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arXiv:hep-th/9310176

Elements of Reality and the Failure of the Product Rule Measurability of Commuting Observables

Lev Vaidman A

Published Oct 27, 1993• 0 citations
arXivHTMLPDF

Abstract

The concept of ``elements of reality" is analyzed within the framework of quantum theory. It is shown that elements of reality fail to fulfill the product rule. This is the core of recent proofs of the impossibility of a Lorentz-invariant interpretation of quantum mechanics. A generalization and extension of the concept of elements of reality is presented. Lorentz-invariance is restored by giving up the product rule. The consequences of giving up the ``and" rule, which must be abandoned together with the product rule, are discussed.

DOI
10.48550/arxiv.hep-th/9310176

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The concept of elements of reality" is analyzed within the framework of quantum theory. It is shown that elements of reality fail to fulfill the product rule. This is the core of recent proofs of the impossibility of a Lorentz-invariant interpretation of quantum mechanics. A generalization and extension of the concept of elements of reality is presented. Lorentz-invariance is restored by giving up the product rule. The consequences of giving up the and" rule, which must be abandoned together with the product rule, are discussed.
We examine a gedanken experiment described by Hardy, which purports to prove that the ``elements of reality'' corresponding to Lorentz-invariant observables cannot themselves be Lorentz invariant without violating quantum mechanics. We consider a number of criticisms of this proof and show that these criticisms are not convincing. We demonstrate that the contradiction, which arises in the gedanken experiment and forms the basis of Hardy's proof, has nothing to do with realism, but is a consequence of the well-known noncovariance of the reduction postulate. A reexamination of the gedanken experiment, using a more appropriate formalism, helps to clarify its implications. We conclude with a brief appraisal of realist interpretations of quantum mechanics in the relativistic domain.
The notion of the Einstein-Podolsky-Rosen (EPR) "element of reality" is much discussed in the literature on the foundations of quantum mechanics. Recently, it has become particularly relevant due to a proposed criterion of the physical reality of a given quantum mechanical observable [A. L. O. Bilobran and R. M. Angelo, Europhys. Lett. 112, 40005 (2015)]. We examine this proposal and its consequently related measure of non-locality [V. S. Gomez and R. M. Angelo, Phys. Rev. A 97, 012123, (2018)] and argue that the criterion is ill-described as quantifying physical reality without introducing serious inconsistency with the basic notions of realism that under-gird enquiry. We agree that this reality criterion demonstrates, along with the famous GHZ results, that general quantum observable values make for poor elements of reality. However, we also argue that this does not mean no such elements of reality are to be found in quantum theory. By arguing for, and adopting, probability distributions as these elements of reality instead, we demonstrate that the criterion of physical reality is actually one of observable predictability. We then examine the relationship of realism-based non-locality to the Bell form and find that, despite the flawed premise, this measure does indeed codify non-locality that is not captured by Bell inequalities.
What is the nature of reality? How should be an answer to this question? At this level, we are so deep that all our concepts are obscure. Quantum theory (QT) is at this level. The quest for interpreting it fails because the clarity of our actual fundamental concepts in ordinary language cannot match the precision of the theory's mathematical formalism. The interpretation of quantum mechanics should define and bring clarity to fundamental concepts; not simply rely over them. Quantum theory requires a new worldview, the one that will make it understandable. We discuss on the formulation and interpretation of closed, and final theories, in other words, completely axiomatic theories that define their own conceptual framework, and closed theories that cannot be made simpler. We claim that everything we can properly say about a theory follows from its interpretation, and that interpreting is saying the meaning of each of its mathematical statements. We argue that quantum mechanics is a closed theory; and, therefore, that part of it is part of the final or fundamental theory of physics (FTP). We propose a new entirely axiomatic formulation of part of quantum theory (including all its mathematical framework) in which new powerful and useful elements are added to the existing formalism. Our interpretation is straightforward: it allows us reading or saying the meaning of each of the theory's axiom, theorem or element. We state that the proposed theory is the FTP. Its interpretation may dissolve most of the existing paradoxes around QT. The principle of excluded middle is not valid in general. Both epistemic and ontic elements are present in the theory since it is a theory of knowledge and truth. This and other fundamental concepts are defined by the theory, which can represent both reality and our knowledge about it. Aesthetics is one of its key features.
The interpretation of quantum mechanics in the context of measurements, and concepts such as state “collapse,” have troubled physicists since the inception of quantum theory. Pushed to their logical extreme, such issues become entangled (in the colloquial sense) with questions of consciousness, reality, etc. Bell’s theorem eliminated possible ways out of this tangle via any (halfway reasonable) classical underlying theory, and the success of quantum mechanics forces us to adopt it as a fundamental theory and face the logical consequences.