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-Lüders 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 ${\sigma }_p\ge |K{|}^{\frac{1}{2}}\hslash$, 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 $r_s\ge 2l_P$, where $l_P$ is the Planck length.

5 pages, 1 figure