Quantum mechanics on a curved Snyder space

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.


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 scale β 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]. As in other cases of deformed Poincaré invariance [4], the momentum space can be identified with a nontrivial manifold, namely a 3-sphere S 3 .
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 [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In particular, it was shown that space is discretized [6], and deformed Heisenberg uncertainty relations hold, implying a lower bound on measurable length [5]. Both classical and quantum dynamics are modified with respect to the standard results, with deviations of order β 2 E 2 , E being the energy of the system [5].
The extension of the Snyder model to a spacetime background of constant curvature was proposed in [2] (see also [20] for a different approach). This generalization was motivated by the necessity of including the cosmological constant Λ ∼ α 2 among the bare parameters of a theory of quantum gravity [2]. In this way, one introduces a third fundamental constant besides the speed of light c and the Snyder parameter β, whence 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 [21]. The properties of SdS space were studied by many authors [22][23][24][25][26][27][28][29][30], mainly in its nonrelativistic version. In this case, both positions and momenta have discrete spectra, and a minimal momentum occurs besides minimal length.
In this paper, we attempt to generalize the results on the algebraic structure of the three-dimensional nonrelativistic Snyder model, investigated in [6], 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 for 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.

The SdS model
The nonrelativistic SdS algebra A depends on two parameters α and β and contains the usual generators of rotationsL k = 1 2 ǫ ijkLij (i, j = 1, . . . , 3), with their standard action on the position and momentum operatorsx i andp i : The commutation relations ofx i andp i satisfy instead a deformation of the Heisenberg algebra, For special values of the parameters, A reduces to the nonrelativistic Snyder algebra (α = 0), or the algebra of isometries of S 3 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 [22], but we shall not discuss it here. As mentioned above, the algebra is invariant for αx µ ↔ βp µ . More generally, it is invariant for rotations in the phase space, 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 [31], which differs from SdS only in the assumption of a standard Heisenberg algebra for positions and momenta, [x i ,p j ] = iKδ ij ,K being a central charge for the rotation group, satisfying [K, Also the 3-dimensional nonrelativistic SdS model enjoys an SO(5) symmetry. In fact, its phase space can be realized on the six-dimensional Grassmannian coset space The space Gr (3,5) can be parametrized by homogeneous coordinates x µ and p µ , that satisfy the constraints [32] α 2 x 2 µ = 1, This parametrization associates a one-parameter set of matrices to each coset. One can then identify the variables x µ and p µ with canonical coordinates of a ten-dimensional phase space and hence reduce it to a six-dimensional phase space parametrized by x i and p i by eliminating the constraints (1.3), using the Dirac formalism [33]. In order to obtain a oneto-one parametrization, one has however to impose a further constraint on the parameters x 4 , x 5 , p 4 , p 5 . This is also required by the Dirac formalism since the constraints (1.3) split into one first class and two second class constraints [25]. Unfortunately, not every constraint leads to the SdS algebra, and one has therefore to choose a suitable gauge [25].

The nonrelativistic Snyder algebra
In this section we review some results on the representation of the Snyder algebra from [5] and [6], that will be useful in the following discussion. The Snyder algebra is the limit of the SdS algebra (1.1)-(1.2) for α → 0. It contains an so(4) subalgebra of so(5) generated byX i =Ĵ 4i andL i = 1 2 ǫ ijkĴjk , while the momentum space is realized as the coset space S 3 = SO(4)/SO (3).
The momentum space can be realized as a 3-sphere, obtained by imposing the constraint P 2 i + P 2 4 = 1/β 2 on a four-vector. This can be shown algebraically, as in [6], or also from a Dirac reduction of the phase space [8]. In the following, we shall mainly concentrate on the operatorsX 2 i andP 2 i and investigate their spectra.

