Narrow Results

The Bell inequality is thought to be a common constraint shared by all models of local hidden variables that aim to describe the entangled states of two qubits. Since the inequality is violated by the quantum mechanical description of these states, it purportedly allows distinguishing in an experimentally testable way the predictions of quantum mechanics from those of models of local hidden variables and, ultimately, ruling the latter out. In this paper, we show, however, that the models of local hidden variables constrained by the Bell inequality all share a subtle, though crucial, feature that is not required by fundamental physical principles and, hence, it might not be fulfilled in the actual experimental setup that tests the inequality. Indeed, the disputed feature neither can be properly implemented within the standard framework of quantum mechanics and it is even at odds with the fundamental principle of relativity. Namely, the proof of the inequality requires the existence of a preferred absolute frame of reference (supposedly provided by the lab) with respect to which the hidden properties of the entangled particles and the orientations of each one of the measurement devices that test them can be independently defined through a long sequence of realizations of the experiment. We notice, however, that while the relative orientation between the two measurement devices is a properly defined physical magnitude in every single realization of the experiment, their global rigid orientation with respect to a lab frame is a spurious gauge degree of freedom. Following this observation, we were able to explicitly build a model of local hidden variables that does not share the disputed feature and, hence, it is able to reproduce the predictions of quantum mechanics for the entangled states of two qubits.

We carryout a comparative study of spin distributions defined over the sphere for bipartite quantum spin assemblies. We analyze Einstein–Podolsky–Rosen–Bohm (EPRB) spin correlations in a spin-s singlet state using these distributions. We observe that in the classical limit of s → ∞, EPRB spin distributions turn out to be delta functions, thus reflecting the perfect anticorrelation property of two-spin vectors associated with a spin-s singlet state.

A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization. Extending the ''probability via expectation'' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotation of unlimited repeatability; hence it can be applied to unique systems such as the universe. Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

A deeper understanding of the origin of quantum correlations is expected to allow a better comprehension of the physical principles underlying quantum mechanics. In this work, we reconsider the possibility of devising "crypto-nonlocal theories", using a terminology firstly introduced by Leggett. We generalize and simplify the investigations on this subject which can be found in the literature. At their deeper level, such theories allow nonlocal correlations which can overcome the quantum limit.

It is shown that the probabilities for the spin singlet can be reproduced through classical resources, with no communication between the distant parties, by using merely shared (pseudo-)randomness. If the parties are conscious beings aware of both the hidden-variables and the random mechanism, then one has a conspiracy. If the parties are aware of only the random variables, they may be induced to believe that they are able to send instantaneous information to one another. It is also possible to reproduce the correlations at the price of reducing the detection efficiency. It is further demonstrated that the same probability decomposition could be realized through action-at-a-distance, provided it existed.

Fractal interpolation functions provide a new light to the natural deterministic approximation and modeling of complex phenomena. The present paper proposes construction of natural cubic spline coalescence hidden variable fractal interpolation surfaces (CHFISs) over a rectangular grid through the tensor product of univariate bases of cardinal natural cubic spline coalescence hidden variable fractal interpolation functions (CHFIFs). Natural cubic CHFISs are self-affine or non-self-affine in nature depending on the IFS parameters of univariate natural cubic spline CHFIFs. An upper bound of the error between the natural cubic spline blended coalescence fractal interpolant and the original function is deduced. Convergence of the natural cubic CHFIS to the original function , and their derivatives are deduced. The effects free variables, constrained free variables and hidden variables are discussed on the natural cubic spline CHFIS with suitably chosen examples.

Certain predictions of quantum theory are not compatible with the notion of local-realism. This was the content of Bell’s famous theorem of the year 1964. Bell proved this with the help of an inequality, famously known as Bell’s inequality. The alternative proofs of Bell’s theorem without using Bell’s inequality are known as “nonlocality without inequality (NLWI)” proofs. We review one such proof namely the Hardy’s proof which due to its simplicity and generality has been considered the best version of Bell’s theorem.

This paper aims at reproducing quantum mechanical (QM) spin and spin entanglement results using a realist, stochastic, and local approach, without the standard QM mathematical formulation. The concrete model proposed includes the description of Stern–Gerlach apparatuses and of Bell test experiments. Single particle trajectories are explicitly evaluated as a function of a few stochastic variables that they assumedly carry on. QM predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. Notably, it is shown that the proposed model, despite being both local and realist, is able to violate the Bell–CHSH inequalities by exploiting the coincidence loophole and thus intrinsically renouncing to one of the Bell’s assumptions.

We formally prove the existence of an enduring incongruence pervading a widespread interpretation of the Bell inequality and explain how to rationally avoid it with a natural assumption justified by explicit reference to a mathematical property of Bell’s probabilistic model. Although the amendment does not alter the relevance of the theorem regarding local realism, it brings back Bell theorem from the realm of philosophical discussions about counterfactual conditionals to the concrete experimental arena.

