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AIP conference proceedings

Fuzzy position and momentum observables

Teiko Heinonen

Published Jan 1, 2005• 0 citations
Publisher

Abstract

The notion of a fuzzy observable is a possible way to describe imprecise measurements. We show that fuzzy versions of the usual position and momentum observables are exactly those observables which satisfy the natural covariance and invariance requirements. We characterize the class of fuzzy position and momentum observables which are informationally equivalent with the canonical position and momentum observables.

DOI
10.1063/1.1874566

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The notion of a fuzzy observable is a possible way to describe imprecise measurements. We show that fuzzy versions of the usual position and momentum observables are exactly those observables which satisfy the natural covariance and invariance requirements. We characterize the class of fuzzy position and momentum observables which are informationally equivalent with the canonical position and momentum observables.
In this thesis the structure and properties of imprecise quantum measurements are investigated. The starting point for this investigation is the representation of a quantum observable as a normalized positive operator measure. A general framework to describe measurement inaccuracy is presented. Requirements for accurate measurements are discussed, and the relation of inaccuracy to some optimality criteria is studied. A characterization of covariant observables is given in the case when they are imprecise versions of a sharp observable. Also the properties of such observables are studied. The case of position and momentum observables is studied. All position and momentum observables are characterized, and the joint positionmomentum measurements are discussed.
The problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so-called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.
The experimentally tested occurrence of probabilities that do not find a representation within the usual Kolmogorov probability theory is mathematically formalized through the notion of the Bell phenomenon for probability measures and for observables. Reference is made to a physically natural definition of observables that includes both the classical and the quantum usual versions. It is shown that even a classical framework can host the Bell phenomenon, provided fuzzy observables are called into play.