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

Loading next page...
 
/lp/springer_journal/uncertainty-principle-on-3-dimensional-manifolds-of-constant-curvature-QNHZ0ZALft
Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off