The Quantum Completeness Problem

Foundations of Physics, 2000

The old Bohr-Einstein debate about the completeness of quantum mechanics (QM) was held on an ontological ground. The completeness problem becomes more tractable, however, if it is preliminarily discussed from a semantic viewpoint. Indeed every physical theory adopts, explicitly or not, a truth theory for its observative language, in terms of which the notions of semantic objectivity and semantic completeness of the physical theory can be introduced and inquired. In particular, standard QM adopts a verificationist theory of truth that implies its semantic nonobjectivity; moreover, we show in this paper that standard QM is semantically complete, which matches Bohr's thesis. On the other hand, one of the authors has provided a Semantic Realism (or SR) interpretation of QM that adopts a Tarskian theory of truth as correspondence for the observative language of QM (which was previously mantained to be impossible); according to this interpretation QM is semantically objective, yet incomplete, which matches EPR's thesis. Thus, standard QM and the SR interpretation of QM come to opposite conclusions. These can be reconciled within an integrationist perspective that interpretes non-Tarskian theories of truth as theories of metalinguistic concepts different from truth.

2006

We present a reformulation of quantum mechanics in terms of probability measures and functions on a general classical sample space and in particular in terms of probability densities and functions on phase space. The basis of our proceeding is the existence of so-called statistically complete observables and the duality between the state spaces and the spaces of the observables, the

2020

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or "observer", or "user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity.

Physical Review, 1935

In loving memory of Asher Peres, we discuss a most important and influential paper written in 1935 by his thesis supervisor and mentor Nathan Rosen, together with Albert Einstein and Boris Podolsky. In that paper, the trio known as EPR questioned the completeness of quantum mechanics. The authors argued that the then-new theory should not be considered final because they believed it incapable of describing physical reality. The epic battle between Einstein and Bohr intensified following the latter's response later the same year. Three decades elapsed before John S. Bell gave a devastating proof that the EPR argument was fatally flawed. The modest purpose of our paper is to give a critical analysis of the original EPR paper and point out its logical shortcomings in a way that could have been done 70 years ago, with no need to wait for Bell's theorem. We also present an overview of Bohr's response in the interest of showing how it failed to address the gist of the EPR argument.

In this paper we define and study the new model for quantum mechanics (QM) – the hybrid epistemic model. The new feature of this model consists in the fact that it does not contain the formal definition of the measurement process but the measurement process is one of possible processes inside of QM. The hybrid-epistemic model of QM is based on two concepts: the quantum state of an ensemble and the properties of individual systems. We show the local nature of EPR correlations in the hybrid-epistemic model of QM in all details. We define precisely the epistemic and the ontic models of QM for the goal to prove that these three models give the same empirical predictions, i.e. that they are empirically equivalent. We show that the no-go theorems (Bell’s theorem, the Leggett-Garg’s theorem and others theorems) cannot be proved in the hybrid-epistemic model of QM. This is one of the main results of this paper. We shall consider the possible inconsistences of the ontic model of QM. We introduce the property-epistemic model of QM which is the special simple case of the hybrid-epistemic model.

Physica Scripta, 2009

Quantum theory (QT) provides statistical predictions for various physical phenomena. To verify these predictions a considerable amount of data has been accumulated in the "measurements" performed on the ensembles of identically prepared physical systems or in the repeated "measurements" on some trapped "individual physical systems". The outcomes of these measurements are in general some numerical time series registered by some macroscopic instruments. The various empirical probability distributions extracted from these time series were shown to be consistent with the probabilistic predictions of QT. More than 70 years ago the claim was made that QT provided the most complete description of "individual" physical systems and outcomes of the measurements performed on "individual" physical systems were obtained in intrinsically random way. Spin polarization correlation experiments (SPCE), performed to test the validity of Bell inequalities, clearly demonstrated the existence of strong long range correlations and confirmed that the beams hitting the far away detectors preserve somehow the memory of their common source which would be destroyed if the individual counts of far away detectors were purely random. Since the probabilities describe the random experiments and are not the attributes of the" individual" physical systems the claim that QT provides a complete description of "individual" physical systems seems not only unjustified but misleading and counter-productive. In this paper we point out that we even don't know whether QT is predictable complete because it has not been tested carefully enough. Namely it was not proven that the time series of existing experimental data did not contain some stochastic fine structures, which could have been averaged out by describing them in terms of the empirical probability distributions. In this paper we advocate various statistical tests which could be used to search for such fine structures in the data and to answer the title question of this paper. In our opinion a proper understanding of the statistical character of QT and of its limitations is crucial in the domains such as: quantum optics and quantum information.

2020

Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbert’s finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzen’s approaches for completeness. This paper investigates both conditions and philosophical background necessary for that modification. The main conclusion is that the concept of infinity as underlying contemporary mathematics cannot be reduced to a single Peano arithmetic, but to at least two ones independent of each other. Intuitionism, quantum mechanics, and Gentzen’s approaches to completeness an even Hilbert’s finitism can be unified from that viewpoint. Math...

In this comment I had shown that there are no EPR paradox and no entangled states of separated particles

Abstr act. The paper addresses Leon Hen.kin's proposition as a " light house", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information t heory, and quantum mechanics: Two st rategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness t o be unified and t hus reduced as a joint property of infinity and of all infinite sets. However, only Henkin's proposition equivalent to an internal posit ion t o infinity is consistent. This can be retraced back to set theory and its axioms, where that of choice is a key. Quantum mechanics is forced to int roduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonst rate that some essential properties of quantum information, entanglement, and quantum comput er originate direct ly from infinit y once it is involved in quant um mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border bet ween them. A special kind of invariance to t he axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted ent irely in the same terms only of set theory. Quantum computer can demonstrat e especially clearly t he privilege of t he internal position, or " observer'' , or " user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity. An essential area of cont emporary knowledge may be synt hesized from a single viewpoint. Both completeness and incompleteness are well distinguishable as to finite-ness: Completeness supposes that any operat ions defined over any finite sets do

2018

The paper addresses Leon Henkin's proposition as a "lighthouse", <br>which can elucidate a vast territory of knowledge uniformly: logic, set theory, <br>information theory, and quantum mechanics: Two strategies to infinity are <br>equally relevant for it is as universal and thus complete as open and thus incomplete. <br>Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' <br>proposition are reformulated so that both completeness and incompleteness to <br>be unified and thus reduced as a joint property of infinity and of all infinite sets. <br>However, only Henkin's proposition equivalent to an internal position to <br>infinity is consistent . This can be retraced back to set theory and its axioms, <br>where tha t of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can <br>demonstrate that some essent...