Local causality in the works of Einstein, Bohm and Bell
Aurélien Drezet
备注：
Accepted for publication in A. Oldofredi (Ed.), Guiding Waves in Quantum Mechanics: 100 Years of de Broglie-Bohm Pilot-Wave Theory, Oxford University Press
In this chapter we discuss the Einstein Podolsky Rosen theorem and its strong relation with Bell's theorem. The central role played by the concept of beable introduced by Bell is emphasized. In particular we stress that beables involved in EPR and Bell theorems are not limited to hidden supplementary variables (e.g., like in the de Broglie-Bohm (dBB) pilot-wave theory) but also include the wave function. In full agreement with Bell this allows us the reformulate the EPR and Bell results as strong theorems concerning nonlocality for quantum mechanics itself and not only for hidden-variables approaches as it is often mistakenly assumed. Furthermore, we clarify some repeated ambiguities concerning `local-realism' and emphasize that neither realism nor determinism nor counterfactual definiteness are prerequisites of EPR and Bell theorems.
Before they met, David Bohm and Roger Penrose each puzzled over the paradox of the arrow of time. After they met, the case for projective physical space became clearer.
The aim of this paper is to give a sharp definition of Bell's notion of local causality. To this end, first we unfold a framework, called local physical theory, integrating probabilistic and spatiotemporal concepts. Formulating local causality within this framework and classifying local physical theories by whether they obey local primitive causality --- a property rendering the dynamics of the theory causal, we then investigate what is needed for a local physical theory, with or without local primitive causality, to be locally causal. Finally, comparing Bell's local causality with the Common …
Bell's Theorem requires any theory which obeys the technical definitions of Free Choice and Local Causality to satisfy the Bell inequality. Invariant set theory is a finite theory of quantum physics which violates the Bell inequality exactly as does quantum theory: in it neither Free Choice nor Local Causality hold, consistent with Bell's Theorem. However, within the proposed theory, the mathematical expressions of both Free Choice and Local Causality involve states which, for number-theoretic reasons, cannot be ontic (cannot lie on the theory's fractal-like invariant set $I_U$ in state space)…
While initial versions of Bell's theorem captured the notion of locality with the assumption of factorizability, in later presentations, Bell argued that factorizability could be derived from the more fundamental principle of local causality. Here we show that, contrary to what is commonly assumed, in order to derive factorizability from the principle of local causality, a non-trivial assumption, similar but strictly independent of settings independence, is required. Loosely speaking, such an extra assumption demands independence between the states of the measurement apparatuses. We conclude t…
Many of the heated arguments about the meaning of "Bell's theorem" arise because this phrase can refer to two different theorems that John Bell proved, the first in 1964 and the second in 1976. His 1964 theorem is the incompatibility of quantum phenomena with the dual assumptions of locality and determinism. His 1976 theorem is the incompatibility of quantum phenomena with the unitary property of local causality. This is contrary to Bell's own later assertions, that his 1964 theorem began with that single, and indivisible, assumption of local causality (even if not by that name). While there a…
This paper investigates dynamical relaxation to quantum equilibrium in the stochastic de Broglie-Bohm-Bell formulation of quantum mechanics. The time-dependent probability distributions are computed as in a Markov process with slowly varying transition matrices. Numerical simulations, supported by exact results for the large-time behavior of sequences of (slowly varying) transition matrices, confirm previous findings that indicate that de Broglie-Bohm-Bell dynamics allows an arbitrary initial probability distribution to relax to quantum equilibrium; i.e., there is no need to make the ad-hoc as…
"Bell's theorem" can refer to two different theorems that John Bell proved, the first in 1964 and the second in 1976. His 1964 theorem is the incompatibility of quantum phenomena with the joint assumptions of Locality and Predetermination. His 1976 theorem is their incompatibility with the single property of Local Causality. This is contrary to Bell's own later assertions, that his 1964 theorem began with the assumption of Local Causality, even if not by that name. Although the two Bell's theorems are logically equivalent, their assumptions are not. Hence, the earlier and later theorems sugges…
This paper introduces an extension of the de Broglie-Bohm-Bell formulation of quantum mechanics, which includes intrinsic particle degrees of freedom, such as spin, as elements of reality. To evade constraints from the Kochen-Specker theorem the discrete spin values refer to a specific basis -- i.e., a single spin vector orientation for each particle; these spin orientations are, however, not predetermined, but dynamic and guided by the (reduced, spin-only) wave function of the system, which is conditional on the realized location values of the particles. In this way, the unavoidable contextua…
The apparent nonlocality of quantum theory has been a persistent concern. Einstein et. al. (1935) and Bell (1964) emphasized the apparent nonlocality arising from entanglement correlations. While some interpretations embrace this nonlocality, modern variations of the Everett-inspired many worlds interpretation try to circumvent it. In this paper, we review Bell's "no-go" theorem and explain how it rests on three axioms, local causality, no superdeterminism, and one world. Although Bell is often taken to have shown that local causality is ruled out by the experimentally confirrmed entanglement …
