Harmonic oscillators in a Snyder geometry

We find that, in presence of the Snyder geometry, the quantization of d isotropic harmonic oscillators can be solved exactly.


Introduction
Recent studies have suggested that a natural cutoff for the ultraviolet behaviour of the physical theories can be obtained modifying the Heisenberg algebra of the canonical commutation rules [1]- [2]- [3]- [4]- [5]. This implies a finite minimal uncertainty ∆x 0 in the position measurement, modifying the structure of space-time at short distances.
Such cutoff can be already applied in the non-relativistic approximation, i.e. in ordinary quantum mechanics. The most elementary example is the harmonic oscillator with a finite minimal uncertainty ∆x 0 , a problem solved in [1], using the momentum representation.
It is the aim of this article the generalization of this result to the case of d isotropic harmonic oscillators quantized with the Snyder algebra. We will show that this problem can be exactly solved, without any approximation, allowing us to discuss how the state degeneracy of d independent harmonic oscillators is removed by the non-commutative deformation.
To reach such aim we have studied the eigenvalue equation in a new representation, the variables ρ i which resolve the Snyder algebra. As an outcome of our research, we have been able to solve, at a mathematic level, a d-dimensional generalization of the well known Gegenbauer equation.

Harmonic oscillator revisited
In [1] the harmonic oscillator has been quantized using the following modified quantization rule: in the momentum representation. The corresponding eigenvalue equation is The idea behind our present paper is to solve this problem in a new representation [2] x This effort will be helpful in the next section to solve exactly the quantization of d harmonic oscillators in the Snyder geometry.
The eigenvalue equation in the variables ρ turns out to be For the ground state it is well known that the solution is To study the excited states we look for a solution of this type from which we obtain the following differential equation for χ(ρ) Let us make the substitution and change the variable ρ → z = √ β ρ, obtaining We easily recognize the Gegenbauer equation, whose polynomial solutions are obtained for The corresponding eigenfunctions are proportional to the Gegenbauer polynomials, satisfying the recurrence equation Similarly to the Hermite polynomials, there is a generating function for the Gegenbauer polynomials The first polynomials are

Harmonic oscillators in d dimensions
The theory of the harmonic oscillator described in the previous section can be generalized in d dimensions. The Hamiltonian of isotropic oscillators of equal mass m and frequency ω in d dimensions and cartesian coordinates can be written as For the generic d case, we extend the commutation rule (2.1) to the Snyder algebra [3] [ This algebra is resolved by the ρ representation [2] x The eigenvalue equation for d oscillators with the same frequency ω and mass m is the following: that, rewritten in the ρ variables, reads The ground state is simply with eigenvalue To study the excited states, we introduce as in the case d = 1 the following ansatz a d-dimensional generalization of the Gegenbauer equation.
Let us introduce the notation ǫ {n i } = µ [ 2mEβ(µ − 1) − d ] and z i = √ βρ i , where the parameter ǫ depends on some quantum numbers n i , from which the equation to be solved is with the energy parameter given by The symmetry of this equation suggests to introduce the following ansatz, i.e. that the polynomial solution space is composed by two parts: i) the solutions P kk of the free differential equation where P kk (z i ) is an homogeneous polynomial in the variables z i of degree k.
In this case it is easy to compute the corresponding energy eigenvalue ii) the solutions P N k with N = k + 2n where To compute the eigenvalue corresponding to P N k (z i ) we apply the following differential operator d i=1 ∂ 2 ∂z i ∂z i to the eigenvalue equation (3.10). With simple algebraic steps we deduce that completed with the energy constraint

This recurrence equation (3.16) is solved by
from which we deduce that the energy eigenvalues depends only on two quantum numbers N and k ( whose difference N − k must be an even positive integer number ) We conclude that, comparing with the case β = 0, the noncommutative deformation (3.2) reduces the states degeneracy from d − 1 degrees of freedom to d − 2.
Now it remains to show that the ansatz with which we have solved the differential equation (3.10) gives all the polynomial solutions. We know that for β → 0 our problem reduces to d independent oscillators; in this case fixing the level N, the number of independent eigenfunctions is given by the formula For β = 0 we must simply count how many independent solutions s d (k) exist of the free differential equation (3.12) defining P kk (z i ). We have computed them until d = 5 oscillators s 2 (k) = 2 k > 0, s 2 (0) = 1 s 3 (k) = 2k + 1 s 4 (k) = (k + 1) 2 In all cases we can check that the following identities hold This completes our proof that we have described a complete basis of the Hilbert space.

Conclusion
In this article we have shown that, even modifying the quantization rule, many problems of quantum mechanics can be exactly solved. In particular we have found that the natural ddimensional extension of the modified Heisenberg algebra (2.1) is surely the Snyder algebra (3.2). We have discussed in detail that the quantization of d isotropic oscillators in noncommutative geometry gives rise to two relevant quantum numbers, from which we can deduce that the residual degeneracy of the states is reduced to d − 2 degrees of freedom. The spectrum contains, besides a linear term in the main quantum number N ( that in the commutative limit is the sum of the single particle quantum numbers n i ), a quadratic term dependent also on a secondary quantum number k, such as N − k is an even positive integer number. In the limit d → 1, our general formula reduces to the single harmonic oscillator spectrum studied in [1].
We therefore expect that the solvability of these examples can be extended to more complex cases. Work is in progress in this direction.