arXiv subjects

Claudio Garola

Publications and source records attributed to Claudio Garola.

At least 19 recordsLinked to original sources

arXiv2020

Kolmogorovian versus non-Kolmogorovian probabilities in contextual theories

Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class T of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing T are compatible with classical mechanics (CM), statistical mechanics (SM) and QM, hence we assume that these theories belong to T. In the case of CM and QM this assumption is irrelevant, as all notions introduced in them as members of T reduce to standard notions. In the case of QM it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian and provides it with an epistemic interpretation. These results were anticipated in a previous paper but are obtained here in a general framework without referring to individual objects, which shows that they hold even if only a minimal (statistical) interpretation of QM is adopted to avoid the problems following from the standard quantum theory of measurement.

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arXiv2019

Throwing bridges: where and how can classical and quantum view be connected?

We have proposed in several recent papers a critical view of some parts of quantum mechanics (QM) that is methodologically unusual because it rests on analysing the language of QM by using some elementary but fundamental tools of mathematical logic. Our approach proves that some widespread beliefs about QM can be questioned and establishes new links with a classical view, which is significant in the debate on the interpretations of QM. We propose here a brief survey of our results, highlighting their common background. We firstly show how quantum logic (QL) can be embedded into classical logic (CL) if the embedding is required to preserve the logical order and not the algebraic structure, and also how QL can be interpreted as a pragmatic sublanguage within a pragmatic extension of CL. Both these results challenge the thesis that CL and QL formalize the properties of different and incompatible notions of truth. We then show that quantum probability admits an epistemic interpretation if contextuality is taken into account as a basic constituent of the language of QM, which overcomes the interpretation of quantum probability as ontic. Finally, we show that the proofs that QM is a contextual theory stand on a supplementary epistemological assumption that is usually unnoticed and left implicit. Dropping such assumption opens the way, at least in principle, to non-contextual interpretations of QM.

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arXiv2018

An epistemic interpretation of quantum probability via contextuality

According to a standard view, quantum mechanics (QM) is a contextual theory and quantum probability does not satisfy Kolmogorov's axioms. We show, by considering the macroscopic contexts associated with measurement procedures and the microscopic contexts (mu-contexts) underlying them, that one can interpret quantum probability as epistemic, despite its non-Kolmogorovian structure. To attain this result we introduce a predicate language L(x), a classical probability measure on it and a family of classical probability measures on sets of mu contexts, each element of the family corresponding to a (macroscopic) measurement procedure. By using only Kolmogorovian probability measures we can thus define mean conditional probabilities on the set of properties of any quantum system that admit an epistemic interpretation but are not bound to satisfy Kolmogorov's axioms. The generalized probability measures associated with states in QM can then be seen as special cases of these mean probabilities, which explains how they can be non-classical and provides them with an epistemic interpretation. Moreover, the distinction between compatible and incompatible properties is explained in a natural way, and purely theoretical classical conditional probabilities coexist with empirically testable quantum conditional probabilities.

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arXiv2015

A Pragmatic Interpretation of Quantum Logic

Scholars have wondered for a long time whether the language of quantum mechanics introduces a quantum notion of truth which is formalized by quantum logic (QL) and is incompatible with the classical (Tarskian) notion. We show that QL can be interpreted as a pragmatic language of assertive formulas which formalize statements about physical systems that are empirically justified or unjustified in the framework of quantum mechanics. According to this interpretation, QL formalizes properties of the metalinguistic notion of empirical justification within quantum mechanics rather than properties of a quantum notion of truth. This conclusion agrees with a general integrationist perspective that interprets nonstandard logics as theories of metalinguistic notions different from truth, thus avoiding incompatibility with classical notions and preserving the globality of logic. By the way, some elucidations of the standard notion of quantum truth are also obtained. Key words: pragmatics, quantum logic, quantum mechanics, justifiability, global pluralism.

