# Uncertainty Principle on 3-Dimensional Manifolds of Constant Curvature

Uncertainty Principle on 3-Dimensional Manifolds of Constant Curvature 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 $$\sigma _p\ge |K|^\frac{1}{2}\hbar$$ σ p ≥ | K | 1 2 ħ , 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 $$r_s\ge 2\,l_P$$ r s ≥ 2 l P , where $$l_P$$ l P is the Planck length. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Foundations of Physics Springer Journals

# Uncertainty Principle on 3-Dimensional Manifolds of Constant Curvature

, Volume 48 (6) – May 23, 2018
10 pages

/lp/springer_journal/uncertainty-principle-on-3-dimensional-manifolds-of-constant-curvature-QNHZ0ZALft
Publisher
Springer US
Subject
Physics; History and Philosophical Foundations of Physics; Quantum Physics; Classical and Quantum Gravitation, Relativity Theory; Statistical Physics and Dynamical Systems; Classical Mechanics; Philosophy of Science
ISSN
0015-9018
eISSN
1572-9516
D.O.I.
10.1007/s10701-018-0173-0
Publisher site
See Article on Publisher Site

### 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 $$\sigma _p\ge |K|^\frac{1}{2}\hbar$$ σ p ≥ | K | 1 2 ħ , 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 $$r_s\ge 2\,l_P$$ r s ≥ 2 l P , where $$l_P$$ l P is the Planck length.

### Journal

Foundations of PhysicsSpringer Journals

Published: May 23, 2018

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