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.
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 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 .
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 , which in the Snyder case is a 3-sphere . 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 .
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  and deformed Heisenberg uncertainty relations hold, implying a lower bound on measurable length . Both classical and quantum dynamics are modified with respect to the standard results, with deviations of order , being the energy of the system .
The extension of the Snyder model to a spacetime background of constant curvature was proposed in  (see also  for a different approach). This generalization was motivated by the necessity of including the cosmological constant among the bare parameters of a theory of quantum gravity . In this way, one introduces a third fundamental constant besides the speed of light and the Snyder parameter , 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 . 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 , 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, and , 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 algebra depends on two parameters and and contains the usual generators of rotations , with their standard action on the position and momentum operators and :The commutation relations of and satisfy instead a deformation of the Heisenberg algebrawhere .
For special values of the parameters, reduces to the nonrelativistic Snyder algebra () or the algebra of isometries of in Beltrami coordinates (). It is possible to define also a noncompact version of the algebra, by analytic continuation to imaginary values of and , with rather different properties , but we will not discuss it here. As mentioned above, the algebra is invariant for . More generally, it is invariant for rotations in the phase space, , . Of course, this symmetry holds also in standard quantum mechanics, with .
The SdS algebra can be considered as a nonlinear realization of a model proposed by Yang , which differs from SdS only in the assumption of a standard Heisenberg algebra for positions and momenta, , being a central charge for the rotation group, satisfying , . With the identifications , , , , the Yang model reproduces an algebra with generators , ().
Also the 3-dimensional nonrelativistic SdS model enjoys an symmetry. In fact, its phase space can be realized on the six-dimensional Grassmannian coset space , with generated by the and by .
The space can be parametrized by homogeneous coordinates and , that satisfy the constraints : This parametrization associates a one-parameter set of matrices to each coset. One can then identify the variables and with canonical coordinates of a ten-dimensional phase space and hence reduce it to a six-dimensional phase space parametrized by and by eliminating the constraints (3), using the Dirac formalism . In order to obtain a one-to-one parametrization, one has however to impose a further constraint on the parameters , , , . This is also required by the Dirac formalism since the constraints (3) split into one first class and two second class constraints . Unfortunately, not every constraint leads to the SdS algebra, and one has therefore to choose a suitable gauge . In particular, the choice yields the algebra (2).
2. The Nonrelativistic Snyder Algebra
The representations of the algebra can be labeled by the eigenvalues of , , and , where is the operator , so that :with , being the eigenvalue of . The eigenvalues of have degeneration .
The momentum space can be realized as a 3-sphere, obtained by imposing the constraint on a four-vector . This can be shown algebraically, as in , or also from a Dirac reduction of the phase space . In the following, we will mainly concentrate on the operators and 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 and defined aswhich act on functions of the Hilbert space, with an arbitrary real parameter and . The operators are symmetric for the measure provided that the functions go to infinity like .
In this representation the operator readsFor , the equation has eigenfunctions  where we have used polar coordinates for , and . The functions are Gegenbauer polynomials with a nonnegative integer parameter and spherical harmonics.
The operator is trivial and its spectrum extends to the real positive line.
2.2. The Representation II
An alternative representation is obtained  by defining with . In this representation the operators are symmetric for the measure and the operator reads It has eigenfunctions where and are Jacobi polynomials with a nonnegative integer. The eigenvalues are given by . Taking , one recovers the eigenvalues (9).
3. The Nonrelativistic SdS Algebra
The representations of the operators and that satisfy the SdS algebra can be obtained from the operators and of the Snyder algebra by taking the linear combinations  with inverse where is a free parameter. Representations with different values of are related by unitary transformations ; therefore in the following we will consider only the case .
The relation between Snyder and SdS representations can be understood by considering the embedding of into , corresponding to the branching . We recall that the vectors of satisfy the constraints (3), while those of satisfy . Taking into account the SdS gauge constraint , it is easy to see that the combination defined as in (15) satisfies the same constraint as the vectors of and then transforms as the Snyder momentum.
3.1. The Momentum Representation I
As for the Snyder model, it is straightforward to check that the solutions with can simply be obtained by multiplying those with vanishing by , so we consider only the case . Then, the solutions of the eigenvalue equation can be deduced from those of (17) by noting that the substitution brings the equation to the same form as (7), with , and hence its eigenfunctions differ only by a phase from those of : The operators and are therefore related by a unitary transformation, and the eigenvalues of are the same as those of , except for a multiplicative constant:
This could have been predicted on the ground of the duality between and . 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 .
3.2. The Momentum Representation II
The operator reads instead This result had been obtained in  for a slightly different operator.
In analogy with the calculations done in the previous section, the eigenvalue equation for can be reduced to the form (12) by introducing a function such that . The solution is therefore with . As in the Snyder case, taking , one recovers the eigenvalues (22).
3.3. The Position Representations
The duality of the SdS algebra for interchange of with 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 in Beltrami coordinates and using transformations analogous to (14).
Position representations can be useful in some problems, like the hydrogen atom, where the potential is a nontrivial function of .
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 . 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 .
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  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.
The authors declare that they have no competing interests.
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