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Quantum Uncertainty and the Spectra of Symmetric Operators

Quantum Uncertainty and the Spectra of Symmetric Operators In certain circumstances, the uncertainty, ΔS[φ], of a quantum observable, S, can be bounded from below by a finite overall constant ΔS>0, i.e., ΔS[φ]≥ΔS, for all physical states φ. For example, a finite lower bound to the resolution of distances has been used to model a natural ultraviolet cutoff at the Planck or string scale. In general, the minimum uncertainty of an observable can depend on the expectation value, t=⟨φ,S φ⟩, through a function ΔS t of t, i.e., ΔS[φ]≥ΔS t , for all physical states φ with ⟨φ,S φ⟩=t. An observable whose uncertainty is finitely bounded from below is necessarily described by an operator that is merely symmetric rather than self-adjoint on the physical domain. Nevertheless, on larger domains, the operator possesses a family of self-adjoint extensions. Here, we prove results on the relationship between the spacing of the eigenvalues of these self-adjoint extensions and the function ΔS t . We also discuss potential applications in quantum and classical information theory. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Quantum Uncertainty and the Spectra of Symmetric Operators

Martin, R.; Kempf, A.
Acta Applicandae Mathematicae , Volume 106 (3) – Sep 5, 2008
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References (30)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer Science+Business Media B.V.
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-008-9302-7
Publisher site
See Article on Publisher Site

Abstract

In certain circumstances, the uncertainty, ΔS[φ], of a quantum observable, S, can be bounded from below by a finite overall constant ΔS>0, i.e., ΔS[φ]≥ΔS, for all physical states φ. For example, a finite lower bound to the resolution of distances has been used to model a natural ultraviolet cutoff at the Planck or string scale. In general, the minimum uncertainty of an observable can depend on the expectation value, t=⟨φ,S φ⟩, through a function ΔS t of t, i.e., ΔS[φ]≥ΔS t , for all physical states φ with ⟨φ,S φ⟩=t. An observable whose uncertainty is finitely bounded from below is necessarily described by an operator that is merely symmetric rather than self-adjoint on the physical domain. Nevertheless, on larger domains, the operator possesses a family of self-adjoint extensions. Here, we prove results on the relationship between the spacing of the eigenvalues of these self-adjoint extensions and the function ΔS t . We also discuss potential applications in quantum and classical information theory.

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: Sep 5, 2008

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