From Snyder space-times to doubly $\kappa$-dependent Yang quantum phase spaces and their generalizations

We propose the doubly $\kappa$-dependent Yang quantum phase space which describes the generalization of $D = 4$ Yang model. We postulate that such model is covariant under the generalized Born map, what permits to derive this new model from the earlier proposed $\kappa$-Snyder model. Our model of $D=4$ relativistic Yang quantum phase space depends on five deformation parameters which form two Born map-related dimensionful pairs: $(M,R)$ specifying the standard Yang model and $(\kappa,\tilde{\kappa})$ characterizing the Born-dual $\kappa$-dependence of quantum space-time and quantum fourmomenta sectors; fifth parameter $\rho$ is dimensionless and Born-selfdual. In the last section, we propose the Kaluza-Klein generalization of $D=4$ Yang model and the new quantum Yang models described algebraically by quantum-deformed $\hat{o}(1,5)$ algebras.

Snyder model [6], which can be described by ô (1,4) algebraic relations, introduces the NC model of quantum relativistic space-time.Such algebraic model does not provide a dynamical input determining e.g. the time evolution of the system.However, there were also introduced quantum dynamical particle models providing the Snyder algebra [6] derived from the constraints obtained from the quantized Dirac brackets (see e.g.[15,16]).The class of relativistic quantum phase spaces with NC quantum space-time coordinates xµ and commuting fourmomenta q µ was proposed already in [6].Further it has been shown (see e.g.[17,18]) that in such class of Snyder phase spaces there is a freedom in commutation relations which can be described by an arbitrary function F (q 2 ) [17].
The D=4 Yang model was introduced in [7] and incorporates in one quantum relativistic phase space algebra the NC quantum space-time coordinates xµ as well as curved NC quantum fourmomenta qµ .This quantum phase space depends on the pair of dimensionful parameters M ([M ] = L −1 ) 1 and R ([R] = L 1 ), describing the curvatures of NC space-time and NC fourmo-menta space which are both related by the Born map [19], [20].
Further new versions of Snyder model were obtained by choosing various realizations of Snyder algebra.From the algebraic point of view these new models remain the same, however due to the physical interpretations of the parameters one obtains effectively new physical models in D=4 dimensions.In particular, one can embed into one algebraic model two types of quantum spacetime noncommutativity: the one characterizing the Snyder model and the other which provides the κ-deformation of Minkowski space-time coordinates2 [21]- [24] with the parameter κ often identified with the Planck mass (see e.g.[25]).Consistent description of the quantum space-times with their noncommutativity described by the sum of Snyder and κ-Minkowski terms3 have been proposed earlier, see e.g.[26]- [30].Moreover, the introduction of supplementary κ-deformation terms in Snyder models has been recently explained [31]- [34] as following from the suitable modification of standard ô(1, 4) realizations.By using two isomorphic realizations of ô(1, 4) algebra one obtains two different D = 4 physical models (Snyder and κ-Snyder).If the linear change of quantum spacetime coordinates contains the Lorentz symmetry generators (for details and explicit transformation formulae see [31]) it describes the passage from Snyder to κ-Snyder models.
In the present paper (see also [35], [36]) we added the κ-Minkowski terms to the Yang model, what provides new Lorentz-covariant quantum relativistic phase spaces described by doubly κ-dependent Yang models4 .Since in Yang models we can introduce the pair of different κ-Minkowski terms, in NC space-time and fourmomenta sectors, we need two independent mass-like parameters κ and κ.We add that for simplicity, instead of doubly κ-dependent Yang models, we will use the shorter notation (κ, κ)-Yang models.It appears that such pair of κ-dependence is related by the generalized Born map B, acting as follows (see also [35], [36]) Further, by using Jacobi identities one can show that in (κ, κ)-Yang models one can still introduce (besides M , R, κ and κ) one additional fifth dimensionless parameter ρ ([ρ] = L 0 ), which can be linked with the so-called TSR ("Triply Special Relativity") model [37] 5 .The plan of our paper is the following.In Sect. 2 we recall algebraic descriptions of Snyder and Yang models, and show how by using the generalized Born map one can derive from κ-Snyder model the (κ, κ)-Yang model.
In Sect. 3 we introduce the algebra of (κ, κ)-Yang model as covariant under the generalized Born map (1).By extending the method presented in [31]- [34] we show that the relations describing (κ, κ)-Yang model can be obtained consistently by the suitable linear transformation of the generators of ô (1,5).We point out that both D = 4 standard Yang model and the new D = 4 (κ, κ)-Yang model can be described by two different realizations of ô(1, 5) algebra linked by the linear map.
In the last Sect.4 we include brief conclusions and we present two new ideas which we believe should be further developed.