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arXiv2015

Outline of a Generalization and a Reinterpretation of Quantum Mechanics Recovering Objectivity

The ESR model has been recently proposed in several papers to offer a possible solution of the problems raising from the nonobjectivity of physical properties in quantum mechanics (QM) (mainly the objectification problem of the quantum theory of measurement). This solution is obtained by embodying the mathematical formalism of QM into a broader mathematical framework and reinterpreting quantum probabilities as conditional on detection rather than absolute. We provide a new and more general formulation of the ESR model and discuss time evolution according to it, pointing out in particular that both linear and nonlinear evolution may occur, depending on the physical environment.

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arXiv2014

A survey of the ESR model for an objective reinterpretation of quantum mechanics

Most scholars concerned with the foundations of quantum mechanics (QM) think that contextuality and nonlocality (hence nonobjectivity of physical properties) are unavoidable features of QM which follow from the mathematical apparatus of QM. Moreover these features are usually considered as basic in quantum information processing. Nevertheless they raise still unsolved problems, as the objectification problem in the quantum theory of measurement. The extended semantic realism (ESR) model offers a possible way out from these difficulties by embedding the mathematical formalism of QM into a broader mathematical formalism and reinterpreting quantum probabilities as conditional on detection rather than absolute. The embedding allows to recover the formal apparatus of QM within the ESR model, and the reinterpretation of QM allows to construct a noncontextual hidden variables theory which justifies the assumptions introduced in the ESR model and proves its objectivity. According to the ESR model both linear and nonlinear time evolution occur, depending on the physical environment, as in QM. In addition, the ESR model, though objective, implies modified Bell's inequalities that do not conflict with QM, supplies different mathematical representations of proper and improper mixtures, provides a general framework in which the local interpretations of the GHZ experiment obtained by other authors 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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arXiv2011

Extended Representations of Observables and States for a Noncontextual Reinterpretation of QM

A crucial and problematical feature of quantum mechanics (QM) is nonobjectivity of properties. The ESR model restores objectivity reinterpreting quantum probabilities as conditional on detection and embodying the mathematical formalism of QM into a broader noncontextual (hence local) framework. We propose here an improved presentation of the ESR model containing a more complete mathematical representation of the basic entities of the model. We also extend the model to mixtures showing that the mathematical representations of proper mixtures does not coincide with the mathematical representation of mixtures provided by QM, while the representation of improper mixtures does. This feature of the ESR model entails that some interpretative problems raising in QM when dealing with mixtures are avoided. From an empirical point of view the predictions of the ESR model depend on some parameters which may be such that they are very close to the predictions of QM in most cases. But the nonstandard representation of proper mixtures allows us to propose the scheme of an experiment that could check whether the predictions of QM or the predictions of the ESR model are correct.

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arXiv2011

Recovering Quantum Logic within an Extended Classical Framework

We present a procedure which allows us to recover classical and nonclassical logical structures as \emph{concrete logics} associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory $\mathcal T$ and classical language $\mathcal L$ expressing $\mathcal T$, an observative sublanguage $L$ of $\mathcal L$ with a notion of truth as correspondence, introducing in $L$ a derived and theory-dependent notion of \emph{C-truth} (\emph{true with certainty}), defining a \emph{physical preorder} induced by C-truth, and finally selecting a set of sentences that are \emph{verifiable} (or \emph{testable}) according to $\mathcal T$, on which a \emph{weak complementation} is induced by $\mathcal T$. The triple consisting of the set of verifiable sentences, physical order and weak complementation is then the desired concrete logic. By applying our procedure we recover a classical logic as the concrete logic associated with classical mechanics and standard quantum logic as the concrete logic associated with quantum mechanics. We also show that our alternative view of standard quantum logic, which can be constructed in a purely formal way, can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Our results then show that some nonstandard logics can be obtained as mathematical structures formalizing the properties of different notions of verifiability in different physical theories. More generally, they strongly support the idea that many nonclassical logics can coexist without conflicting with classical logic (\emph{global pluralism}), for they formalize metalinguistic notions that do not coincide with the notion of truth (described by Tarski's truth theory).