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 [7,6] it was realized by operatorsP i andX i defined aŝ which act on functions ψ(P i ) of the Hilbert space, with µ an arbitrary real parameter. The operators are symmetric for the measure (2.4) In this representation the operatorX 2 i readŝ For µ = 0, the equationX 2 i φ = X 2 i φ has eigenfunctions [6] φ nlm = const. × sin l χ C (l+1) n (cos χ) Y l m (P θ , P ϕ ), (2.6) where we have used polar coordinates P ρ , P θ , P ϕ for P i , and χ = arctan βP ρ . The functions C (a) n are Gegenbauer polynomials with n a nonnegative integer parameter, and Y l m (P θ , P ϕ ) spherical harmonics.
It is easy to see that if µ = 0, the eigenfunctions (2.6) are simply multiplied by cos µ χ. The eigenvalues are of course independent of µ and read X 2 i = β 2 (n 2 + 2nl + 2n + l), (2.7) with 0 ≤ l ≤ n. They can easily be identified with (2.1) by setting n = 2j − l. The operatorP 2 i =P 2 ρ is trivial and its spectrum extends to the real positive line.

The nonrelativistic SdS algebra
The representations of the operatorsx i andp i that satisfy the SdS algebra can be obtained from the operatorsX i andP i of the Snyder algebra by taking the linear combinations [22]x where λ is a free parameter. Representations with different values of λ are related by unitary transformations [22], therefore in the following we shall consider only the case λ = 0.
The relation between Snyder and SdS representations can be understood by considering the embedding of S 3 into Gr (3,5), corresponding to the branching SO(5) → SO(4). We recall that the vectors of Gr(3, 5) satisfy the constraints (1.3), while those of S 3 satisfy β 2 (P 2 i + P 2 4 ) = 1. Taking into account the SdS gauge constraint αx 5 + βp 5 = 0, it is easy to see that the combination P µ = p µ + α β x µ defined as in (3.2) satisfies the same constraint as the vectors of SO(4)/SO(3) and then transforms as the Snyder momentum.

The momentum representation I
Setting λ = 0, from (3.1) and (2.2) one obtains the representation As for the Snyder model, the eigenfunctions can be written in terms ofx 2 i ,L 2 andL 3 . Clearly,x 2 i =X 2 i , and hence the equation has the same eigenfunctions (2.6) and eigenvalues (2.7) as in the Snyder model. The calculation of the operatorp 2 i is a bit more involved. From (3.1) one has for In the representation of section 2.2, From (2.5) and (3.6) follows then As for the Snyder model, it is straightforward to check that the solutions with µ = 0 can simply be obtained by multiplying those with vanishing µ by cos µ χ, so we consider only the case µ = 0. Then, the solutions of the eigenvalue equationp 2 i φ = p 2 i φ can be deduced from those of (3.4) by noting that the substitution φ = (1 + β 2 P 2 ρ ) − i 2αβ ψ brings the equation to the same form as (2.5), with X 2 i → β 2 α 2 p 2 i , and hence its eigenfunctions differ only by a phase from those ofx 2 i : The operatorsx 2 i andp 2 i are therefore related by a unitary transformation, and the eigenvalues ofp 2 i are the same as those ofx 2 i , except for a multiplicative constant: This could have been predicted on the ground of the duality betweenx i andp 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.

The momentum representation II
Also for SdS one can adopt the alternative representation of section [22]. Starting from (2.8), one obtains, for λ = 0, As before, for λ = 0, the operatorx 2 i coincides withX 2 i and its eigenfunctions and eigenvalues are given respectively by (2.11) and (2.7).
The operatorp 2 i reads instead (3.11) This result had been obtained in [22] for a slightly different operator.

The position representations
The duality of the SdS algebra for interchange ofx i withp 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 S 3 in Beltrami coordinates and using transformations analogous to (3.1).
From (3.3) and (3.10) one obtains in this way the action of the momentum and position operators on the Hilbert space of functions of X i . We report them in the case λ = 0, where they read, respectively, Position representations can be useful in some problems, like the hydrogen atom, where the potential is a nontrivial function of X i .

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 [22]. 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.
As suggested in ref. [6] for the flat 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.