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Summary: Published as: Aerts, D., Durt, T. and Van Bogaert, B., 1993, "A physical example of quantum fuzzy
sets and the classical limit", Tatra Mt. Math. Publ., 1, 5 - 15.
A physical example of quantum fuzzy sets, and the classical limit.
Diederik Aerts, Thomas Durt and Bruno Van Bogaert,
Theoretical Physics (TENA), Free University of Brussels,
Pleinlaan 2, 1050 Brussels, Belgium.
Abstract : We present an explicit physical example of an experimental situation on a physical entity
that gives rise to a fuzzy set. The fuzziness in the example is due to fluctuations of the experimental
apparatus, and not to an indeterminacy about the states of the physical entity, and is described by
a varying parameter . For zero value of the parameter (no fluctuations), the example reduces to
a classical mechanics situation, and the corresponding fuzzy set is a quasi-crisp set. For maximal
value (maximal fluctuations), the example gives rise to a quantum fuzzy set, more precisely a spin-
model. In between, we have a continuum of fuzzy situations, neither classical, nor quantum. We
believe that the example can make us understand the nature of the quantum mechanical fuzziness
and probability, and how these are related to the classical situation.
Introduction.
After the now rather commonly accepted failure of 'local' hidden variable theories
to deliver a model that could substitute for quantum mechanics (ref. 1, 2, 3, 4, 5, 6, 7),
it is often thought that an 'understandable' explanation for the probabilities of quantum
mechanics is now impossible, and that only the rather vague view of the presence of
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