Realizations of the Extended Snyder Model

. We present the exact realization of the extended Snyder model. Using similarity transformations, we construct realizations of the original Snyder and the extended Snyder models. Finally, we present the exact new realization of the κ -deformed extended Snyder model.


Introduction
The first example of NC geometry was presented in [36].Fundamental length scale could be identified in natural way with Planck length L p = Gℏ/c 3 ≈ 1.62 × 10 −35 m [11].The length scale enters the theory through commutators of spacetime coordinates in [1,2,8,9].Deformations of spacetime symmetries-gravity, group-valued momenta, and noncommutative fields were presented in [3].
Coproduct and star product in the Snyder model were calculated in [6,12] using ideas from development of NC geometry [20].However, in the Snyder model, the algebra generated by position operators is not closed and the bialgebra resulting from implementation of the coproduct is not a Hopf algebra.In particular, the coproduct is noncoassociative and the star product is nonassociative as well [6].
A closed Lie algebra can be obtained if one adds generators of Lorentz algebra [12] to position generators.In this way one can define a Hopf algebra with a coassoaciative coproduct.If Lorentz generators are added as extended coordinates, we call this algebra extended Snyder algebra, and the theory based on this the extended Snyder model [26].
Our goal is to construct realizations of the Snyder algebra (1.1)- (1.3) in terms of the Heisenberg algebra generated by coordinates x µ and momenta p µ satisfying the commutation relations In Section 2, we start with the original Snyder realization with M µν = x µ p ν − x ν p µ and use similarity transformations to construct a family of realizations of Snyder model.In Section 3, we apply this method to construct realizations of the extended Snyder model in which the Lorentz generators are realized by M µν = xµν + x µ p ν − x ν p µ , where xµν are additional tensorial generators.In Section 4, we present the exact new realization of the κ-deformed extended Snyder model.Finally, in Section 5, we give the discussion and conclusion.

Realizations of the Snyder model
The original Snyder realization in terms of x µ and p ν is given by where x • p = x α p α . 1 Further realizations of the Snyder model can be obtained by similarity transformations by the operator S = e iG , where Note that for β 2 = 0 we have G = 0 and S = id and G is Lorentz invariant and linear in the coordinates x α .
Theorem 2.1.Using similarity transformation defined by S = e iG , where G is given by (2.3), we obtain the corresponding realizations of Snyder model where In order to prove the above theorem, first we prove the following propositions.Note that if F 0 (u) = 0, then φ 3 (u) = 0, hence, for simplicity in what follows we assume that F 0 (u) = 0 and G , where S = e iG and G = (x • p)F (u).Then where Proof .By defining the iterated commutator and using the Hadamard formula, we have We prove relation (2.5) by induction on n.Using the Leibniz rule for adjoint representation and it is easy to see that for n = 1 we have where g 11 (u) = F and g 21 (u) = 2 Ḟ .In following, we denote g ij ≡ g ij (u).Assume that the relation holds for some n > 1.Then by the induction assumption, we have where, using the Leibniz rule and (2.8), we obtain and Let us denote , where g 10 = 1, and Then, substituting (2.9) into (2.7), it follows that (2.5) holds.Now, we expand n! and prove by induction on n that Note that for n = 0, we have g 10 = id(1) = 1 and for n = 1 it is easy to verify that Suppose that relation (2.11) is true for some n > 1.By the induction assumption and from (2.10), we have which proves our claim (2.11) and consequently (2.6) holds.
, where S = e iG and G = (x • p)F (u).Then where Proof .Analogous to the proof of the previous proposition, first by using the Hadamard formula, we find Then, by induction on n, we prove that (ad iG ) n (p µ ) = p µ g 3n . (2.14) After short computation, for n = 1 we have where g 31 = −F .Assume that relation (2.14) holds for some n > 1.Then by the induction assumption, we find where which proves claim (2.14).Finally, if we denote , where g 30 = 1, then (2.12) holds.Also, we prove by induction on n that Note that for n = 0, we have g 30 = id(1) = 1, and for n = 1, we get Suppose that relation (2.16) holds for some n > 1.Then by the induction assumption and from (2.15), we have Therefore, (2.16) holds for every n, which implies that (2.13) holds.■ Now, using results proven in the previous propositions, we can finally prove our main result given by Theorem 2.1. and Finally, using (2.5) and (2.12), it follows from (2.18) that ( Remark 2.6.When φ 1 (u) is fixed and φ 2 (u) is given with (2.4), then φ 3 (u) depends on F 0 and can be arbitrary.There is a family of realizations with fixed φ 1 (u) and arbitrary φ 3 (u).
Remark 2.7.A Hermitian realization can be obtained starting with the hermitian form of (2.1), that is and instead of G writing 1 2 G + G † .Then result of Theorem 2.1 is obtained in hermitian form 1  2 xµ + x † µ .

