Generalizations of Snyder model to curved spaces

We consider generalizations of the Snyder algebra to a curved spacetime background with de Sitter symmetry. As special cases, we obtain the algebras of the Yang model and of triply special relativity. We discuss the realizations of these algebras in terms of canonical phase space coordinates, up to fourth order in the deformation parameters. In the case of triply special relativity we also find exact realization, exploiting its algebraic relation with the Snyder model.


Introduction
Noncommutative geometries have often been advocated as plausible candidates for describing physics at the Planck scale [1]. The first model of noncommutative spacetime was suggested by Snyder [2] in 1947. Soon after the publication of Snyder's paper, C.N. Yang proposed a model that combined noncommutativity with spacetime curvature [3]. Yang's model was based on the fifteen-dimensional SO (1,5) algebra. The generators of this algebra were identified with the coordinates of a phase space with de Sitter symmetry and with the generators of Lorentz transformation. The remaining generator rotates positions into momenta, but its physical meaning was not specified.
More recently, Kowalski-Glikman and Smolin [4] proposed a model inspired by that of Yang, which realizes the same symmetries in a nonlinear way, reducing to fourteen the number of generators. They called this model triply special relativity (TSR) because it contains three fundamental constants, identified with the speed of light, the Planck length and the cosmological constant, generalizing in this way the idea advanced in doubly special relativity theories [5] of deforming the Poincaré symmetry by the introduction of a new fundamental constant. A particularly interesting property of TSR [6] is that it realizes the Born duality [7] for the exchange of position and momentum operators. Another interesting consequence of this model is the prediction of the existence of both a minimal length and a minimal momentum [8].
Later, one of us showed that this model can be realized exactly in terms of coordinates and momenta only [8], and introduced the alternative denomination of Snyder-de Sitter (SdS) spacetime. In [8,9] it was also shown that TSR algebra can be obtained from the Snyder algebra by a nonunitary transformation.
While we are not aware of other papers dealing with the Yang model, except the recent proposal of a supersymmetric extension of the algebra [10], a number of articles have investigated aspects of TSR. Most of them treat its classical limit, either in a nonrelativistic or relativistic setting [11]. Also the quantum field theory of a self-interacting scalar field in SdS spacetime has been investigated in [12].
It is known that a fruitful approach to noncommutative geometry is based on Hopf algebras [13], that describe the symmetries of the quantum spacetime. A powerful tool in this formalism are realizations of Hopf algebras in terms of the Heisenberg algebra, that were introduced in [14 -16]. Only recently this approach has been considered in the context of Yang and TSR models in [17]. In this paper, it has been proposed that TSR and a slight generalization of the Yang model can be treated in a unified way in this formalism.
In the present paper, we discuss general perturbative realizations of the unified model proposed in [17], in terms of the standard Heisenberg algebra. We also exploit the relation of TSR with the Snyder model to write down some exact realizations. This results should consent to define a star product and a twist following the approach of [15][16][17]. This topic is currently being investigated.

The model
We consider a noncommutative algebra of the form [17] [x µ , with real parameters α and β and η µν the flat metric. We interpret the Hermitian operatorsx µ =x † µ and p µ =p † µ as coordinates of the phase space and M µν = M † µν as generators of Lorentz transformations. The rank-2 tensor g µν depends onx µ ,p µ and M µν , with g † µν = g µν . The algebra (1)-(3) is invariant under Born duality, α ↔ β,x µ ↔p µ , M µν ↔ M µν , g µν ↔ g νµ . In the limit β → 0 it contains as a subalgebra the de Sitter algebra, in the limit α → 0 the Snyder algebra.
The Jacobi identities imply Depending on the form of g µν , one can recover well known models. For example, the Yang model is characterized by the choice [3] g µν = h(x 2 ,x·p +p·x,p 2 ) η µν , while the TSR model is characterized by [4]

Hermitian realizations
We are interested in finding Hermitian realizations of the above models in phase space, in terms of canonical variables x µ and p µ , In particular, we shall look for representations where the generators M µν and g µν can be written in terms of x µ and p µ . Therefore, we assumex where F , G, H, K, h i are Lorentz-invariant functions of x 2 , x·p + p·x and p 2 .
In the following, we shall consider these realizations in a perturbative expansion in α and β. At second order, we make the ansatẑ We may also go to the next order, with the ansatẑ where c i , d i are constants. Born dual realizations of (12)- (14) are obtained byx µ ↔p µ , α ↔ β and x µ ↔ p µ .

