# Quantum Mechanics on a Curved Snyder Space

Authors: Salvatore Mignemi; Rina Štrajn
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

We study the representations of the three-dimensional Euclidean Snyder-de Sitter algebra. This algebra generates the symmetries of a model admitting two fundamental scales (Planck mass and cosmological constant) and is invariant under the Born reciprocity for exchange of positions and momenta. Its representations can be obtained starting from those of the Snyder algebra and exploiting the geometrical properties of the phase space that can be identified with a Grassmannian manifold. Both the position and momentum operators turn out to have a discrete spectrum.


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## Body

## 1. Introduction

The Snyder-de Sitter (SdS) model, or triply special relativity [1], was introduced as a generalization of the Snyder model [2] to a curved background.The Snyder model has been the first example of noncommutative geometry proposed in the literature and is based on a deformation of the Heisenberg algebra by a fundamental invariant scaleβ with dimension of inverse energy. It was introduced in the hope of achieving a regularization of field theory through the new scale, whose presence, contrary to what one may expect, does not affect the Lorentz invariance but only deforms the translation symmetry [3].The Snyder model can also be considered as an example of doubly special relativity, namely, a theory where a new fundamental scale is introduced by deforming special relativity [4, 5]. Models of this sort can be described in terms of a curved momentum space [6], which in the Snyder case is a 3-sphere S3. In general, the nontrivial geometry of the momentum space entails some remarkable consequences on the definition of locality that loses its absolute meaning and becomes observer dependent, suggesting the possibility that locality is a relative property [7].The properties of the Snyder space and its dynamics, both in the nonrelativistic and relativistic version, have been investigated in several papers and in various contexts, both classical and quantum [8–26]. In particular, it was shown that space is discretized [9] and deformed Heisenberg uncertainty relations hold, implying a lower bound on measurable length [8]. Both classical and quantum dynamics are modified with respect to the standard results, with deviations of order β2E2, E being the energy of the system [8].The extension of the Snyder model to a spacetime background of constant curvature was proposed in [1] (see also [27] for a different approach). This generalization was motivated by the necessity of including the cosmological constant Λ~α2 among the bare parameters of a theory of quantum gravity [1]. In this way, one introduces a third fundamental constant besides the speed of light c and the Snyder parameter β, which explains the name of triply special relativity originally given to the theory. The most relevant feature of this generalization is its duality for the interchange between positions and momenta that realizes Born reciprocity principle [28]. The properties of SdS space were studied by many authors [29–37], mainly in its nonrelativistic version: it turns out that both positions and momenta have discrete spectra, and a minimal momentum occurs besides minimal length. More general theories with curved momentum space on a curved spacetime background have also been studied from the point of view of relative locality in [38, 39], especially in relation with cosmology and geodesic motion.In this paper, we attempt to generalize the results on the algebraic structure of the three-dimensional nonrelativistic Snyder model, investigated in [9], to the case of a curved background. Although the nonrelativistic limit is physically less interesting than the relativistic theory, it can shed some light on the structure of the theory and serve as a first step in the construction of a quantum field theory. Of course, in this limit only two fundamental constants, α and β, are left.In the SdS case, the algebraic structure is less useful than in the Snyder case, because for SdS the spectrum of the position operator cannot be derived directly from the algebraic structure of the theory. It is however possible to find a relation between the representations of Snyder and of SdS, which enables one to find analytically the spectrum of the square of the position and momentum operators. Because of the Born reciprocity these spectra are essentially identical.


