Uncertainty principle on 3-dimensional manifolds of constant curvature

Abstract

We consider the Heisenberg uncertainty principle of position and momentum in 3-dimensional spaces of constant curvature $K$. The uncertainty of position is defined coordinate independent by the geodesic radius of spherical domains in which the particle is localized after a von Neumann-L\"uders projection. By applying mathematical standard results from spectral analysis on manifolds, we obtain the largest lower bound of the momentum deviation in terms of the geodesic radius and $K$. For hyperbolic spaces, we also obtain a global lower bound $\sigma_p\geq |K|^\frac{1}{2}\hbar$, which is non-zero and independent of the uncertainty in position. Finally, the lower bound for the Schwarzschild radius of a static black hole is derived and given by $r_s\geq 2\,l_P$, where $l_P$ is the Planck length.