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arXiv2010

Generalized Observables, Bell's Inequalities and Mixtures in the ESR Model for QM

The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) into a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide in this review an overall view on the present status of our research on this topic. We attain in a new, shortened way a mathematical representation of the generalized observables introduced by the ESR model and a generalization of the projection postulate of elementary QM. Basing on these results we prove that the Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, a modified BCHSH inequality and quantum predictions hold together in the ESR model because they refer to different parts of the picture of the physical world supplied by the model. Then we show that a new mathematical representation of mixtures must be introduced in the ESR model which does not coincide with the standard representation in QM and avoids some deep problems that arise from the representation of mixtures provided by QM. Finally we get a nontrivial generalization of the Luders postulate, which is justified in a special case by introducing a reasonable physical assumption on the evolution of the compound system made up of the measured system and the measuring apparatus.

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arXiv2009

Embedding Quantum Mechanics Into a Broader Noncontextual Theory: A Conciliatory Result

The extended semantic realism (ESR) model embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here an improved version of this model and show that it predicts that, whenever idealized measurements are performed, a modified Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality holds if one takes into account all individual systems that are prepared, standard quantum predictions hold if one considers only the individual systems that are detected, and a standard BCHSH inequality holds at a microscopic (purely theoretical) level. These results admit an intuitive explanation in terms of an unconventional kind of unfair sampling and constitute a first example of the unified perspective that can be attained by adopting the ESR model.

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arXiv2009

A Hilbert Space Representation of Generalized Observables and Measurement Processes in the ESR Model

The extended semantic realism (ESR) model recently worked out by one of the authors embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here a Hilbert space representation of the generalized observables introduced by the ESR model that satisfy a simple physical condition, propose a generalization of the projection postulate, and suggest a possible mathematical description of the measurement process in terms of evolution of the compound system made up of the measured system and the measuring apparatus.

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arXiv2008

Reinterpreting Quantum Probabilities in a Realistic and Local Framework: The Modified BCHSH Inequalities

Most physicists uphold that the tests of the Bell inequalities (BI) performed up to now confirm the predictions of standard quantum mechanics (SQM) and refute local realism. But some scholars criticize this conviction, defending local realism in various ways. We present here a new viewpoint based on an improved version of the extended semantic realism (ESR) model that has been recently worked out by one of the authors. The ESR model embodies the mathematical formalism of SQM into a more general framework in which not only local realism but also objectivity of physical properties holds, and the probabilities of SQM are reinterpreted as conditional instead of absolute. Hence the ESR model provides some predictions that are formally identical to those of SQM but have a different physical interpretation, and further predictions that differ also formally from those of SQM. In particular, we show here that the BI introduced by Clauser, Horne, Shimony and Holt (BCHSH inequalities) must be replaced by modified BCHSH inequalities. These depend on detection probabilities which may be such that the new inequalities are never violated by the conditional expectation values predicted by the model. The condition that no violation occurs implies the existence of upper bounds on detection probabilities, which makes the ESR model falsifiable. These results admit an intuitive explanation in terms of unfair sampling but basically differ from the seemingly similar results obtained by other approaches in which the efficiency problem is discussed in order to vindicate some kind of local realism.

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arXiv2006

On the Physical Interpretation of Partial Traces: Two Nonstandard Viewpoints

Mixed states are introduced in physics in order to express our ignorance about the actual state of a physical system and are represented in standard quantum mechanics (QM) by density operators. Such operators also appear if one considers a (pure) entangled state of a compound system $Ω$ and performs partial traces on the projection operator representing it. Yet, they do not represent mixed states (or proper mixtures) of the subsystems in this case, but improper mixtures, since the coefficients in the convex sums expressing them never bear the ignorance interpretation. Hence, one cannot attribute states to the subsystems of a compound physical system in QM (subentity problem). We discuss here two alternative proposals that can be worked out within the Brussels and Lecce approaches. We firstly summarize the general framework provided by the former, which suggests that improper mixtures could be considered as new pure states. Then, we show that improper mixtures can be considered as true (yet nonpure) states also according to the latter. The two proposals seem to be compatible notwithstanding their different terminologies.