### 1.1. The SdS Model

The nonrelativistic SdS algebraA depends on two parameters α and β and contains the usual generators of rotations L^k=1/2ϵijkL^iji,j=1,…,3, with their standard action on the position and momentum operators x^i and p^i:(1)L^i,L^j=iϵijkL^k,L^i,x^j=-iϵijkx^k,L^i,p^j=-iϵijkp^k.The commutation relations of x^i and p^i satisfy instead a deformation of the Heisenberg algebra(2)x^i,x^j=iβ2ϵijkL^k,p^i,p^j=iα2ϵijkL^k,x^i,p^j=iδij+α2x^ix^j+β2p^jp^i+αβx^jp^i+p^ix^j,where L^ij=1/2(x^ip^j+p^jx^i-x^jp^j-p^ix^j).For special values of the parameters,A reduces to the nonrelativistic Snyder algebra (α=0) or the algebra of isometries of S3 in Beltrami coordinates (β=0). It is possible to define also a noncompact version of the algebra, by analytic continuation to imaginary values of α and β, with rather different properties [29], but we will not discuss it here. As mentioned above, the algebra is invariant for αxμ↔βpμ. More generally, it is invariant for rotations in the phase space, αxi→αxicos⁡θ+βpisin⁡θ, βpi→-αxisin⁡θ+βpicos⁡θ. Of course, this symmetry holds also in standard quantum mechanics, with α=β=1.The SdS algebra can be considered as a nonlinear realization of a model proposed by Yang [40], which differs from SdS only in the assumption of a standard Heisenberg algebra for positions and momenta, [x^i,p^j]=iK^δij, K^ being a central charge for the rotation group, satisfying K^,x^i=iα2p^i, K^,p^i=-iβ2x^i. With the identifications L^ij=J^ij, αx^i=J^4i, βp^i=J^5i, K^=J^45, the Yang model reproduces an so5 algebra with generators J^μν, (μ,ν=1,…,5).Also the 3-dimensional nonrelativistic SdS model enjoys anSO(5) symmetry. In fact, its phase space can be realized on the six-dimensional Grassmannian coset space Gr(3,5)=SO(5)/SO(3)×SO(2), with SO(3) generated by the Jij and SO(2) by J45.The spaceGr(3,5) can be parametrized by homogeneous coordinates xμ and pμ, that satisfy the constraints [41]: (3)α2xμ2=1,β2pμ2=1,xμpμ=0.This parametrization associates a one-parameter set of matrices to each coset. One can then identify the variables xμ and pμ with canonical coordinates of a ten-dimensional phase space and hence reduce it to a six-dimensional phase space parametrized by xi and pi by eliminating the constraints (3), using the Dirac formalism [42]. In order to obtain a one-to-one parametrization, one has however to impose a further constraint on the parameters x4, x5, p4, p5. This is also required by the Dirac formalism since the constraints (3) split into one first class and two second class constraints [32]. Unfortunately, not every constraint leads to the SdS algebra, and one has therefore to choose a suitable gauge [32]. In particular, the choice αx5+βp5=0 yields the algebra (2).