II. FROM κ-SNYDER TO DOUBLY κ-DEPENDENT YANG MODELS
The NC algebraic structures in D = 4 Snyder and Yang type models are described by Lie algebras ô(1, 4) (D = 4 dS algebra) and ô(1, 5) (D = 5 dS algebra which is isomorphic to D = 4 Euclidean conformal algebra).In Snyder model, by using D = 4 dS algebra generators Mab = ( Mµν , M4µ ) (a, b = 0, 1, . . ., 4), one postulates the following identification of NC space-time coordinates (µ, ν = 0, 1, 2, 3): where M denotes the inverse of the elementary length parameter which plays the role of dimensionful mass-like deformation parameter, frequently identified with the Planck mass.The following set of algebraic relations describes the D = 4 Snyder model [6] 6 with η µν = diag(−1, 1, 1, 1): where relations (4) express the Lorentz covariance of Snyder model and ( 5) describes the relativistic Lorentz extension of quantum mechanical nonrelativistic angular momentum ô(3) algebra, withdependence used as in standard books on Quantum Mechanics (see e.g.[40,41]).The Yang model is obtained if we supplement the relations (3-5) by the algebraic relations for quantum relativistic fourmomenta qµ : [ Mµν , qρ ] = i (η µρ qν − η νρ qµ ) (7) where in astrophysical applications R describes the cosmological D = 4 dS radius.From the relations (2-7) it follows that we obtain ô(1, 5) Lie algebra with MAB = ( Mµν , M4µ , M5µ , M45 ) (A, B = 0, 1, . . .5) if we assign the generators M5µ , M45 to the quantum fourmomenta variables qµ as follows, see e.g.[39]: where in D = 4 the rescaled generator r describes the quantum internal ô(2) symmetry acting on the ô(2) doublet representation (x µ , qµ ): The dimensionful factors R 2 and M 2 imply that the generator r is dimensionless, and determines the following basic relativistic Heisenberg algebra relation: where the case with r = 1 corresponds to the canonical commutation relations.One should observe that the Yang model algebra described by the relations (3-7) and (9-10) is covariant under the following Born map B [19,20,42] where B is a pseudo-involution satisfying the relation B 4 = 1, which permits to define the Yang model as the Born-map extension of the Snyder model.κ-Snyder model, proposed in [26,27], has the following two-parameter extension of the relations (3-4): Recently various properties of this model were investigated in [31][32][33][34].The constant dimensionless four-vector a µ permits to select three types of the κ-deformations of quantum Minkowski spaces: time-like (or standard one) if a µ a µ = −1, tachyonic if a µ a µ = 1 and light-like if a µ a µ = 0, corresponding to the metric signature we have chosen.If we put M → ∞ in (12) we obtain the generalized a µ -dependent κ-deformed Minkowski space-time, with x = a µ xµ describing the unique NC quantum coordinate.
The main aim of this paper is to introduce the new doubly κ-dependent Yang model, obtained from the κ-Snyder model by adding the κ-Minkowski type terms to quantum fourmomenta sector.We add them by postulating that: The relation ( 14) can be obtained from relation (12) by the use of the generalized Born map B which one gets by adding to (11) the following relations7 Further, after using the generalized Born map B given by the relations ( 11), (15), one gets from (13) the following κ-dependence of covariance relations (7) for quantum fourmomenta Finally, it follows that the relations ( 9), (10), due to their selfduality under the map (11), remain the same in (κ, κ)-Yang model.

