ν M µλ ) . (5)The main point is that commutation relations (1),(2) and (5) define a Lie algebragenerated by Lorentz generators M µν and ˆ x λ . The necessary and sufficient conditionfor consistency of an extended algebra, which includes generators M µν , D µ and ˆ x λ , is that Jacobi identity holds for all combinations of the generators M µν , D µ and ˆ x λ . Particularly, the algebra generated by D µ and ˆ x ν is a deformed Heisenberg-Weylalgebra and we require that this algebra has to be of the form,[ D µ , ˆ x ν ] = Φ µν ( D ) , (6)where Φ µν ( D ) are some functions of generators D µ , which are required to satisfy theboundary condition Φ µν (0) = η µν . This condition means that deformed NC space,together with the corresponding coordinates, reduces to ordinary commutative spacein the limiting case of vanishing deformation parameters, a µ , s → . One certain type of noncommutativity, which interpolates between Snyder spaceand κ -Minkowski space, has already been investigated in 12 , ,
14 in the context ofLagrangian particle dynamics. However, in these papers algebra generated by NCcoordinates and Lorentz generators is not linear and is not closed in the generators ofthe algebra. Thus, it is not of Lie-algebra type. In contrast to this, here we consideran algebra (1),(2),(5), which is of Lie-algebra type, that is, an algebra which is linearin all generators and Jacobi identities are satisfied for all combinations of generatorsof the algebra. Besides that, it is important to note that, once having relations (1)and (2), there exists only one possible choice for the commutation relation between M µν and ˆ x λ , which is consistent with Jacobi identities and makes Lie algebra toclose, and this unique choice is given by the commutation relation (5).une 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla Kappa-deformed Snyder spacetime In subsequent considerations we shall be interested in possible realizations of thespace-time algebra (1) in terms of canonical commutative space-time coordinates X µ , [ X µ , X ν ] = 0 , (7)which, in addition, with derivatives D µ close the undeformed Heisenberg algebra,[ D µ , X ν ] = η µν . (8)From the beginning, the generators D µ are considered as deformed derivatives con-jugated to ˆ x through the commutation relation (6). However, in the whole paper werestrict ourselves to natural choice 17 in which deformed derivatives are identifiedwith the ordinary derivatives, D µ ≡ ∂∂X µ .Thus, our aim is to find a class of covariant Φ αµ ( D ) realizations,ˆ x µ = X α Φ αµ ( D ) , (9)satisfying the boundary conditions Φ αµ (0) = η αµ , and commutation relations(1) and (5). In order to complete this task, we introduce the standard coordinaterepresentation of the Lorentz generators M µν ,M µν = X µ D ν − X ν D µ . (10)All other commutation relations, defining the extended algebra, are then automat-ically satisfied, as well as all Jacobi identities among ˆ x µ , M µν , and D µ . This isassured by the construction (9) and (10). In the next section we turn to problem offinding an explicit Φ µν ( D ) realizations (9).
3. Realizations of NC coordinates
Let us define general covariant realizations:ˆ x µ = X µ ϕ + i ( aX ) (cid:0) D µ β + ia µ D β (cid:1) + i ( XD ) (cid:0) a µ γ + i ( a − s ) D µ γ (cid:1) , (11)where ϕ , β i and γ i are functions of A = ia α D α and B = ( a − s ) D α D α . We furtherimpose the boundary condition that ϕ (0 ,
0) = 1 as the parameters of deformation a µ → s → . In this way we assure that ˆ x µ reduce to ordinary commutativecoordinates in the limit of vanishing deformation.It can be shown that Eq.(5) requires the following set of equations to be satisfied, ∂ϕ∂A = − , ∂γ ∂A = 0 , β = 1 , β = 0 , γ = 0 . Besides that, the commutation relation (1) leads to the additional two equations, ϕ ( ∂ϕ∂A + 1) = 0 , (12)( a − s )[2( ϕ + A ) ∂ϕ∂B − γ ( A ∂ϕ∂A + 2
B ∂ϕ∂B ) + γ ϕ ] − a ∂ϕ∂A − s = 0 . (13)une 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla S.Meljanac et al.