## 1.1. The SdS Model

The nonrelativistic SdS algebraA depends on two parameters α and β and contains the usual generators of rotations L^k=1/2ϵijkL^iji,j=1,…,3, with their standard action on the position and momentum operators x^i and p^i:(1)L^i,L^j=iϵijkL^k,L^i,x^j=-iϵijkx^k,L^i,p^j=-iϵijkp^k.The commutation relations of x^i and p^i satisfy instead a deformation of the Heisenberg algebra(2)x^i,x^j=iβ2ϵijkL^k,p^i,p^j=iα2ϵijkL^k,x^i,p^j=iδij+α2x^ix^j+β2p^jp^i+αβx^jp^i+p^ix^j,where L^ij=1/2(x^ip^j+p^jx^i-x^jp^j-p^ix^j).For special values of the parameters,A reduces to the nonrelativistic Snyder algebra (α=0) or the algebra of isometries of S3 in Beltrami coordinates (β=0). It is possible to define also a noncompact version of the algebra, by analytic continuation to imaginary values of α and β, with rather different properties [29], but we will not discuss it here. As mentioned above, the algebra is invariant for αxμ↔βpμ. More generally, it is invariant for rotations in the phase space, αxi→αxicos⁡θ+βpisin⁡θ, βpi→-αxisin⁡θ+βpicos⁡θ. Of course, this symmetry holds also in standard quantum mechanics, with α=β=1.The SdS algebra can be considered as a nonlinear realization of a model proposed by Yang [40], which differs from SdS only in the assumption of a standard Heisenberg algebra for positions and momenta, [x^i,p^j]=iK^δij, K^ being a central charge for the rotation group, satisfying K^,x^i=iα2p^i, K^,p^i=-iβ2x^i. With the identifications L^ij=J^ij, αx^i=J^4i, βp^i=J^5i, K^=J^45, the Yang model reproduces an so5 algebra with generators J^μν, (μ,ν=1,…,5).Also the 3-dimensional nonrelativistic SdS model enjoys anSO(5) symmetry. In fact, its phase space can be realized on the six-dimensional Grassmannian coset space Gr(3,5)=SO(5)/SO(3)×SO(2), with SO(3) generated by the Jij and SO(2) by J45.The spaceGr(3,5) can be parametrized by homogeneous coordinates xμ and pμ, that satisfy the constraints [41]: (3)α2xμ2=1,β2pμ2=1,xμpμ=0.This parametrization associates a one-parameter set of matrices to each coset. One can then identify the variables xμ and pμ with canonical coordinates of a ten-dimensional phase space and hence reduce it to a six-dimensional phase space parametrized by xi and pi by eliminating the constraints (3), using the Dirac formalism [42]. In order to obtain a one-to-one parametrization, one has however to impose a further constraint on the parameters x4, x5, p4, p5. This is also required by the Dirac formalism since the constraints (3) split into one first class and two second class constraints [32]. Unfortunately, not every constraint leads to the SdS algebra, and one has therefore to choose a suitable gauge [32]. In particular, the choice αx5+βp5=0 yields the algebra (2).


## 2. The Nonrelativistic Snyder Algebra

In this section we review some results on the representation of the Snyder algebra from [8, 9] that will be useful in the following discussion.The Snyder algebra is the limit of the SdS algebra (1)-(2) for α→0. It contains an so(4) subalgebra of so(5) generated by X^i=J^4i and L^i=1/2ϵijkJ^jk, while the momentum space is realized as the coset space S3=SO(4)/SO(3).The representations of theso(4) algebra can be labeled by the eigenvalues of A^i2, L^i2, and L^3, where A^i is the operator 1/2(L^i+β-1X^i), so that X^i2=β2(4A^i2-L^i2):(4)L^i2j,l,m=ll+1j,l,m,L^3j,l,m=mj,l,m,X^i2j,l,m=β24jj+1-ll+1j,l,m,with 0≤l≤2j, j(j+1) being the eigenvalue of A^i2. The eigenvalues of X^i2 have degeneration 2l+1.The momentum space can be realized as a 3-sphere, obtained by imposing the constraintPi2+P42=1/β2 on a four-vector P. This can be shown algebraically, as in [9], or also from a Dirac reduction of the phase space [11]. In the following, we will mainly concentrate on the operators X^i2 and P^i2 and investigate their spectra.


