Abstract
Scholars concerned with the foundations of quantum mechanics (QM) usually think that contextuality (hence nonobjectivity of physical properties, which implies numerous problems and paradoxes) is an unavoidable feature of QM which directly follows from the mathematical apparatus of QM. Based on some previous papers on this issue, we criticize this view and supply a new informal presentation of the extended semantic realism (ESR) model which embodies the formalism of QM into a broader mathematical formalism and reinterprets quantum probabilities as conditional on detection rather than absolute. Because of this reinterpretation a hidden variables theory can be constructed which justifies the assumptions introduced in the ESR model and proves its objectivity. When applied to special cases the ESR model settles long-standing conflicts (it reconciles Bell’s inequalities with QM), provides a general framework in which previous results obtained by other authors (as local interpretations of the GHZ experiment) are recovered and explained, and supports an interpretation of quantum logic which avoids the introduction of the problematic notion of quantum truth.
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Notes
The word “realism” has a variety of meanings in the literature. The adjective “realistic” used here refers to a very weak form of realism which is compatible with many possible interpretations of QM, among which the standard (Copenhagen) interpretation (Garola and Sozzo 2010a).
If the language of physics is suitably formalized, the physical property assigned by (a) is expressed by a first order predicate, while the property assigned by (b) is expressed by a predicate of higher order (Garola and Pykacz 2004).
Just as physical properties, states can be expressed by first order predicates if the language of physics is suitably formalized, but they must not be confused with physical properties. They bear indeed a different physical interpretation, and the property of “being in a given state” is not a physical property in the sense specified above (Garola and Pykacz 2004). The difference between states and physical properties is crucial, for instance, in the state property systems introduced by the Brussels approach to QM (Aerts 1999a, b).
The conjunction of value definiteness of \(E\) in \(S\) and adequacy condition for exact measurements implies that the result of an exact measurement of \(E\) on any physical object in the state \(S\) is prefixed, though it may depend on the whole measurement context in absence of an assumption of noncontextuality. Therefore this conjunction, extended to all physical properties and states, seems to express what many scholars mean by the word “realism” when using the phrase “local realism” in the context of Bell’s theorem and associated experiments. Other scholars, however, have criticized this terminology (Norsen 2007) and we avoid using it in the present section.
The observational language has a semantic interpretation provided by assignment rules which make some symbols of this language correspond to macroscopic entities, as preparing or registering devices, outcomes of measurements and so on, that belong to the part of the world studied by the theory. Moreover, the semantic interpretation adopts, often implicitly, a theory of truth, which defines truth values for some (not necessarily all) statements of the observational language. Finally, the correspondence rules make some parts of the observational language correspond to parts of the theoretical language, thus providing a partial and indirect interpretation of the theoretical language.
We recall that some scholars opposed this conclusion because the limited efficiencies of the detectors used in the experiments obliged the experimentalists to introduce additional assumptions to show that the predictions of QM which imply nonlocality were fulfilled (detection loophole). In particular, a fair sampling assumption was usually introduced which states that the sample of all physical objects (pairs of particles) that are detected in the experiments is representative of all physical objects that are emitted. Several models then devised to show that also local explanations of the obtained results are possible. These alternative proposals, however, usually aimed to defend some forms of “local realism” (see footnote 4) against QM. Hence they are radically different from the alternative proposed in this paper, which recovers value definiteness and locality by reinterpreting quantum probabilities but also recovers the mathematical formalism of QM embodying it into a broader mathematical framework (Sect. 5).
Idealized measurements correspond in the ESR model to the first kind, ideal measurements of QM, which are exact measurements that satisfy the Lüders postulate (Beltrametti and Cassinelli 1981). To be precise, they are maximally efficient measurements that satisfy the generalized Lüders postulate (Garola and Sozzo 2009, 2011a).
The idea of introducing a “no show” outcome is not new in the literature. It was introduced indeed in various h.v. models that were dubbed by Fine “prism models” (Fine 1982a, b, 1989; Szabó 2000). These models, however, mainly aim to provide explanations of the results obtained in Aspect’s and similar experiments avoiding nonlocality. The ESR model constitutes instead a general theory, which recovers the aforesaid idea into a broader conceptual framework.
The distinction between proper and improper mixtures (d’Espagnat 1976; Busch et al. 1991) is controversial, and some authors maintain that only improper mixtures exist in QM (Fano 1957; Park 1968a, b; Beltrametti and Cassinelli 1981; Ballentine 1998). From a mathematical point of view, all mixtures are represented by density operators in QM. Nevertheless the two kinds of mixtures are not experimentally equivalent (Timpson and Brown 2005) and are empirically interpreted on different classes of preparation procedures (Garola and Sozzo 2012). The different representations of the two kinds of mixtures in the ESR model follow from this empirical difference (hence the possibility of distinguishing proper from improper mixtures at a physical level has a mathematical counterpart, which is epistemologically satisfactory).
We have seen in footnote 6 that some scholars opposed this conclusion because the efficiencies of the real measurement devices used in the experiments were very low, so that additional assumptions were needed to obtain the derived inequalities for real measurements which were disproved by the experimental data. Indeed these data could then simply show that the additional assumptions were false, not that Bell?s inequalities were violated (Santos 2004). More recent experiments, however, seem to overcome this criticism introducing measurements with efficiencies close to 1 (Genovese 2005). Other scholars criticize instead the standard proofs of Bell’s inequalities because such proofs do not require only the assumption of value definiteness but rest on a hidden Bell’s postulate (HBP) which states that “an experiment involving several incompatible measurements can be written on a single probability space, independently of the measurement context” (Adenier 2009). Furthermore, an attempt of going beyond QM has been recently done by Khrennikov with his prequantum classical statistical field theory (PCSFT), combined with a threshold signal detection model (TSD) which can be considered as a measurement theory for PCSFT (Khrennikov 2012a, b). Khrennikov’s theory is contextual but local and shares some important features with the ESR model (as a fundamental non-detectability which occurs when measurements are performed).
It must be noted, however, that the above objection does not invalidate the procedures that lead one to recover QL within an extended classical framework. It only means that the extension of a property does not bear a physical interpretation if an orthodox view of QM is accepted. This remark invalidates the standard belief that reconciling QL with CL is also mathematically impossible. We add that the methods summarized by steps 1–4 can also be applied to classical mechanics. In this case one obtains a classical logical structure as concrete logic if one assumes that all sentences of \(L\) are verifiable. Nevertheless one can obtain nonclassical structures if the notion of verification is suitably restricted. In particular, one can obtain again QL in this way, which shows that this logical structure is not specific of QM, as pointed out by several authors (Aerts 1999b).
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Acknowledgments
The authors are greatly indebted with Prof. Jaroslaw Pykacz and Dr. Sandro Sozzo for reading the manuscript and providing useful remarks and suggestions.
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Garola, C., Persano, M. Embedding Quantum Mechanics into a Broader Noncontextual Theory. Found Sci 19, 217–239 (2014). https://doi.org/10.1007/s10699-013-9341-z
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DOI: https://doi.org/10.1007/s10699-013-9341-z