Realizations of the extended Snyder model
Different realizations of the Snyder algebra can be obtained introducing additional tensorial generators xµν = −x νµ .This alternative approach was suggested in [12] and it was studied perturbatively from a different point of view in [26,31,32] based on the results in [23].The In this case, we consider realizations of the Lorentz generators of the form where x µν are commuting variables.3).A short computation using (2.8) yields and Similarly, by using (3.7), we check that (3.3) and (3.4) satisfy (1.2)-(1.3),therefore (3.3) and (3.4) is a realization of the extended Snyder model.■ In order to obtain a family of realizations of the extended Snyder model, we use similarity transformations from Section 2, by S = e iG where G = (x • p)F (u).First, note that and Finally, by using results given in Section 2, (2.19) and (2.22), we obtain a family of realizations of the extended Snyder model where φ 1 (u) and φ 2 (u) satisfy (2.4).Note that realizations (3.3), (3.4), (3.10) and (2.4) are the exact results written in closed form.
In the following paper we present the exact new result for the κ-deformed extended Snyder model that is written in closed form and different from the perturbative results discussed in [27,28].
Then one particular realization of the algebra (4.1)-( 4.3) is given by Furthermore, from (2.8) we get x and In similar way, by using (4.6)-(4.10),we show that (4.4) and (4.5) satisfy (4.2)-(4.3).■ For a µ = 0, we get the realization of the extended Snyder model found in Section 3 For β 2 = 0, we find This is a new result corresponding to the κ-Poincaré algebra with additional tensorial generators xµν .The most general realizations of xµ in all cases in this section are obtained by using the most general corresponding similarity transformations.Construction of Hermitian realizations in Sections 3 and 4 can be obtained simply by changing xµ with 1 2 xµ + x † µ , as in Remark 2.7.

Conclusion and discussion
In Section 2, we defined similarity transformations (2.3) and using Propositions 2.2 and 2.3, we proved realizations of the Snyder model (2.4) in Theorem 2.1.This result was obtained in [5,6] without using similarity transformations.In Section 3, we gave a proof of Theorem 3.1 (equations (3.3)-(3.4))that includes additional tensorial generators xµν and it is a generalization of the original Snyder realization.This is a new exact result leading to an associative star product and coassociative coproduct [26].Also, we obtained exact results for the realizations of the extended Snyder model with functions φ 1 (u) and φ 2 (u) (3.10) using Propositions 2.2 and 2.3.In Section 4, we proved Theorem 4.1 (equations (4.4) and (4.5)) and this is a new exact result for the κ-deformed extended Snyder model.The physical role of the additional tensorial generators xµν is not completely clear, except that they mathematically lead to an associative star product and coassociative coproduct [12,26].Some attempts for applications of the extended Snyder model were made in [12,16,26] and of the κ-deformed extended Snyder model in [17,27,28].Possible applications of the generalizations of the Snyder model to curved spaces were discussed in [29,30].The future prospect of our investigation is the construction of the star product and twist.