### 2.1. The Representation I

One can define several different representations of the Snyder algebra on a Hilbert space. Usually they are given in momentum representation. In [9, 10] it was realized by operators P^i and X^i defined as(5)P^i=Pi,X^i=i∂∂Pi+iβ2PiPj∂∂Pj+μ,L^i=-iϵijkPj∂∂Pk,which act on functions ψ(Pi) of the Hilbert space, with μ an arbitrary real parameter and -∞<Pi<∞. The operators are symmetric for the measure (6)d3P1+β2Pi22-μ,provided that the functions ψ(Pi) go to infinity like (1/Pi2)μ-1/2.In this representation the operatorX^i2 reads(7)X^i2=-1+β2Pρ22∂2∂Pρ2+2Pρ∂∂Pρ-μβ221+β2Pρ2Pρ∂∂Pρ+1+μβ2Pρ2+3+L^i2Pρ2.For μ=0, the equation X^i2ϕ=Xi2ϕ has eigenfunctions [9] (8)ϕnlm=const.×sinl⁡χCnl+1cos⁡χYmlPθ,Pφ,where we have used polar coordinates Pρ,Pθ,Pφ for Pi, and χ=arctan⁡βPρ. The functions Cn(a) are Gegenbauer polynomials with n a nonnegative integer parameter and Yml(Pθ,Pφ) spherical harmonics.It is easy to see that ifμ≠0, the eigenfunctions (8) are simply multiplied by cosμχ. The eigenvalues are of course independent of μ and read (9)Xi2=β2n2+2nl+2n+l,with 0≤l≤n. They can easily be identified with (4) by setting n=2j-l.The operatorP^i2=P^ρ2 is trivial and its spectrum extends to the real positive line.


### 2.2. The Representation II

An alternative representation is obtained [8] by defining (10)P^i=Pi1-β2Pk2,X^i=i1-β2Pk2∂∂Pi,with Pk2<1/β2. In this representation the operators are symmetric for the measure (11)d3P1-β2Pk2,and the operator X^i2 reads (12)X^i2=-1-β2Pρ2∂2∂Pρ2-2-3β2Pρ2Pρ∂∂Pρ+1-β2Pρ2L^i2Pρ2.It has eigenfunctions (13)ϕqlm=const.×sinl⁡ηcos⁡ηPq1/2,l+1/2cos⁡2ηYmlPθ,Pφ,where η=arcsin⁡βPρ and Pq(a,b) are Jacobi polynomials with q a nonnegative integer. The eigenvalues are given by β22q+l+22-ll+1-1. Taking q=n-1/2, one recovers the eigenvalues (9).


## 2.1. The Representation I

One can define several different representations of the Snyder algebra on a Hilbert space. Usually they are given in momentum representation. In [9, 10] it was realized by operators P^i and X^i defined as(5)P^i=Pi,X^i=i∂∂Pi+iβ2PiPj∂∂Pj+μ,L^i=-iϵijkPj∂∂Pk,which act on functions ψ(Pi) of the Hilbert space, with μ an arbitrary real parameter and -∞<Pi<∞. The operators are symmetric for the measure (6)d3P1+β2Pi22-μ,provided that the functions ψ(Pi) go to infinity like (1/Pi2)μ-1/2.In this representation the operatorX^i2 reads(7)X^i2=-1+β2Pρ22∂2∂Pρ2+2Pρ∂∂Pρ-μβ221+β2Pρ2Pρ∂∂Pρ+1+μβ2Pρ2+3+L^i2Pρ2.For μ=0, the equation X^i2ϕ=Xi2ϕ has eigenfunctions [9] (8)ϕnlm=const.×sinl⁡χCnl+1cos⁡χYmlPθ,Pφ,where we have used polar coordinates Pρ,Pθ,Pφ for Pi, and χ=arctan⁡βPρ. The functions Cn(a) are Gegenbauer polynomials with n a nonnegative integer parameter and Yml(Pθ,Pφ) spherical harmonics.It is easy to see that ifμ≠0, the eigenfunctions (8) are simply multiplied by cosμχ. The eigenvalues are of course independent of μ and read (9)Xi2=β2n2+2nl+2n+l,with 0≤l≤n. They can easily be identified with (4) by setting n=2j-l.The operatorP^i2=P^ρ2 is trivial and its spectrum extends to the real positive line.


