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English

Generalized Localizability

Published inHelvetica physica acta, vol. 40, no. 5, p. 559-570
Publication date1967
Abstract

It is well known that particles of restmass zero with spin ≠ 0 belonging to an irreducible representation of the Lorentzgroup do not admit position operators. Yet such particles exist in nature (for instance the neutrino and the photon) and they are localizable in the experimental sense of this term. The mathematical description of localizability is generalized in this paper so as to be applicable in cases where the conventional position operator does not exist. The generalization consists in omitting the hypothesis that all observations of position measurements are compatible with one another. Compatibility is maintained only for space domains which do not overlap or for which one is contained inside the other. For the other cases of overlapping domains compatibility cannot be justified on empirical grounds and it can be dropped. The resulting mathematical object is a generalized system of imprimitivities and it is the appropriate concept for the mathematical description of certain localizable systems. We give a standard method for constructing such systems based on a theorem of Neumark. A particle which is localizable in this sense is called weakly localizable. A further weakening of the conditions leads us to the notion of nearly localizable systems. In this case there exists no states which localize the particle exactly in a given space domain, but only states which approximate this property to an arbitrary degree of accuracy (in the topology induced by the states). We have verified that particles of mass m = 0 and spin 1/2 (neutrinos) are nearly localizable. In addition we have verified that the photons are in fact also weakly localizable.

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Citation (ISO format)
JAUCH, Joseph-Maria, PIRON, Constantin. Generalized Localizability. In: Helvetica physica acta, 1967, vol. 40, n° 5, p. 559–570. doi: 10.5169/seals-113783
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ISSN of the journal0018-0238
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