The important result of this paper is that all above required conditions are solvedby a general form of realization which in a compact form can be written asˆ x µ = X µ ( − A + f ( B )) + i ( aX ) D µ − ( a − s )( XD ) D µ γ , (14)where γ is necessarily restricted to be γ = − f ( B ) df ( B ) dB f ( B ) − B df ( B ) dB . (15)From the above relation we see that γ is not an independent function, but insteadis related to generally an arbitrary function f ( B ), which has to satisfy the boundarycondition f (0) = 1. Also, it is readily seen from the realization (14) that ϕ in (11)is given by ϕ = − A + f ( B ) . Various choices of the function f ( B ) lead to differentrealizations of NC space-time algebra (1). The particularly interesting cases are for f ( B ) = 1 and f ( B ) = √ − B .It is now straightforward to show that the deformed Heisenberg-Weyl algebra(6) takes the form[ D µ , ˆ x ν ] = η µν ( − A + f ( B )) + ia µ D ν + ( a − s ) D µ D ν γ (16)and that the Lorentz generators M µν can be expressed in terms of NC coordinatesas M µν = (ˆ x µ D ν − ˆ x ν D µ ) 1 ϕ . (17)We also point out that in the special case when realization of NC space (1) ischaracterized by the function f ( B ) = √ − B , there exists a unique element Z satisfying: [ Z − , ˆ x µ ] = − ia µ Z − + sD µ , [ Z, D µ ] = 0 . (18)From these two equations it follows[ Z, ˆ x µ ] = ia µ Z − sD µ Z , ˆ x µ Z ˆ x ν = ˆ x ν Z ˆ x µ . (19)The element Z is a generalized shift operator 16 and its expression in terms of A and B can be shown to have the form Z − = − A + √ − B. (20)As a consequence, the Lorentz generators can be expressed in terms of Z as M µν = (ˆ x µ D ν − ˆ x ν D µ ) Z, (21)and one can also show that the relation[ Z − , M µν ] = − i ( a µ D ν − a ν D µ ) (22)holds. In the rest of paper we shall only be interested in the realizations determinedby f ( B ) = √ − B .une 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla Kappa-deformed Snyder spacetime
4. Leibniz rule and coalgebra
The symmetry underlying deformed Minkowski space, characterized by the com-mutation relations (1), is the deformed Poincar´e symmetry which can most conve-niently be described in terms of quantum Hopf algebra. As was seen in relations(2),(3) and (4), the algebraic sector of this deformed symmetry is the same as thatof undeformed Poincar´e algebra. However, the action of Poincar´e generators on thedeformed Minkowski space is deformed, so that the whole deformation is containedin the coalgebraic sector. This means that the Leibniz rules, which describe theaction of M µν and D µ generators, will no more have the standard form, but insteadwill be deformed and will depend on a given Φ µν realization.Generally we find that in a given Φ µν realization we can write 16 , e ik ˆ x | > = e iK µ ( k ) X µ (23)and e ik ˆ x ( e iqX ) = e iP µ ( k,q ) X µ , (24)where k ˆ x = k α X β Φ βα ( D ) . In (23) we have introduced the vacuum state | > ≡ x µ and D µ , µ = 0 , , ..., n − , and allows for infinite series in D µ . This vacuum state is defined by φ ( X ) | > ≡ φ ( X ) · φ ( X ) , (25) D µ | > ≡ D µ , M µν | > = 0 , (26)with φ ( X ) belonging to a space of ordinary functions in commutative coordinates.It is also understood that NC coordinates ˆ x, appearing in (23) and (24) refer tosome particular realization (14), i.e. they are assumed to be represented by (14).The quantities K µ ( k ) are readily identified as K µ ( k ) = P µ ( k,
0) and P µ ( k, q )can be found by calculating the expression P µ ( k, − iD ) = e − ik ˆ x ( − iD µ ) e ik ˆ x , (27)where it is assumed that at the end of calculation the identification q = − iD has tobe made. One way to explicitly evaluate the above expression is by using the BCHexpansion perturbatively, order by order. To avoid this tedious procedure, we canturn to much more elegant method to obtain the quantity P µ ( k, − iD ). This consistsin writing the differential equation dP ( t ) µ ( k, − iD ) dt = Φ µα ( iP ( t ) ( k, − iD )) k α , (28)satisfied by the family of operators P ( t ) µ ( k, − iD ) , defined as P ( t ) µ ( k, − iD ) = e − itk ˆ x ( − iD µ ) e itk ˆ x , ≤ t ≤ , (29)and parametrized with the free parameter t which belongs to the interval 0 ≤ t ≤ . The family of operators (29) represents the generalization of the quantityune 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla S.Meljanac et al. P µ ( k, − iD ) , determined by (27), namely, P µ ( k, − iD ) = P (1) µ ( k, − iD ) . Note alsothat solutions to differential equation (28) have to satisfy the boundary condition P (0) µ ( k, − iD ) = − iD µ ≡ q µ . The function Φ µα ( D ) in (28) is deduced from (14) andreads as Φ µα ( D ) = η µα ( − A + f ( B )) + ia µ D α − ( a − s ) D µ D α γ . (30)In all subsequent considerations we shall restrict ourselves to the case where f ( B ) = √ − B. Then we have γ = 0 and consequently Eq.