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Mathematics

Action at a Distance in Quantum Theory

Jerome Blackman

Published May 6, 2015• 0 citations
Publisher

Abstract

The purpose of this paper is to present a consistent mathematical framework that shows how the EPR (Einstein. Podolsky, Rosen) phenomenon fits into our view of space time. To resolve the differences between the Hilbert space structure of quantum theory and the manifold structure of classical physics, the manifold is taken as a partial representation of the Hilbert space. It is the partial nature of the representation that allows for action at a distance and the failure of the manifold picture.

DOI
10.3390/math3020329

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The purpose of this paper is to present a consistent mathematical framework that shows how the EPR (Einstein. Podolsky, Rosen) phenomenon fits into our view of space time. To resolve the differences between the Hilbert space structure of quantum theory and the manifold structure of classical physics, the manifold is taken as a partial representation of the Hilbert space. It is the partial nature of the representation that allows for action at a distance and the failure of the manifold picture.
In their celebrated paper titled Can quantum mechanical description of physical reality be considered complete?, Einstein, Podolsky and Rosen (EPR) showed for the first time the existence of `Spooky action-at-a-distance'. Though the result of their paper is unquestionable, but the conclusion of the same became sensational because of its challenge to quantum mechanical formalism whether it's complete or not in describing the physical reality of Nature. Bohr's physical and philosophical reply to that conclusion justified the completeness of quantum mechanics. Here, a simple algebraic way is presented for the results of these two classic papers in such a way that the actual reason behind why quantum world necessarily exhibits the action-at-a-distance and how Bohr defended against the incompleteness of the quantum formalism will become clear.This approach naturally reveals what physical assumption of EPR went wrong while considering the entangled quantum system and also provides the missing mathematical argument in Bohr's reply.
The relation between quantum theory and special relativity is peculiar. On the one hand it is close and essential. Steven Weinberg [1], for example, takes the position that the standard model is an inevitable consequence of the marriage of the two theories. On the other, the action at a distance implied by the famous paper by Einstein, Podolsky, and Rosen [2] is certainly contrary to the implications of special relativity in spite of the fact that it does not allow transmission of information. The setting of quantum theory is a Hilbert spaceH and a path inH whose development in time is governed by a one parameter group of unitary transformations while the setting for special relativity can be thought of as Minkowski space with the time development not absolute, but dependent on the inertial system of the observer. The problem is to show how these two very different viewpoints can be reconciled in a way that includes an explanation of the "action at a distance" conundrum.
The remarkable quantum phenomenon of 'non-locality' discovered by Einstein, Podolsky and Rosen (EPR) is the centre of considerable experimental and theoretical activity these days. Nevertheless, many (perhaps most) physicists do not feel the phenomenon is especially paradoxical, or even worthy of interest. This article attempts to reveal, in a simple way, just where the fascination of the EPR paradox lies, and to answer some of the arguments invoked by critics to 'explain away' the paradox.
From a phenomenological and experimental viewpoint, quantum mechanics is remarkably successful in explaining effects and processes in the microscopic world. Recently, however, a growing body of literature has revisited the theory's own formal foundations, specifically addressing issues of self-consistency and completeness. As a major historical example of these foundational challenges, the Einstein-Podolsky-Rosen (EPR) argument famously claimed that the theory is incomplete. While many valid criticisms and refutations of the EPR logic exist, they generally rely on a combination of technical results and conceptual objections to EPR's interpretive assumptions, thus transcending the plain quantum framework itself. Motivated by these trends of examining the theory's structural limits, here we revisit the EPR argument. Focusing on elements essential to the EPR reasoning, we first analyze general quantum correlations for EPR states: (i) the fact that their observables are always associated with non-commuting operators, and (ii) the nature of the correlated information obtained through measurement. From these two points alone, and relying strictly on the core rules of quantum mechanics, we demonstrate how to overturn the alleged EPR incompleteness without invoking any extraneous propositions. Consequently, we show that the standard formalism alone suffices to resolve such type of skepticism regarding the theory's physical reach. This offers, at least in a paradigmatic instance, a powerful indication of a structurally well-founded, self-consistent quantum theory.
The phenomenon of nonlocality in quantum mechanics is one of its most fundamental features and is most strikingly exemplified in the discussion of the Einstein, Podolsky and Rosen (EPR) type of experiment. The generality of Bell's inequality and the results of experiments done up to now show that local hidden-variable theories are ruled out as a means to resolve the famous EPR paradox. What remains for further consideration are the nonlocal and the Einstein-separable hidden-variable models. Finally, an alternative and possibly successful approach in trying to 'explain' nonlocality might involve ideas of backward causation.
This thesis argues that the apparent clash between Einstein-Podolsky-Rosen reasoning and the empirical violation of Bell inequalities is not best explained by "action at a distance," but by a structural conflict between the classical, distributive logic tacitly built into locality arguments and the non-distributive lattice that governs quantum propositions. It develops the Tractarian-Stone picture of classical logical space, and sets it against the Hilbert-space framework. Because span strictly exceeds union, distributivity generally fails, and with it the classical route to factorization that locality arguments require. Methodologically, the thesis contributes (i) a semantics-first comparison of classical Boolean lattices with the orthomodular lattice of closed subspaces, showing how Boolean "centers" of compatibility persist locally within a non-distributive structure; (ii) a reconstruction of the counterfactual step that isolates exactly which independence assumptions are doing empirical work; and (iii) an operational reading that connects finite-data tests to locality via matching conditions without invoking signaling. The result is a unified account in which violations of Bell inequalities register the failure of classical distributivity to extend to quantum disjunction, not evidence for hidden superluminal influence. Locality survives in its minimal, experimentally accessible form - no signaling - while classical factorization is revealed as a special case valid only inside compatible sublattices.