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Journal of Mathematical Physics

On the problem of a local extension of the quantum formalism

Thomas D. Angelidis

Published May 1, 1993• 7 citations
Publisher

Abstract

The suggested impossibility of a consistent local extension of the quantum formalism is reviewed in the context of the Einstein–Podolsky–Rosen–Bohm (EPRB) ideal experiment. A consistent local theory Th(𝒢) extending the quantum formalism is proposed. The postulates of Th(𝒢) stipulate a finer state specification than that given by the quantum state ‖γ1,γ2≳. This describes how the spontaneous annihilation process itself breaks the spherical symmetry of the singlet state ‖γ1,γ2≳ into a pair of back-to-back photons having only axial symmetry about the direction of their motion. The theory Th(𝒢) shows that local action suffices to explain all that the quantum formalism predicts for the EPRB ideal experiment as well as some other results.

DOI
10.1063/1.530181

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The suggested impossibility of a consistent local extension of the quantum formalism is reviewed in the context of the Einstein–Podolsky–Rosen–Bohm (EPRB) ideal experiment. A consistent local theory Th(𝒢) extending the quantum formalism is proposed. The postulates of Th(𝒢) stipulate a finer state specification than that given by the quantum state ‖γ1,γ2≳. This describes how the spontaneous annihilation process itself breaks the spherical symmetry of the singlet state ‖γ1,γ2≳ into a pair of back-to-back photons having only axial symmetry about the direction of their motion. The theory Th(𝒢) shows that local action suffices to explain all that the quantum formalism predicts for the EPRB ideal experiment as well as some other results.
A realistic view of the canonical EinsteinndashndashndashPodolskyndashndashndashRosenndashndashndashBohm experiment based on quantum theory and its consequences
A realistic view of the Einstein–Podolsky–Rosen–Bohm experiment with spins based on quantum theory is presented. This view implies that there is no action at a distance. It also implies that the measurement result A ( B ) for particle 1 (2) depends on both magnet angles, and hence the probability of obtaining the result A ( B ) also depends on both magnet angles. In the light of these realistic implications, it is clear that what is wrong at least with local realistic theory is not the locality or no-action-at-a-distance assumption itself but rather the formal implementation of that assumption.
The best case for thinking that quantum mechanics is nonlocal rests on Bell's Theorem, and later results of the same kind. However, the correlations characteristic of Einstein–Podolsky–Rosen (EPR)–Bell (EPRB) experiments also arise in familiar cases elsewhere in quantum mechanics (QM), where the two measurements involved are timelike rather than spacelike separated; and in which the correlations are usually assumed to have a local causal explanation, requiring no action-at-a-distance (AAD). It is interesting to ask how this is possible, in the light of Bell's Theorem. We investigate this question, and present two options. Either (i) the new cases are nonlocal too, in which case AAD is more widespread in QM than has previously been appreciated (and does not depend on entanglement, as usually construed); or (ii) the means of avoiding AAD in the new cases extends in a natural way to EPRB, removing AAD in these cases too. There is a third option, viz., that the new cases are strongly disanalogous to EPRB. But this option requires an argument, so far missing, that the physical world breaks the symmetries which otherwise support the analogy. In the absence of such an argument, the orthodox combination of views—action-at-a-distance in EPRB, but local causality in its timelike analogue—is less well established than it is usually assumed to be. 1 Introduction 1.1 Background 1.2 Outline of the argument2 The Experiments 2.1 Standard EPRB 2.2 Sideways EPRB 2.3 Comparing the experiments 2.4 The need for beables3 The Symmetry Considerations 3.1 The action symmetry 3.2 Time-symmetry in SEPRB4 The Basic Trilemma 4.1 An intuitive defence of Option III?5 Avoiding the Trilemma?6 The Classical Objection7 Defending Option III 7.1 The free will argument 7.2 Independence and consistency8 Entanglement and Epistemic Perspective
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.
Maudlin has claimed that no local theory can reproduce the predictions of standard quantum mechanics that violate Bell’s inequality for Bohm’s version (two spin-half particles in a singlet state) of the Einstein-Podolsky-Rosen problem. It is argued that, on the contrary, standard quantum mechanics itself is a counterexample to Maudlin’s claim, because it is local in the appropriate sense (measurements at one place do not influence what occurs elsewhere there) when formulated using consistent principles in place of the inconsistent appeals to “measurement” found in current textbooks. This argument sheds light on the claim of Blaylock that counterfactual definiteness is an essential ingredient in derivations of Bell’s inequality.