For the probabilistic description of all the joint von Neumann measurements on a D-dimensional quantum system, we present the specific example of a context-invariant quasi hidden variable model, proved in Loubenets to exist for each Hilbert space. In this model, a quantum observable X is represented by a variety of random variables satisfying the functional condition required in quantum foundations but, in contrast to a contextual model, each of these random variables equivalently models X under all (...) joint von Neumann measurements, regardless of their contexts. This, in particular, implies the specific local qHV model for an N-qudit state and allows us to derive the new exact upper bound on the maximal violation of 2\2-setting Bell-type inequalities of any type under N-partite joint von Neumann measurements on an N-qudit state. For d = 2, this new upper bound coincides with the maximal violation by an N-qubit state of the Mermin–Klyshko inequality. Based on our results, we discuss the conceptual and mathematical advantages of context-invariant and local qHV modelling. (shrink)

In the quantum logic approach, Bell inequalities in the sense of Pitowski are related with quasi hidden variables in the sense of Deliyannis. Some properties of hidden variables on effect algebras are discussed.

In the quantum logic approach, Bell inequalities in the sense of Pitowski are related with quasi hidden variables in the sense of Deliyannis. Some properties of hidden variables on effect algebras are discussed.

We recently showed that it is possible to deal withcollections of indistinguishable elementary particles (in thecontext of quantum mechanics) in a set-theoretical framework, byusing hidden variables. We propose in the presentpaper another axiomatics for collections of indiscernibleswithout hidden variables, where hidden predicates are implicitlyassumed. We also discuss the possibility of a quasi-settheoretical picture for quantum theory. Quasi-set theory, basedon Zermelo-Fraenkel set theory, was developed for dealing withcollections of indistinguishable, but, not identical objects.

We recently showed that it is possible to deal withcollections of indistinguishable elementary particles (in thecontext of quantum mechanics) in a set-theoretical framework, byusing hidden variables. We propose in the presentpaper another axiomatics for collections of indiscernibleswithout hidden variables, where hidden predicates are implicitlyassumed. We also discuss the possibility of a quasi-settheoretical picture for quantum theory. Quasi-set theory, basedon Zermelo-Fraenkel set theory, was developed for dealing withcollections of indistinguishable, but, not identical objects.

Two EPR arguments are reviewed, for their own sake, and for the purpose of clarifying the status of "stochastic" hidden variables. The first is a streamlined version of the EPR argument for the incompleteness of quantum mechanics. The role of an anti-instrumentalist ("realist") interpretation of certain probability statements is emphasized. The second traces out one horn of a central foundational dilemma, the collapse dilemma; complex modal reasoning, similar to the original EPR, is used to derive determinateness (of all (...) spin components of two spin- 1 / 2 particles in the singlet state) from just (a form of) weak locality, result definiteness, and an assumption on propensities based on conservation. Theories meeting these conditions are therefore constrained by the Bell inequalities. Neither controversial assumptions of "strong locality" ("factorability") nor of determinism are employed in the derivation. The categories of "stochastic hidden variables" are then analyzed; one can focus on "quasi-definite" theories, without loss of generality. A means of excluding these is proposed, based on a demand that certain ideal cases be accurately treated. Theorems from quantum measurement theory, sometimes cited as showing that such cases are not physically possible, are found inapplicable. (shrink)

We use a simple relational framework to develop the key notions and results on hidden variables and non-locality. The extensive literature on these topics in the foundations of quantum mechanics is couched in terms of probabilistic models, and properties such as locality and no-signalling are formulated probabilistically. We show that to a remarkable extent, the main structure of the theory, through the major No-Go theorems and beyond, survives intact under the replacement of probability distributions by mere relations.

Although skeptical of the prohibitive power of no-hidden-variables theorems, John Bell was himself responsible for the two most important ones. I describe some recent versions of the lesser known of the two (familiar to experts as the "Kochen-Specker theorem") which have transparently simple proofs. One of the new versions can be converted without additional analysis into a powerful form of the very much better known "Bell's Theorem," thereby clarifying the conceptual link between these two results of Bell.