(Y )
AB ] = L 0 (dimensionless), in consistency with relation (18), with Mµν describing D = 4 Lorentz algebra and the scalar r providing the generator of the ô(2) internal symmetries.Relations (18) are describing the (κ, κ)-Yang model if we insert the following components of the D = 6 metric tensor: where AB ] = L 0 ) in consistency with relations (18).The algebra (18) for any choice of symmetric metric g AB satisfies two important properties: i) By direct calculation one can show that the Lie algebra (18) satisfies Jacobi identities.

ii) For any nondegenerate symmetric metric g (Y )
AB with the signature described by diagonal matrix η AB one can find (6 × 6)-dimensional linear map S = S AB satisfying the relation One can also relate the Lie algebras ( 17) and ( 18) by the following maps We point out that the map ( 23) describes a simple linear transformation between realizations of D=6 dimensional algebra ô (1,5).In D = 4 physical dimensions, it describes the passage from the standard D=4 Yang model (i.e.algebra (17)) to new D=4 doubly-κ-dependent Yang model, where coordinates and momenta do not commute and their commutator is proportional to the sum of Lorentz generators and κ-Minkowski terms in space-time sector (and respectively κ-Minkowski type terms in fourmomenta sector), i.e. the algebraic relations ( 18)-( 21).Additionally, we observe that the matrix S satisfying relations ( 22), ( 23) is not unique, with arbitrariness described by the pseudoorthogonal matrix O, where OηO T = η.For concrete choice (22) of the matrix g (Y ) we choose 6 × 6 matrix S parametrized as follows 9 with parameters a, b, c, d satisfying the conditions We can pass to lower triangular S matrix if we put d = 0 in the formulae (24,25).In such a case the set of equations ( 25) has the following solutions: with ǫ, ǫ ′ = ±1.In (κ, κ)-Yang models one can select nine classes of double κ-dependence based on the Lorentzcovariant normalized length values of the dimensionless fourvectors a µ , b µ which are related with the fourvectors g µ and h µ from the matrix (24) if we choose 9 For the simplicity of formulae in (24), we introduce the shorthand notation and The nine classes of double κ-dependence are obtained by the choices of the parameters ǫ, ǫ ′ = (±1, 0), where a µ a µ = ǫ and b µ b µ = ǫ ′ .The missing algebraic relations of (κ, κ)-Yang model, which describe the modified D = 4 Heisenberg algebra sector of ô(6; g AB ) by the generalization of relations ( 9), (10) look as follows: Additionally, we have i.e. we see that the internal and Lorentzian generators do not commute with each other.
It can be checked that the relations ( 30)-( 33) are self dual under the generalized Born map (11), extended by (15).Moreover, it can be shown that the relations ( 30)-( 33) can be derived as the general solutions of the Jacobi identities for generators xµ , qµ , r and Mµν .

IV. OUTLOOK
In this paper we have proposed the new relativistic quantum phase spaces by introducing κextensions of the standard Yang model which define doubly κ-dependent Yang models (i.e.(κ, κ)-Yang models).In such models there appear additional κ-Minkowski terms, linked by the generalized Born map (11), (15), which provide the standard κ-Minkowski type terms in commutativity relations between quantum relativistic space-time coordinates xµ (see (12), ( 13)) and introduce the new κ-Minkowski type terms in quantum fourmomenta commutation relations (see ( 14), ( 16)).
The doubly κ-dependent Yang model, proposed in this paper, is described by the following five parameters (M, R, κ, κ, ρ): -mass-like parameter M ([M ] = L −1 ) describing the constant curvature in quantum space-time sector (e.g.M can be identified with the Planck mass) -parameter R ([R] = L 1 ) defining the constant curvature in quantum fourmomenta sector (e.g.R can be linked to the radius of the Universe) -parameters κ and κ which describe two independent modifications of quantum space-time and quantum fourmomenta sectors, respectively -the fifth dimensionless parameter ρ ([ρ] = L 0 ) parametrising, in the commutator [x µ , qν ], the term proportional to M µν 10 .
We have shown in Sect. 3 that the algebraic structure of doubly κ-dependent Yang model can be derived from the ô(1, 5) algebra if we select the suitable realizations of its generators by proper choice of the matrix S (see (22)(23)(24)).Finally, we propose two ways which could lead to the valuable generalizations of Snyder and Yang models.
-Firstly, we propose the generalization of Yang models to the Kaluza-Klein geometries in D = (1, 3 + 2N ) with Lorentzian signature and ô(2N ) internal symmetries (in particular if N = 1 we obtain the standard Yang model with Born-extended internal Abelian ô(2) symmetry).Such models should be useful in studies of the new unification models of gravity and particle physics which employ the higher-dimensional Lorentz algebras ô(1, 3 + 2N ) (e.g. for N = 5 see [45], for N = 7 see [46]).