We continue our investigation of the phenomenological implications of the ''deformed'' commutation relations ͓x i ,p j ͔ϭiប͓(1ϩ␤ p 2 )␦ i j ϩ␤Јp i p j ͔. These commutation relations are motivated by the fact that they lead to the minimal length uncertainty relation which appears in perturbative string theory. In this paper, we consider the effects of the deformation on the classical orbits of particles in a central force potential. Comparison with observation places severe constraints on the value of the minimum length.

The energy spectrum of the Coulomb potential with minimal length commutation relations ͓X i , P j ͔ = iប͕␦ ij ͑1+␤P 2 ͒ + ␤ЈP i P j ͖ is determined both numerically and perturbatively for arbitrary values of ␤Ј / ␤ and angular momenta ᐉ. The constraint on the minimal length scale from precision hydrogen spectroscopy data is of the order of a few GeV −1 , weaker than previously claimed. DOI: 10.1103/PhysRevA.72.012104 PACS number͑s͒: 03.65.Ge, 02.40.Gh, 31.15.Md, 32.10.Fn Quantum gravity incorporates Newton's constant as a dimensional parameter that could manifest itself as a minimal length in the system. Recent string theoretic considerations suggest that this length scale might imply an ultravioletinfrared ͑UV-IR͒ correspondence, contrary to the normal perceptions on momentum and spatial separations. Large momenta are now directly tied to large spatial dimensions, which then implies the existence of a minimal length. Earlier studies have focused upon its amelioration of ultraviolet divergences ͓1͔, but did not take into full account the UV-IR correspondence.There are various ways of implementing such an idea, but the simplest is to suppose that coordinates no longer commute in D-dimensional space. This, in turn, leads to a deformation of the canonical commutation relations. In our previous works, we adopted the equivalent hypothesis that the fundamental commutation relations between position and momentum are no longer constant multiples of the identity. In this paper, we report on constraints on the minimal length hypothesis from precision measurements on hydrogenic atoms. This system has a potential that is singular at the origin, and is therefore particularly sensitive to whether there is a fundamental minimal length. Considerations based upon higher-dimensional theories suggest that such lengths may be large ͓2͔.To set the context, we note that if in one dimension we havewhere ␤ is a small parameter, then the resulting uncertainty relation ͑⌬X͒͑⌬P͒ ജ iប͕1+␤͑⌬P͒ 2 ͖ exhibits a form of the UV-IR correspondence, and gives as minimal length ⌬X ജប ͱ ␤ ͓3͔.We had examined the harmonic oscillator system under this hypothesis in ͓4͔, but no real constraint can be obtained on the minimal length, presumably because of the softness of the potential at the origin. An interesting approach is to take the classical limit ប → 0 of the commutation relations; it yields an unbelievably strong bound, but its robustness might be questioned ͓5͔.We will work in arbitrary D Ͼ 1 dimensions, where ͑1͒ takes the tensorial formwhich, assuming that the momenta commute ͓P i , P j ͔ =0, leads via the Jacobi identity to the nontrivial position commutation relationsThe position and momentum operators can be represented bywhere the operators x i and p j satisfy the canonical commutation relations ͓x i , p j ͔ = iប␦ ij . The simplest representation is momentum diagonal,In this representation the eigenvalue equation for the distance squared operator R 2 = X i X i can be solved exactly. With z = ͑␤ + ␤Ј͒p 2 − 1 ͑␤ + ␤Ј͒p 2 + 1 , ͑5͒ the eig...

We revisit the classical model for voter dynamics in a two-party system with two basic modifications. In contrast to the original voter model studied in regular lattices, we implement the opinion formation process in a random network of agents in which interactions are no longer restricted by geographical distance. In addition, we incorporate the rapidly changing nature of the interpersonal relations in the model. At each time step, agents can update their relationships. This update is determined by their own opinion, and by their preference to make connections with individuals sharing the same opinion, or rather with opponents. In this way, the network is built in an adaptive manner, in the sense that its structure is correlated and evolves with the dynamics of the agents. The simplicity of the model allows us to examine several issues analytically. We establish criteria to determine whether consensus or polarization will be the outcome of the dynamics and on what time scales these states will be reached. In finite systems consensus is typical, while in infinite systems a disordered metastable state can emerge and persist for infinitely long time before consensus is reached.

We propose an exactly solvable model for the dynamics of voters in a two-party system. The opinion formation process is modeled on a random network of agents. The dynamical nature of interpersonal relations is also reflected in the model, as the connections in the network evolve with the dynamics of the voters. In the infinite time limit, an exact solution predicts the emergence of consensus, for arbitrary initial conditions. However, before consensus is reached, two different metastable states can persist for exponentially long times. One state reflects a perfect balancing of opinions, the other reflects a completely static situation. An estimate of the associated lifetimes suggests that lack of consensus is typical for large systems.