It is a widespread belief that the Kochen-Specker theorem imposes a contextuality constraint on the ontology of beables in quantum hidden variables theories. On the other hand, after Bell’s influential critique, the importance of von Neumann’s wrongly called ‘impossibility proof’ has been severely questioned. However, Max Jammer, Jeffrey Bub and Dennis Dieks have proposed insightful reassessments of von Neumann’s theorem: what it really shows is that hidden variables theories cannot represent their beables by means of Hermitian (...) operators in Hilbert space. Hereby I show that i) the very same constraint can be derived from Gleason’s theorem, and that ii) if we consider the import of von Neumann’s and Gleason’s theorems, the relevance of the Kochen-Specker theorem for hidden variables theories gets substantially weakened: it does not force them to be contextual in any interesting sense of the term. (shrink)

Noncontextual hidden variables theories, assigning simultaneous values to all quantum mechanical observables, are inconsistent by theorems of Gleason and others. These theorems do not exclude contextual hidden variables theories, in which a complete state assigns values to physical quantities only relative to contexts. However, any contextual theory obeying a certain factorisability conditions implies one of Bell's Inequalities, thereby precluding complete agreement with quantum mechanical predictions. The present paper distinguishes two kinds of contextual theories, ‘algebraic’ and ‘environmental’, (...) and investigates when factorisability is reasonable. Some statements by Fine about the philosophical significance of Bell's Inequalities are then assessed. (shrink)

Entangled states whose Wigner functions are non-negative may be viewed as being accounted for by local hidden variables (LHV). Recently, there were studies of Bell’s inequality violation (BIQV) for such states in conjunction with the well known theorem of Bell that precludes BIQV for theories that have LHV underpinning. We extend these studies to teleportation which is also based on entanglement. We investigate if, to what extent, and under what conditions may teleportation be accounted for via LHV theory. (...) Our study allows us to expose the role of various quantum requirements. These are, e.g., the uncertainty relation among non-commuting operators, and the no-cloning theorem which forces the complete elimination of the teleported state at its initial port. (shrink)

This article takes up a suggestion that the reason we cannot find certain hidden variable theories for quantum mechanics, as in Bell’s theorem, is that we require them to assign joint probability distributions on incompatible observables. These joint distributions are problematic because they are empirically meaningless on one standard interpretation of quantum mechanics. Some have proposed getting around this problem by using generalized probability spaces. I present a theorem to show a sense in which generalized probability spaces can’t serve (...) as hidden variable theories for quantum mechanics, so the proposal for getting around Bell’s theorem fails. 1 Introduction2 Bell’s Theorem and Classical Probability Spaces2.1 Bell’s derivation of the Bell inequalities2.2 Pitowsky’s derivation of the Bell inequalities3 Incompatible Observables4 Generalized Probability Spaces5 A ‘No-Go’ Theorem6 Conclusions. (shrink)

To relax the apparent tension between nonlocal hidden variables and relativity, we propose that the observable proper time is not the same quantity as the usual proper-time parameter appearing in local relativistic equations. Instead, the two proper times are related by a nonlocal rescaling parameter proportional to |ψ|2, so that they coincide in the classical limit. In this way particle trajectories may obey local relativistic equations of motion in a manner consistent with the appearance of nonlocal quantum correlations. (...) To illustrate the main idea, we first present two simple toy models of local particle trajectories with nonlocal time, which reproduce some nonlocal quantum phenomena. After that, we present a realistic theory with a capacity to reproduce all predictions of quantum theory. (shrink)

It is argued that among possible nonlocal hidden-variable theories a particular class (called here “crypto-nonlocal” or CN) is relatively plausible on physical grounds. CN theories have the property that (for example) the two photons emitted in an atomic cascade process are indistinguishable in their individual statistical properties from photons emitted singly, and that in the latter case the effects of nonlocality are unobservable. It is demonstrated that all CN theories are constrained by inequalities which are violated by the quantum-mechanical (...) predictions; these inequalities bear no simple relation to Bell's inequalities, and an explicit example is constructed of a CN theory which violates the latter. It is also shown that while existing experiments cannot rule out general CN theories, they do rule out (subject to a few caveats such as the usual ones concerning the well-known “loopholes”) the subclass in which the photon polarizations are linear. (shrink)

I show that for any quantum dynamics and any choice of observables as hidden variables an adequate hidden variable theory always exists. I argue that hidden variable theories have no more problems in reconciling non-locality with relativity than no-hidden-variable theories.

A realist separable hidden variables theory in conformity with Einstein's principle of causality is developed in this paper to explain the Einstein-Podolsky-Rosen paradox, and the experimental results (including those in Aspect's four polarizers experiment) obtained so far with a view to test the non-separability of quantum mechanics.

