Uncertainty principle on 3-dimensional manifolds of constant curvature

TL;DR

This paper extends the Heisenberg uncertainty principle to 3D manifolds of constant curvature, deriving bounds on momentum deviation related to geometry, with implications for black hole physics.

Contribution

It introduces a coordinate-independent uncertainty measure on curved manifolds and derives new bounds on momentum uncertainty and black hole Schwarzschild radius.

Findings

Lower bound on momentum deviation depends on geodesic radius and curvature.

For hyperbolic spaces, a non-zero, curvature-dependent lower bound is established.

Black hole Schwarzschild radius is bounded below by twice the Planck length.

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.