Short distance versus long distance physics: The classical limit of the minimal length uncertainty relation

We continue our investigation of the phenomenological implications of the “deformed” commutation relations $[x_i,p_j]=iħ\left[\right(1+\beta p^2){\delta }_{\mathrm{ij}}+{\beta }^{\prime }{p}_i{p}_j].$ 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.