A hidden-variables model is presented which, by using a hypothesis of “directionalization” of the photons at the deflectors, is able to reproduce all the quantum mechanical predictions for the Orsay realization of the Einstein-Podolsky-Rosen-Bohm experiment, even for ideal polarizers, detectors, time-coincidence windows, and “event-ready” setups. The model also holds for the no-enhancement assumption. The requirements for an experiment aimed to discriminate between quantum mechanics and the new model are discussed. Under some plausible assumptions, such experiment is achievable with (...) the current technology. It would be essentially a remake of the Orsay one with improved deflectors and a time-resolved measurement of the photon flows toward each detector. (shrink)

Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as ``physically real" and classical mechanics, like quantum physics, is indeterministic.

Bell's problem of the possibility of a local hidden variable theory of quantum phenomena is considered in the context of the general problem of representing the statistical states of a quantum mechanical system by measures on a classical probability space, and Bell's result is presented as a generalization of Maczynski's theorem for maximal magnitudes. The proof of this generalization is shown to depend on the impossibility of recovering the quantum statistics for sequential probabilities in a classical representation without introducing (...) a randomization process for the hidden variables. Hidden variable theories that exclude such a randomization process are termed “strict,” and it is shown that the class of local hidden variable theories is included in the class of strict theories. A counterargument by Freedman and Wigner is evaluated with reference to Clauser's extension of a hidden variable model proposed by Bell. (shrink)

The modal interpretation of quantum mechanics has two variants: the Copenhagen variant (CV) and the anti-Copenhagen variant (ACV). Healey uses the Bell-Wigner locality condition to criticize the latter, which I do not advocate. 2 The conclusions of Healey's admirably written article are therefore welcome to me. But if I had wished to advocate the ACV, I do not think that his arguments would have dissuaded me. Specifically, as I shall explain, we should distinguish between Physical Locality and Metaphysical Locality. The (...) first principle is unexceptionable, but Healey needs the second for his reductio. At the end I shall briefly indicate why I reject the ACV and advocate instead the Copenhagen variant. (shrink)

Interpretations of the quantum mechanical formalism true to the spirit of scientific realism satisfy not only principles of scientific realism but also principles of causality that guide realist constructions. Formally, such interpretations are hidden variables theories and are commonly believed to be ruled out by the most recent no-hidden-variables argument expressed by Bell's theorem. This dissertation investigates the possibility of constructing indeterministic hidden variables theories in light of Bell's result. A pair of arguments in (...) the literature lead to the conclusion that there is no generality to be gained in moving to the stochastic case. The author uses limitations on measurements discovered by Wigner as the basis for a response to both. Proofs of the existence of the limitations are analyzed and it is shown that, contrary to claims in the literature, we cannot always assume that measurements subject to the limitations are perfectly revealing and, therefore, that the assumption of strict anti-correlations, an assumption crucial to one of those arguments, is unjustified in the Bell case. The author suggests how Bell's 1971 inequality must be modified to incorporate these limitations . This result is used against the second argument, due to Fine, which relies on Bell's 1971 inequality. A thought experiment due to Jarrett is reviewed and it is argued that the independence assumptions incorporated into hidden variables theories are not without additional and controversial assumptions about causality related to constraints imposed by the special theory of relativity. The author concludes that Bell's theorem tells us nothing about determinism or special relativity but does tell us that there can be no interpretation of quantum mechanics that does not violate either one of three principles of scientific realism or one of three traditional principles of causality. (shrink)

The usual definition of (non-contextual) hidden variables is found to be too restrictive, in the sense that, according to it, even some classical systems do not admit hidden variables. A more general concept is introduced and the term “approximate hidden variables” is used for it. This new concept avoids the aforementioned problems, since all classical systems admit approximate hidden variables. Standard quantum systems do not admit approximate hidden variables, unless the (...) corresponding Hilbert space is 2-dimensional. However, an appropriate non-standard quantum system, which arises by focussing on momentum and position and neglecting the remaining observables, admits approximate hidden variables. Systems associated with JBW-algebras (resp. von Neumann algebras) and satisfying some mild conditions admit approximate hidden variables iff they are classical, that is, iff the corresponding JBW-algebra (resp. von Neumann algebra) is associative (resp. commutative). (shrink)