## 2.2. The Representation II

An alternative representation is obtained [8] by defining (10)P^i=Pi1-β2Pk2,X^i=i1-β2Pk2∂∂Pi,with Pk2<1/β2. In this representation the operators are symmetric for the measure (11)d3P1-β2Pk2,and the operator X^i2 reads (12)X^i2=-1-β2Pρ2∂2∂Pρ2-2-3β2Pρ2Pρ∂∂Pρ+1-β2Pρ2L^i2Pρ2.It has eigenfunctions (13)ϕqlm=const.×sinl⁡ηcos⁡ηPq1/2,l+1/2cos⁡2ηYmlPθ,Pφ,where η=arcsin⁡βPρ and Pq(a,b) are Jacobi polynomials with q a nonnegative integer. The eigenvalues are given by β22q+l+22-ll+1-1. Taking q=n-1/2, one recovers the eigenvalues (9).


## 3. The Nonrelativistic SdS Algebra

The representations of the operatorsx^i and p^i that satisfy the SdS algebra can be obtained from the operators X^i and P^i of the Snyder algebra by taking the linear combinations [29] (14)x^i=X^i+λβαP^i,p^i=1-λP^i-αβX^i,with inverse (15)P^i=p^i+αβx^i,X^i=1-λx^i-λβαp^i,where λ is a free parameter. Representations with different values of λ are related by unitary transformations [29]; therefore in the following we will consider only the case λ=0.The relation between Snyder and SdS representations can be understood by considering the embedding ofS3 into Gr(3,5), corresponding to the branching SO(5)→SO(4). We recall that the vectors of Gr(3,5) satisfy the constraints (3), while those of S3 satisfy β2(Pi2+P42)=1. Taking into account the SdS gauge constraint αx5+βp5=0, it is easy to see that the combination Pμ=pμ+α/βxμ defined as in (15) satisfies the same constraint as the vectors of SO(4)/SO(3) and then transforms as the Snyder momentum.


### 3.1. The Momentum Representation I

Settingλ=0, from (14) and (5) one obtains the representation (16)x^i=i∂∂Pi+iβ2PiPj∂∂Pj+μ,p^i=Pi-iαβ∂∂Pi+β2PiPj∂∂Pj+μ.As for the Snyder model, the eigenfunctions can be written in terms of x^i2, L^2, and L^3.Clearly,x^i2=X^i2, and hence the equation (17)x^i2ψ=xi2ψhas the same eigenfunctions (8) and eigenvalues (9) as in the Snyder model.The calculation of the operatorp^i2 is a bit more involved. From (14) one has, for λ=0, (18)p^i2=P^i2-αβX^iP^i+P^iX^i+α2β2X^i2.In the representation of Section 2.2, (19)X^iP^i+P^iX^i=2i1+β2Pρ2Pρ∂∂Pρ+3i+iβ21+2μPρ2.From (7) and (19) it follows then that(20)p^i2=-α2β21+β2Pρ22∂2∂Pρ2+1+β2Pρ21+β2Pρ2+iβαPρ22Pρ∂∂Pρ-L^i2Pρ2-αβ3i+iβ2-βαPρ2+μα221+β2Pρ2Pρ∂∂Pρ+1+μβ2Pρ2+3-2iβαPρ2.As for the Snyder model, it is straightforward to check that the solutions withμ≠0 can simply be obtained by multiplying those with vanishing μ by cosμχ, so we consider only the case μ=0. Then, the solutions of the eigenvalue equation p^i2ϕ=pi2ϕ can be deduced from those of (17) by noting that the substitution ϕ=(1+β2Pρ2)-i/2αβψ brings the equation to the same form as (7), with Xi2→β2/α2pi2, and hence its eigenfunctions differ only by a phase from those of x^i2: (21)ϕnlm=const.×sinl⁡χcosi/αβ⁡χCnl+1cos⁡χYmlPθ,Pφ.The operators x^i2 and p^i2 are therefore related by a unitary transformation, and the eigenvalues of p^i2 are the same as those of x^i2, except for a multiplicative constant: (22)pi2=α2n2+2nl+2n+l.This could have been predicted on the ground of the duality betweenx^i and p^i. It follows that in the SdS model also the eigenvalues of the momentum square (and hence of the energy) are quantized and that they do not depend on β.