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arXiv2006

Physical propositions and quantum languages

The word \textit{proposition} is used in physics with different meanings, which must be distinguished to avoid interpretational problems. We construct two languages $\mathcal{L}^{\ast}(x)$ and $\mathcal{L}(x)$ with classical set-theoretical semantics which allow us to illustrate those meanings and to show that the non-Boolean lattice of propositions of quantum logic (QL) can be obtained by selecting a subset of \textit{p-testable} propositions within the Boolean lattice of all propositions associated with sentences of $\mathcal{L}(x)$. Yet, the aforesaid semantics is incompatible with the standard interpretation of quantum mechanics (QM) because of known no-go theorems. But if one accepts our criticism of these theorems and the ensuing SR (semantic realism) interpretation of QM, the incompatibility disappears, and the classical and quantum notions of truth can coexist, since they refer to different metalinguistic concepts (\textit{truth} and \textit{verifiability according to QM}, respectively). Moreover one can construct a quantum language $\mathcal{L}_{TQ}(x)$ whose Lindenbaum-Tarski algebra is isomorphic to QL, the sentences of which state (testable) properties of individual samples of physical systems, while standard QL does not bear this interpretation.

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arXiv2006

On the Notion of Proposition in Classical and Quantum Mechanics

The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set $P^{f}$ of all physical propositions is partially ordered and contains a subset $P^{f}_{T}$ of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, $P^{f}_{T}$ turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language $L_{TQ}(x)$, and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.

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arXiv2005

Quantum Machine and SR Approach: a Unified Model

The Geneva-Brussels approach to quantum mechanics (QM) and the semantic realism (SR) nonstandard interpretation of QM exhibit some common features and some deep conceptual differences. We discuss in this paper two elementary models provided in the two approaches as intuitive supports to general reasonings and as a proof of consistency of general assumptions, and show that Aerts' quantum machine can be embodied into a macroscopic version of the microscopic SR model, overcoming the seeming incompatibility between the two models. This result provides some hints for the construction of a unified perspective in which the two approaches can be properly placed.

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arXiv2005

Truth and Completeness in Quantum Mechanics: A Semantic Viewpoint

The Einstein, Podolski and Rosen (EPR) argument aiming to prove the incompleteness of quantum mechanics (QM) was opposed by most EPR's contemporary physicists and is not accepted within the standard interpretation of QM, which maintains that QM is a complete theory. An analysis of the semantic implications of the opponent positions shows that they imply different notions of truth. The introduction of a nonclassical notion of truth within the standard interpretation is usually justified by referring to known theorems that should prove that QM is a contextual and nonlocal theory. However, these theorems are based on a doubtful implicit epistemological assumption. If one renounces it, one can provide an alternative interpretation of QM that it realistic in a semantic sense. Within this interpretation the EPR viewpoint is recovered and QM is considered a (semantically) incomplete, noncontextual and local theory. Furthermore, the new interpretation provides several suggestions for constructing a more general theory embedding QM and for connecting QM with classical physics and relativity.

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arXiv2005

A Pragmatic Interpretation of Quantum Logic

Scholars have wondered for a long time whether quantum mechanics (QM) subtends a quantum concept of truth which originates quantum logic (QL) and is radically different from the classical (Tarskian) concept of truth. We show in this paper that QL can be interpreted as a pragmatic language of pragmatically decidable assertive formulas, which formalize statements about physical systems that are empirically justified or unjustified in the framework of QM. According to this interpretation, QL formalizes properties of the metalinguistic concept of empirical justification within QM rather than properties of a quantum concept of truth. This conclusion agrees with a general integrationist perspective, according to which nonstandard logics can be interpreted as theories of metalinguistic concepts different from truth, avoiding competition with classical notions and preserving the globality of logic. By the way, some elucidations of the standard concept of quantum truth are also obtained.

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