Quantum uncertainty and the spectra of symmetric operators

Original · EN

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 ΔSₜ of t, i.e., ΔS [ϕ]≥ ΔSₜ, 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 ΔSₜ. We also discuss potential applications in quantum and classical information theory.