(28) reads dP ( t ) µ dt = k µ (cid:20) aP ( t ) + q a − s )( P ( t ) ) (cid:21) − a µ kP ( t ) , (31)where we have used an abbreviation P ( t ) µ ≡ P ( t ) µ ( k, − iD ) . The solution to differentialequation (31), which obeys the required boundary conditions, looks as P ( t ) µ ( k, q ) = q µ + (cid:0) k µ Z − ( q ) − a µ ( kq ) (cid:1) sinh( tW ) W (32)+ (cid:20) (cid:0) k µ ( ak ) − a µ k (cid:1) Z − ( q ) + a µ ( ak )( kq ) − sk µ ( kq ) (cid:21) cosh( tW ) − W . In the above expression we have introduced the following abbreviations, W = q ( ak ) − sk , (33) Z − ( q ) = ( aq ) + p a − s ) q (34)and it is understood that quantities like ( kq ) mean the scalar product in a Minkowskispace with signature η µν = diag ( − , , · · · , P ( t ) µ ( k, q ) , therequired quantity P µ ( k, q ) simply follows by setting t = 1 and finaly we also get K µ ( k ) = (cid:20) k µ ( ak ) − a µ k (cid:21) cosh W − W + k µ sinh WW . (35)Furthermore, we define the star product by the relation, e ikX ⋆ e iqX ≡ e iK − ( k )ˆ x ( e iqX ) = e i D µ ( k,q ) X µ , (36)where D µ ( k, q ) = P µ ( K − ( k ) , q ) , (37)with K − ( k ) being the inverse of the transformation (35).We claim that the function D µ ( k, q ) is related to the coproduct △ D µ for thetranslation generators. In particular, this relation is given by i D µ ( − iD ⊗ , ⊗ ( − iD )) = △ D µ . (38)This can be seen as follows 15 , ,
17. We start with the general definition of the starproduct f ⋆ g = m ⋆ ( f ⊗ g ) , une 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla Kappa-deformed Snyder spacetime where m ⋆ denotes deformed multiplication in the commutative algebra of smoothfunctions. If we take a derivative of both sides in the above definition of the starproduct, we get D µ ( f ⋆ g ) = m ⋆ ( △ D µ ( f ⊗ g )) . By applying this to (36), we find (38).Thus, the function D µ ( k, q ) determines the deformed Leibniz rule and thecorresponding coproduct △ D µ . It also gives a generalized momentum addition rule.However, in the general case of deformation, when both parameters a µ and s aredifferent from zero, it is quite a difficuilt task to obtain a closed form for △ D µ , sowe give it in a form of a series expansion up to second order in the deformationparameter a, △ D µ = D µ ⊗ + ⊗ D µ − iD µ ⊗ aD + ia µ D α ⊗ D α −
12 ( a − s ) D µ ⊗ D (39) − a µ ( aD ) D α ⊗ D α + 12 a µ D ⊗ aD + 12 sD µ D α ⊗ D α + O ( a ) . Now that we have a coproduct, it is a straightforward procedure 15 ,
17 to con-struct a star product between arbitrary two functions f and g of commuting coor-dinates, generalizing in this way relation (36) that holds for plane waves. Thus, thegeneral result for the star product, valid for the NC space (1), has the form( f ⋆ g )( X ) = lim Y → XZ → X e X α [ i D α ( − iD Y , − iD Z ) − D αY − D αZ ] f ( Y ) g ( Z ) . (40)Although star product is a binary operation acting on the algebra of functionsdefined on the ordinary commutative space, it encodes features that reflect non-commutative nature of space (1). Note also that in the case when s is different fromzero, the star product (40) is nonassociative.The general results obtained so far can be specialized to some particular cases,of which Snyder space and κ -Minkowski space are particularly interesting.For a = 0 , we have a Snyder type of noncommutativity,[ˆ x µ , ˆ x ν ] = sM µν . (41)In this situation, our realization (14) reduces precisely to that obtained in 29. Fora special choice when f ( B ) = 1 , we have the realizationˆ x µ = X µ − s ( XD ) D µ , (42)which is the case that was also considered in 34. In other interesting situation when f ( B ) = √ − B, the general result (14) reduces toˆ x µ = X µ p sD . (43)This choice of f ( B ) is the one for which most of our results, through all over thepaper, are obtained and which is the main subject of our investigations. It is alsoune 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla S.Meljanac et al. considered by Maggiore 35 ,