### 3.2. The Momentum Representation II

Also for SdS one can adopt the alternative representation of Section2.2 [29]. Starting from (10), one obtains, for λ=0, (23)x^i=i1-β2Pk2∂∂Pi,p^i=Pi1-β2Pk2-iαβ1-β2Pk2∂∂Pi.As before, forλ=0, the operator x^i2 coincides with X^i2 and its eigenfunctions and eigenvalues are given, respectively, by (13) and (9).The operatorp^i2 reads instead (24)p^i2=-α2β21-β2Pρ2∂2∂Pρ2+2-3β2+2iβ/αPρ2Pρ∂∂Pρ-1+2iαβPρ2+3iα/β1-β2Pρ2+α2L^2β2Pρ2.This result had been obtained in [29] for a slightly different operator.In analogy with the calculations done in the previous section, the eigenvalue equation forp^i2 can be reduced to the form (12) by introducing a function ψ such that ϕ=(1-β2Pρ2)i/2αβψ. The solution is therefore (25)ϕqml=const.×sinl⁡ηcos1+i/2αβ⁡ηPql+1/2,1/2cos⁡2ηYmlPθ,Pφ,with η=arcsin⁡βPρ. As in the Snyder case, taking q=n+1/2, one recovers the eigenvalues (22).


### 3.3. The Position Representations

The duality of the SdS algebra for interchange ofx^i with p^i permits to define position representations by simply exchanging the roles of the phase space coordinates. Alternatively, such representations can be obtained starting from those of the symmetries of S3 in Beltrami coordinates and using transformations analogous to (14).From (16) and (23) one obtains in this way the action of the momentum and position operators on the Hilbert space of functions of Xi. We report them in the case λ=0, where they read, respectively, (26)p^i=i∂∂Xi+iα2XiXk∂∂Xk+μ,x^i=Xi-iβα∂∂Xi+α2XiXk∂∂Xk+μ,p^i=i1-α2Xk2∂∂Xi,x^i=Xi1-α2Xk2-iβα1-α2Xk2∂∂Xi.Position representations can be useful in some problems, like the hydrogen atom, where the potential is a nontrivial function ofXi.


## 3.1. The Momentum Representation I

Settingλ=0, from (14) and (5) one obtains the representation (16)x^i=i∂∂Pi+iβ2PiPj∂∂Pj+μ,p^i=Pi-iαβ∂∂Pi+β2PiPj∂∂Pj+μ.As for the Snyder model, the eigenfunctions can be written in terms of x^i2, L^2, and L^3.Clearly,x^i2=X^i2, and hence the equation (17)x^i2ψ=xi2ψhas the same eigenfunctions (8) and eigenvalues (9) as in the Snyder model.The calculation of the operatorp^i2 is a bit more involved. From (14) one has, for λ=0, (18)p^i2=P^i2-αβX^iP^i+P^iX^i+α2β2X^i2.In the representation of Section 2.2, (19)X^iP^i+P^iX^i=2i1+β2Pρ2Pρ∂∂Pρ+3i+iβ21+2μPρ2.From (7) and (19) it follows then that(20)p^i2=-α2β21+β2Pρ22∂2∂Pρ2+1+β2Pρ21+β2Pρ2+iβαPρ22Pρ∂∂Pρ-L^i2Pρ2-αβ3i+iβ2-βαPρ2+μα221+β2Pρ2Pρ∂∂Pρ+1+μβ2Pρ2+3-2iβαPρ2.As for the Snyder model, it is straightforward to check that the solutions withμ≠0 can simply be obtained by multiplying those with vanishing μ by cosμχ, so we consider only the case μ=0. Then, the solutions of the eigenvalue equation p^i2ϕ=pi2ϕ can be deduced from those of (17) by noting that the substitution ϕ=(1+β2Pρ2)-i/2αβψ brings the equation to the same form as (7), with Xi2→β2/α2pi2, and hence its eigenfunctions differ only by a phase from those of x^i2: (21)ϕnlm=const.×sinl⁡χcosi/αβ⁡χCnl+1cos⁡χYmlPθ,Pφ.The operators x^i2 and p^i2 are therefore related by a unitary transformation, and the eigenvalues of p^i2 are the same as those of x^i2, except for a multiplicative constant: (22)pi2=α2n2+2nl+2n+l.This could have been predicted on the ground of the duality betweenx^i and p^i. It follows that in the SdS model also the eigenvalues of the momentum square (and hence of the energy) are quantized and that they do not depend on β.


