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some results on unsharp localization in quantum mechanics

CATTANEO G., Tiziana Marsico, Giuseppe Antonio NISTICO'

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

Some no go theorems in standard quantum mechanics on Hibert spaces, on the non-localizability of zero mass, non-zero spin particles by sharp observables, i.e., projection valued (PV) measures, are introduced. The solution by unsharp observables, i.e., POV measures is presented and the relationship with the induced inner and outer PV measures (or PV rough approximations) is discussed.

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Some no go theorems in standard quantum mechanics on Hibert spaces, on the non-localizability of zero mass, non-zero spin particles by sharp observables, i.e., projection valued (PV) measures, are introduced. The solution by unsharp observables, i.e., POV measures is presented and the relationship with the induced inner and outer PV measures (or PV rough approximations) is discussed.
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Abstract The well-definedness of particles of any kind depends on the limits, approximations, or other conditions that may or may not be involved, for example, whether there are interactions and whether ostensibly related energy is localizable. In particular, their theoretical status differs between its non-relativistic and relativistic versions: One can properly define interacting elementary particles in single-system non-relativistic quantum mechanics, at least in the case of non-zero mass systems; by contrast, one is severely challenged to define even these properly in the relativistic quantum field theories that now underlie the study of particle physics. Here, the impact of localizability on this status is reviewed in relation to the work of Paul Busch on positive operator valued measures that significantly probes the relevance of quantum unsharpness to it.
The quantum mechanical measurement problem is the difficulty of dealing with the indefiniteness of the pointer observable at the conclusion of a measurement process governed by unitary quantum dynamics. There has been hope to solve this problem by eliminating idealizations from the characterization of measurement. We state and prove two ‘insolubility theorems’ that disappoint this hope. In both the initial state of the apparatus is taken to be mixed rather than pure, and the correlation of the object observable and the pointer observable is allowed to be imperfect. In the insolubility theorem for sharp observables, which is only a modest extension of previous results, the object observable is taken to be an arbitrary projection valued measure. In the insolubility theorem for unsharp observables, which is essentially new, the object observable is taken to be a positive operator valued measure. Both theorems show that the measurement problem is not the consequence of neglecting the ever-present imperfections of actual measurements.
In the conventional axiomatic formalism of quantum mechanics, a physical measurement of a quantum observable is mathematically represented by the spectral decomposition of the Hermitian operator associated with the observable. This constitutes the scenario of a L\"uders (projective) measurement, which includes a von Neumann (rank-one projective) measurement as a special and prominent instance. In this context, the measurement is called sharp in the sense that each measurement operator is an orthogonal projection (eigenprojection of the observable). In the modern operational formalism, a measurement is represented by a positive-operator valued measure (POVM), which consists of a family of non-negative definite operators (measurement operators, effects) summing to the identity. In this scenario, a measurement is called unsharp (fuzzy) if some measurement operators are not orthogonal projections. A natural question arises as to how to quantify unsharpness of a measurement. In this work, we address this issue in terms of uncertainty. For this purpose, we study a family of observables associated with a measurement and their uncertainty. By exploiting the difference between the (total) measurement uncertainty and the observable uncertainty, we are led to some information-theoretic quantifiers of unsharpness. We reveal their basic properties and illustrate them through some important measurements. In particular, we characterize L\"uders measurements and equiangular POVMs as extreme measurements in terms of unsharpness.
A commutative positive operator valued (POV) measure F with real spectrum is characterized by the existence of a projection valued measure E (the sharp reconstruction of F) with real spectrum such that F can be interpreted as a randomization of E. This paper focuses on the relationships between this characterization of commutative POV measures and Neumark’s extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of F and the sharp reconstruction of F. The relevance of this result to the theory of nonideal quantum measurement and to the definition of unsharpness is analyzed.