36. For this case when f ( B ) = √ − B, the exact resultfor the coproduct (37) can be obtained and it is given by △ D µ = D µ ⊗ Z − + ⊗ D µ + sD µ D α Z − + 1 ⊗ D α , (44)where Z − = p sD . (45)Another interesting case is when the parameter s is equal to zero. This cor-responds to κ -deformed space investigated in 7 , , ,
10 and 16 ,
17. The general form(14) for the realizations in the case of κ -deformed space reduces toˆ x µ = X µ (cid:16) − A + √ − B (cid:17) + i ( aX ) D µ , (46)with B = a D , and the coproduct takes on the form △ D µ = D µ ⊗ Z − + ⊗ D µ + ia µ ( D α Z ) ⊗ D α − ia µ (cid:3) Z ⊗ iaD, (47)where the generalized shift operator (20) is here specialized to Z − = − iaD + p − a D . (48)The result (47) is exact and is also written in a closed form, as is the coproduct(44) for the case of Snyder space.
5. Conclusion
In this paper we have investigated a Lie-algebraic type of deformations of Minkowskispace and analyzed the impact these deformations have on the modification of coal-gebraic structure of the symmetry algebra underlying Minkowski space. By findinga coproduct, we were able to see how coalgebra, which encodes the deformation ofMinkowski space, modifies and to which extent the Leibniz rule is deformed withrespect to ordinary Leibniz rule. Since the coproduct is related to the star product,we were also able to write how star product looks like on NC spaces characterizedby the general class of deformations of type (1). We have also found many differ-ent classes of realizations of NC space (1) and specialized obtained results to somespecific cases of particular interest.The deformations that we have considered are characterized by the commonfeature that the algebraic sector of the quantum (Hopf) algebra, which is describedby the Poincar´e algebra, is undeformed, while, on the other hand, the correspondingcoalgebraic sector is affected by deformations.There is a vast variety of possible physical applications which could be expectedto originate from the modified geometry at the Planck scale, which in turn reflectsitself in a noncommutative nature of the configuration space. Which type of noncom-mutativity is inherent to configuration space is still not clear, but it is reasonableto expect that more wider is the class of noncommutativity taken into account,une 10, 2018 23:57 WSPC/INSTRUCTION FILE ws-mpla
Kappa-deformed Snyder spacetime more likely is that it will reflect true properties of geometry and relevant featuresat Planck scale. In particular, NC space considered in this paper is an interpolationof two types of noncommutativity, κ -Minkowski and Snyder, and as such is morelikely to reflect geometry at small distances then are each of these spaces alone,at least, it includes all features of both of these two types of noncommutativity,at the same time. As was already done for κ -type noncommutativity, it would beas well interesting to investigate the effects of combined κ -Snyder noncommutativ-ity on dispersion relations 37 , ,
38, black hole horizons 40, Casimir energy 41 andviolation of CP symmetry, the problem that is considered in 42 in the context ofSnyder-type of noncommutativity.Work that still remains to be done includes an elaboration and developmentof methods for physical theories on NC space considered here, particularly, thecalculation of coproduct for the Lorentz generators, △ M µν , S -antipode, differentialforms 22 , ,
44, Drinfeld twist 45 , , , ,
49, twisted flip operator 49 , , ,
52 and R -matrix 53 ,
52. We shall address these issues in the forthcoming papers, togetherwith a number of physical applications, such as the field theory for scalar fields54 , , ,
57 and its twisted statistics properties, as a natural continuation of ourinvestigations put forward in previous papers 49 , Acknowledgments
We thank A. Borowiec and Z. ˇSkoda for useful comments. This work was supportedby the Ministry of Science and Technology of the Republic of Croatia under contractNo. 098-0000000-2865.
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