## 3.2. The Momentum Representation II

Also for SdS one can adopt the alternative representation of Section2.2 [29]. Starting from (10), one obtains, for λ=0, (23)x^i=i1-β2Pk2∂∂Pi,p^i=Pi1-β2Pk2-iαβ1-β2Pk2∂∂Pi.As before, forλ=0, the operator x^i2 coincides with X^i2 and its eigenfunctions and eigenvalues are given, respectively, by (13) and (9).The operatorp^i2 reads instead (24)p^i2=-α2β21-β2Pρ2∂2∂Pρ2+2-3β2+2iβ/αPρ2Pρ∂∂Pρ-1+2iαβPρ2+3iα/β1-β2Pρ2+α2L^2β2Pρ2.This result had been obtained in [29] for a slightly different operator.In analogy with the calculations done in the previous section, the eigenvalue equation forp^i2 can be reduced to the form (12) by introducing a function ψ such that ϕ=(1-β2Pρ2)i/2αβψ. The solution is therefore (25)ϕqml=const.×sinl⁡ηcos1+i/2αβ⁡ηPql+1/2,1/2cos⁡2ηYmlPθ,Pφ,with η=arcsin⁡βPρ. As in the Snyder case, taking q=n+1/2, one recovers the eigenvalues (22).


## 3.3. The Position Representations

The duality of the SdS algebra for interchange ofx^i with p^i permits to define position representations by simply exchanging the roles of the phase space coordinates. Alternatively, such representations can be obtained starting from those of the symmetries of S3 in Beltrami coordinates and using transformations analogous to (14).From (16) and (23) one obtains in this way the action of the momentum and position operators on the Hilbert space of functions of Xi. We report them in the case λ=0, where they read, respectively, (26)p^i=i∂∂Xi+iα2XiXk∂∂Xk+μ,x^i=Xi-iβα∂∂Xi+α2XiXk∂∂Xk+μ,p^i=i1-α2Xk2∂∂Xi,x^i=Xi1-α2Xk2-iβα1-α2Xk2∂∂Xi.Position representations can be useful in some problems, like the hydrogen atom, where the potential is a nontrivial function ofXi.


## 4. Conclusions

We have investigated some properties of the nonrelativistic SdS algebra and in particular its Hilbert space representations. This algebra is notable because it generates deformed commutation relations without breaking the Lorentz invariance. Since position and momentum are related by a duality, their operators have identical spectra, except for a multiplicative constant, and both are discrete due to the compactness of the algebra. Of course many other unitary equivalent representations exist besides the ones considered in this paper that lead to the same physical results.Our discussion may be useful for the study of nontrivial systems in SdS space. At present, only the free particle and the harmonic oscillator have been discussed [29]. An interesting system to investigate would be the hydrogen atom. However, preliminary calculations seem to indicate that it leads to third-order differential equations that are difficult to study.The results of this paper may also be extended without much effort to the case of negative coupling constants in the algebra (2). In that case the algebra is no longer compact and the spectra of positions and momenta are expected to be continuous [29].Another interesting development would be the investigation of the model from the point of view of relative locality, starting from the discussion of theories with nontrivial momentum space geometry in curved spacetimes initiated in [38, 39].Finally, as suggested in [9] in the discussion of the flat space limit, our results may also be employed for the construction of a quantum field theory on curved Snyder space, by exploiting the lattice-like structure that has emerged from our investigation.



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