Recently it was shown that certain fluid-mechanical ‘pilot-wave’ systems can strikingly mimic a range of quantum properties, including single particle diffraction and interference, quantization of angular momentum etc. How far does this analogy go? The ultimate test of quantumness of such systems is a Bell-test. Here the premises of the Bell inequality are re-investigated for particles accompanied by a pilot-wave, or more generally by a resonant ‘background’ field. We find that two of these premises, namely outcome independence and measurement independence, (...) may not be generally valid when such a background is present. Under this assumption the Bell inequality is possibly violated. A class of hydrodynamic Bell experiments is proposed that could test this claim. Such a Bell test on fluid systems could provide a wealth of new insights on the different loopholes for Bell’s theorem. Finally, it is shown that certain properties of background-based theories can be illustrated in Ising spin-lattices. (shrink)

In the present paper we give a precise definition of a hidden-variable theory for quantum mechanics, whereby we adopt the weakest possible definition of a hidden-variable theory, which is compatible with the assumption that the bounded observables of a quantum mechanical system are represented by the elements of the real part Ar of a W*-algebra A (of the most general type) and the states are represented by the “normal states” (in the mathematical sense) of A. We then go (...) on to show that an example put forward by Bell in 1966 satisfies our definition (Sec. 2). Finally we make use of Bell's famous theorem to show that for a sufficiently non-commutative W*-algebra A no hidden-variable theory in our sense exists (Theorem 3.3 and its corollaries). (shrink)

We propose to experimentally test non-deterministic time evolution in quantum mechanics by consecutive measurements of non-commuting observables on the same prepared state. While in the standard theory the measurement outcomes are uncorrelated, in a super-deterministic hidden variables theory the measurements would be correlated. We estimate that for macroscopic experiments the correlation time is too short to have been noticed yet, but that it may be possible with a suitably designed microscopic experiment to reach a parameter range where one (...) would expect a super-deterministic modification of quantum mechanics to become relevant. (shrink)

In this paper I discuss two objections raised against von Fintel’s (1994) and Stanley and Szabó’s (2000a) hidden variable approach to quantifier domain restriction (QDR). One of them concerns utterances of sentences involving quantifiers for which no contextual domain restriction is needed, and the other concerns multiple quantified contexts. I look at various ways in which the approaches could be amended to avoid these problems, and I argue that they fail. I conclude that we need a more flexible account (...) of QDR, one that allows for the hidden variables in the LF responsible for QDR to vary in number. Recanati’s (2002; 2004) approach to QDR, which makes use of the apparatus of “variadic functions”, is flexible enough to account successfully for the two phenomena discussed. I end with a few comments on what I take to be the most promising way to construe variadic functions. (shrink)

o (2000), 243). In particular, the idea is that binding interactions between the relevant expressions and natural lan- guage quantifiers are best explained by the hypothesis that those expressions harbor hidden but bindable variables. Recently, however, Herman Cappelen and Ernie Lepore have rejected such binding arguments for the presence of hid- den variables on the grounds that they overgeneralize — that, if sound, such arguments would establish the presence of hidden variables in all sorts of (...) ex- pressions where it is implausible that they exist (Cappelen and Lepore (2005), Cappelen and Lepore (2002)).1 In what follows we respond to Cappelen’s and Lepore’s attempted reductio by bringing out crucial disanalogies between cases where the binding argument is successful and cases where it is not. But we have a deeper purpose than merely to respond to Cappelen and Lepore: we think the attempted reductio goes wrong by not taking sufficiently seriously the nature of the binding relation that holds between quantifiers and arguments/variables, and that our criticism will serve to highlight the nature and importance of this relation. (shrink)

A local hidden variable theory of quantum mechanics is formulated by adapting Gell-Mann and Hartle’s many-histories formulation. The resulting theory solves the measurement problem by exploiting the independence loophole in Bell’s theorem; it violates the independence of hidden variable values and measuring device settings. Although the theory is problematic in some respects, it provides a concrete example via which the tenability of this approach can be better evaluated.

One implication of Bell’s theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., (...) a finite collection of probability zero events whose disjunction exhausts the space of experimental possibilities. (shrink)

Hidden variables are usually presented as potential completions of the quantum description. We describe an alternative role for these entities, as aids to calculation in quantum mechanics. This is illustrated by the computation of the time-dependence of a massless relativistic spinor field obeying Weyl’s equation from a single-valued continuum of deterministic trajectories (the “hidden variables”). This is achieved by generalizing the exact method of state construction proposed previously for spin 0 systems to a general Riemannian manifold (...) from which the spinor construction is extracted as a special case. The trajectories form a non-covariant structure and the Lorentz covariance of the spinor field theory emerges as a kind of collective effect. The method makes manifest the spin 1/2 analogue of the quantum potential that is tacit in Weyl’s equation. This implies a novel definition of the “classical limit” of Weyl’s equation. (shrink)

J. Solomon [Journal de Physique 4, 34 (1933)] produced an argument of great generality claiming to demonstrate the impossibility of hidden variables in quantum theory, an argument which M. Jammer [The Philosophy of Quantum Mechanics(Wiley, New York, 1974)] said raised a number of questions. For the first time, this argument is discussed, a simple hidden variable model violating the argument is analysed in detail, and the error in the proof is located.