A sharper formulation is presented for an interpretation of quantum mechanics advocated by the author. We claim that only those quantum theories should be considered for which an ontological basis can be constructed. In terms of this basis, the entire theory can be considered as being deterministic. An example is illustrated: massless, noninteracting fermions are ontological. Subsequently, as an essential element of the deterministic interpretation, we put forward conservation laws concerning the ontological nature of a variable, and the uncertainties concerning the realization of states. Quantum mechanics can then be treated as a device that combines statistics with mechanical, deterministic laws, such that uncertainties are passed on from initial states to final states.

We compare quantum mechanics as a theory involving probabilities to the framework of Kolmogorov's probability theory with emphasis on the connections of these theories to actual experiments. We find crucial differences in the way incompatible experiments are defined and treated in these two approaches and show that these differences are the origin for difficulties and apparent contradictions that are encountered when considering so called no-go proofs particularly that of John Bell. For example, Bell was convinced that in a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, violations of Einstein-locality must occur. Based on our comparative study, we show that rather the opposite is true and that a precise space-time treatment based on relativity uncovers contradictions in the assumptions for Bell's no-go proof and resolves the difficulties.

It is argued that local realism is a fundamental principle, which might be rejected only if experiments clearly show that it is untenable. Forty years after Bell's work no experiment has provided a valid, loophole-free, violation of local realism which, in my opinion, is thus reinforced. I study a simple, but wide, family of local realistic models and derive new inequalities almost insensitive to the detection loophole. I argue that quantum mechanics, with some change in the theory of measurement, might be compatible with local realism.

The development of quantum physics over the last century has stimulated a rapprochement between contemporary science and the Jewish mystical tradition or Kabbalah. Here, evidence is adduced from the classical mystical texts (e.g. the Zohar) and the works of leading Kabbalists of the 16^th-18^th centuries supporting an hypothesis that pivotal concepts elaborated by the influential physicist, David Bohm (1917-1992) are intrinsic to the Jewish mystical worldview. Specifically, we will demonstrate striking parallelisms between the “implicate-explicate orders” and “holographic universe” of Bohmian mechanics and the Kabbalistic principles of Hitlabshut (ensheathment), Hitkallelut (interinclusion) and Hitkashrut (interpenetration). Possible implications of these homologies for epistemology, religion and modern physics are discussed.

Statistical inference from large-scale data can benefit from sources of heterogeneity. We discuss recent progress of the mathematical formalization and theory for exploiting heterogeneity towards predictive stability and causal inference in high-dimensional models. The topic is directly motivated by a broad range of applications and we will show an illustration from molecular biology with gene knock out experiments.

Why, despite all efforts to the contrary, have attempts at unification based on the supposedly more fundamental quantum theory failed miserably? The truth is that the essential idea or concept of the quantum itself has never been fully understood. What is the quantum, or rather, what is its ultimate nature? Science may be able to work adequately with the quantum; in a sense science is quite articulate in the language of the quantum, i.e., its mathematical interpretation of the quantum mechanics, but science has no idea of the true physical nature of the quantum. Scientists and philosophers have wasted energy and efforts on irrelevant issues such as the debate over determinism and indeterminism instead of carefully analyzing the physical source of the quantum. Only with a true understanding of the physical nature of the quantum will the unification of the quantum and relativity ever become a reality.

To date, quantum mechanics has proven to be our most successful theoretical model. However, it is still surrounded by a “mysterious halo”, which can be summarized in a simple but challenging question: Why quantum phenomena are not understood under the same logic as classical ones? Although this is an open question (probably without an answer), from a pragmatist's point of view there is still room enough to further explore the quantum world, marveling ourselves with new physical insights. We just need to look back in the historical evolution of the quantum theory and thoroughly reconsider three key issues: (1) how this theory has developed since its early stages at a conceptual level, (2) what kind of experiments can be performed at present in a laboratory, and (3) what nonstandard conceptual models are available to extract some extra information. This contribution is aimed at providing some answers (and, perhaps, also raising some issues) to these questions through one of such models, namely Bohmian mechanics, a hydrodynamic formulation of the quantum theory, which is currently trying to open new pathways of understanding. Specifically, the Chapter constitutes a brief and personal overview on the historic and contextual evolution of this quantum formulation, its physical meaning and interest (leaving aside metaphysical issues), and how it may help to overcome some preconceived paradoxical aspects of the quantum theory.

Too many physicists believe the ‘phallacy’ that the quantum is more fundamental than relativity without any valid supporting evidence, so the earliest attempts to unify physics based on the continuity of relativity have been all but abandoned. This belief is probably due to the wealth of pro-quantum propaganda and general ‘phallacies in fysics’ that were spread during the second quarter of the twentieth century, although serious ‘phallacies’ exist throughout physics on both sides of the debate. Yet both approaches are basically flawed because both relativity and the quantum theory are incomplete and grossly misunderstood as they now stand. Had either side of the quantum versus relativity controversy sought common ground between the two worldviews, total unification would have been accomplished long ago. The point is, literally, that the discrete quantum, continuous relativity, basic physical geometry, theoretical mathematics and classical physics all share one common characteristic that has never been fully explored or explained – a paradoxical duality between a dimensionless point (discrete) and an extended length (continuity) in any dimension – and if the problem of unification is approached from an understanding of how this paradox relates to each paradigm, all of physics and indeed all of science could be unified under a single new theoretical paradigm.