Quantum Uncertainty and the Spectra of Symmetric Operators |
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Authors: | R T W Martin A Kempf |
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Institution: | (1) Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada |
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Abstract: | In certain circumstances, the uncertainty, ΔSφ], of a quantum observable, S, can be bounded from below by a finite overall constant ΔS>0, i.e., ΔSφ]≥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., ΔSφ]≥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.
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Keywords: | Self-adjoint extensions of symmetric operators Generalized observables Finite minimum uncertainty Spectra of symmetric operators |
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