Yang model
The original Yang model was characterized by an algebra where g µν was considered as an independent generator. Here we adopt instead the definition (7), where g µν is a Hermitian operator, written in terms of a Lorentz-invariant function h of the phase space variablesx µ andp µ . Clearly, at zeroth order, h (0) = 1. At second order in α and β we can set with g i real parameters. One easily sees that the realization (12) satisfies the Yang algebra if and At this order, the simplest realization of the Yang algebra is given by the choice a 1 = b 1 = a 4 = b 4 = 0.
To fourth order, we can assume Inserting (13)- (14) in the Yang algebra, one gets the independent conditions Taking into account (16), it follows that at this order there are six further independent parameters c 1 , c 2 , c 3 , c 4 , c 7 and c 10 , with The simplest Hermitian realization of the Yang model up to fourth order in α, β is thereforê with Note that h,x µ andp µ satisfy [h,x µ ] = iβ 2p µ and [h,p µ ] = −iα 2x µ .

Triply special relativity
The TSR algebra is defined by (1)-(3) with g µν given by (8). An important relation following from (8) is From this, after some manipulations, one can obtain an equivalent form of the algebra, written explicitly in terms ofx µ andp µ only, that was first proposed in [8,9]. In those papers, the Lorentz generators were defined as or equivalently, using (21), It follows that g µν can be written as We call this SdS realization of the TSR algebra. Using (22) and (24) we can obtain second-order realizations of the SdS algebra in terms of the canonical Heisenberg algebra by inserting (10) and (12) in the defining relations. It is easy to see that the SdS algebra is satisfied if (25)

Further developments
The relation of the TSR algebra with the Snyder algebra was first noticed in [8,9]. Here we exploit it using a different derivation. We proceed as follows: the expression (8) of g µν can be written as where we have defined P µ =p µ + α βx µ . Sincex † µ =x µ ,p † µ =p µ , M † µν = M µν , it follows that P † µ = P µ , g † µν = g µν , and consequently from (28) [P µ , P ν ] = 0. Hence, Together with [x µ ,x ν ] = iβ 2 M µν this recalls the commutation relations of the Snyder model [2]. One can hence derive a realization of the SdS algebra from the realizations of the Snyder model discussed in [15,18], in terms of canonical variables X µ , P ν satisfying [X µ , X ν ] = [P µ , P ν ] = 0, [X µ , P ν ] = iη µν , namely, This realization is not included in the class investigated in the previous section, becausep µ contains terms proportional to X µ that were neglected in the ansatz (12), but is exact, although it is not symmetric in X and P and is not well defined in the limit β → 0. A realization that is regular for vanishing β, but singular for α → 0 can be obtained by duality, starting from a representation of de Sitter algebra in Beltrami coordinates [19]. One haŝ where X µ and P µ still satisfy canonical commutation relations, but are not the same as in (30).
Further realizations can be obtained by similarity transformations starting from the ones found above. In fact, let us consider the commutation relations (1)-(3) and act on them with a unitary operator S from the left and S −1 from the right, defininĝ Thenx ′ µ andp ′ µ satisfy the same commutation relations asx µ andp µ . We may write S = e iG , with G = G(x 2 , x·p + p·x , p 2 ), G † = G and [M µν , G] = 0. In this way we generate infinitely many realizations ofx andp in terms of x and p, satisfying the same algebra.

Conclusions
We have discussed a general quantum algebra that depends on two parameters α and β, usually identified with a minimum length and the cosmological constant, and includes as special cases Yang [3] and TSR [4] algebras. This algebra is relevant for quantum gravity research, because it combines the effects of noncommutativity with those of the curvature of spacetime, a subject that has attracted a large interest recently [21].
We have found realizations of these quantum algebras on canonical phase space. The form of the algebra (1)-(3) is much more general than the special cases we have considered, and we are now constructing more general models of this class. In particular, even considering algebras not more than quadratic in the generators, several possibilities are available.
A more difficult problem is to construct a quasi-Hopf algebra associated to these models. It seems that star product and twist have not been constructed so far for Yang and TSR models. Star products should be nonassociative as for the Snyder model [15,17,18]. Star products, related to noncommutative coordinatesx µ whose realizations depend on the parameters α and β and on x µ , p µ are under construction, exploiting the method proposed in [15,18,20]. This construction implies a generalization of the Hopf algebroid approach [22].
An interesting field of application of our results is QFT. A field theory based on the SdS algebra has been discussed in [12], where it was also remarked its similitude with the Grosse-Wulkenhaar model [23]. This model is of primary relevance because it gives rise to a renormalizable and exactly solvable theory, which, in analogy with SdS field theory, can be thought as a field theory in noncommutative curved space [24].