In the contemporary discussion of hidden variable interpretations of quantum mechanics, much attention has been paid to the “no hidden variable” proof contained in an important paper of Kochen and Specker. It is a little noticed fact that Bell published a proof of the same result the preceding year, in his well-known 1966 article, where it is modestly described as a corollary to Gleason's theorem. We want to bring out the great simplicity of Bell's formulation of this result (...) and to show how it can be extended in certain respects. (shrink)

I will argue that the Peres-Mermin square does not necessarily rule out a value-definite (deterministic) noncontextual hidden variable model if the operators are not given a physical interpretation satisfying the following two requirements: (i) each operator is uniquely realized by a single physical measurement; (ii) commuting operators are realized by simultaneous measurements. To underpin this claim, I will construct three hidden variable models for three different physical realizations of the Peres-Mermin square: one violating (i), another violating (ii), and (...) a third one violating both (i) and (ii). (shrink)

This paper is motivated by the desire to formulate a relativistically covariant hidden-variable particle trajectory interpretation of the quantum theory of the vector field that is formulated in such a way as to allow the inclusion of gravity. We present a methodology for calculating the flows of rest energy and a conserved density for the massive vector field using the time-like eigenvectors and eigenvalues of the stress-energy-momentum tensor. Such flows may be used to define particle trajectories which follow the (...) flow. This work extends our previous work which used a similar procedure for the scalar field. The massive, spin-one, complex vector field is discussed in detail and the flows of energy-momentum are illustrated in a simple example of standing waves in a plane. (shrink)

A hidden-variable model for quantum–mechanical spin, as represented by the Pauli spin operators, is proposed for systems illustrating the well-known no-hidden-variables arguments by Peres (Phys Lett A 151:107–108, 1990) and Mermin (Phys Rev Lett 65:3373–3376, 1990) and by Greenberger et al. (Bell’s theorem, quantum theory, and conceptions of the universe, Kluwer, Dordrecht, 1989). Both arguments rely on an assumption of non-contextuality; the latter argument can also be phrased as a non-locality argument, using a locality assumption. The model (...) suggested here is compatible with both assumptions. This is possible because the scalar values of spin observables are replaced by vectors that are components of orientations. (shrink)

We construct, for any finite dimension n, a new hidden measurement model for quantum mechanics based on representing quantum transition probabilities by the volume of regions in projective Hilbert space. For n=2 our model is equivalent to the Aerts sphere model and serves as a generalization of it for dimensions n .≥ 3 We also show how to construct a hidden variables scheme based on hidden measurements and we discuss how joint distributions arise in our (...) class='Hi'>hidden variables scheme and their relationship with the results of Fine [J. Math. Phys. 23 1306 (1982)]. (shrink)

Bell’s theorem requires the assumption that hidden variables are independent of future measurement settings. This independence assumption rests on surprisingly shaky ground. In particular, it is puzzlingly time-asymmetric. The paper begins with a summary of the case for considering hidden variable models which, in abandoning this independence assumption, allow a degree of ‘backward causation’. The remainder of the paper clarifies the physical significance of such models, in relation to the issue as to whether quantum mechanics provides a (...) complete description of physical reality. (shrink)

Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann’s ‘no hidden variables’ proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense ‘like Bohm’s theory.’ His seminal result provides a positive answer to the question. I argue that Bell’s analysis misconstrues von Neumann’s argument. What (...) von Neumann proved was the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (deterministic) states in which the quantum observables are represented as the ‘beables’ of the theory, to use Bell’s term. That is, the quantum probabilities could not reflect the distribution of pre-measurement values of beables, but would have to be derived in some other way, e.g., as in Bohm’s theory, where the probabilities are an artefact of a dynamical process that is not in fact a measurement of any beable of the system. (shrink)

We prove the existence of hidden variables, or, what we call generalized common causes, for finite sequences of pairwise correlated random variables that do not have a joint probability distribution. The hidden variables constructed have upper probability distributions that are nonmonotonic. The theorem applies directly to quantum mechanical correlations that do not satisfy the Bell inequalities.