'Locality' is a fraught word, even within the restricted context of Bell's theorem. As one of us has argued elsewhere, that is partly because Bell himself used the word with different meanings at different stages in his career. The original, weaker, meaning for locality was in his 1964 theorem: that the choice of setting by one party could never affect the outcome of a measurement performed by a distant second party. The epitome of a quantum theory violating this weak notion of locality (and hence exhibiting a strong form of nonlocality) is Bohmian mechanics. Recently, a new approach to quantu…
John Stewart Bell's famous 1964 theorem is widely regarded as one of the most important developments in the foundations of physics. It has even been described as "the most profound discovery of science." Yet even as we approach the 50th anniversary of Bell's discovery, its meaning and implications remain controversial. Many textbooks and commentators report that Bell's theorem refutes the possibility (suggested especially by Einstein, Podolsky, and Rosen in 1935) of supplementing ordinary quantum theory with additional ("hidden") variables that might restore determinism and/or some notion of a…
This paper presents a \emph{non-instant field model} for electrodynamics that permits a causal explanation of the \emph{Aharonov-Bohm effect} and a \emph{covariant quantization} of the respective Maxwell equations via the \emph{Gupta-Bleuler method}. Our model satisfies the following \emph{correspondence principle}: if $A^μ$, $\vE$, $\vB$ denote the four potential, the electric field and the magnetic field of the non-instant field model, then the respective classical quantities are $\A[A^μ]$, $\A[\vE]$, $\A[\vB]$, where $\A$ is a covariant time averaging operator. Here $\A[A^μ]$ is interpreted…
In papers published in the 25 years following his famous 1964 proof John Bell refined and reformulated his views on locality and causality. Although his formulations of local causality were in terms of probability, he had little to say about that notion. But assumptions about probability are implicit in his arguments and conclusions. Probability does not conform to these assumptions when quantum mechanics is applied to account for the particular correlations Bell argues are locally inexplicable. This account involves no superluminal action and there is even a sense in which it is local, but it…
Bell's theorem shows that the reasonable relativistic causal principle known as "local causality" is not compatible with the predictions of quantum mechanics. It is not possible maintain a satisfying causal principle of this type while dropping any of the better-known assumptions of Bell's theorem. However, another assumption of Bell's theorem is the use of classical logic. One part of this assumption is the principle of "ontic definiteness", that is, that it must in principle be possible to assign definite truth values to all propositions treated in the theory. Once the logical setting is cla…
This paper aims to motivate Bell's notion of local causality by means of Bayesian networks. In a locally causal theory any superluminal correlation should be screened off by atomic events localized in any so-called \textit{shielder-off region} in the past of one of the correlating events. In a Bayesian network any correlation between non-descendant random variables are screened off by any so-called \textit{d-separating set} of variables. We will argue that the shielder-off regions in the definition of local causality conform in a well defined sense to the d-separating sets in Bayesian networks…
In a Bell test, the set of observed probability distributions complying with the principle of local realism is fully characterized by Bell inequalities. Quantum theory allows for a violation of these inequalities, which is famously regarded as Bell nonlocality. However, finding the maximal degree of this violation is, in general, an undecidable problem. Consequently, no algorithm can be used to derive quantum analogs of Bell inequalities, which would characterize the set of probability distributions allowed by quantum theory. Here we present a family of inequalities, which approximate the set …
The starting point of the present paper is Bell's notion of local causality and his own sharpening of it so as to provide for mathematical formalisation. Starting with Norsen's (2007, 2009) analysis of this formalisation, it is subjected to a critique that reveals two crucial aspects that have so far not been properly taken into account. These are (i) the correct understanding of the notions of sufficiency, completeness and redundancy involved; and (ii) the fact that the apparatus settings and measurement outcomes have very different theoretical roles in the candidate theories under study. Bot…
Between 1964 and 1990, the notion of nonlocality in Bell's papers underwent a profound change as his nonlocality theorem gradually became detached from quantum mechanics, and referred to wider probabilistic theories involving correlations between separated beables. The proposition that standard quantum mechanics is itself nonlocal (more precisely, that it violates `local causality') became divorced from the Bell theorem per se from 1976 on, although this important point is widely overlooked in the literature. In 1990, the year of his death, Bell would express serious misgivings about the mathe…
Bell's [Physics 1 (1964) 195-200] theorem is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell's inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804-1807] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The paper starts with a proof of a strong, finite sample, version of Bell's inequality and thereby also of Bell's theorem, which states that quantum theory is incompatible with the con…
In this short note, I derive the Bell-CHSH inequalities as an elementary result in the present-day theory of statistical causality based on graphical models or Bayes' nets, defined in terms of DAGs (Directed Acyclic Graphs) representing direct statistical causal influences between a number of observed and unobserved random variables. I show how spatio-temporal constraints in loophole-free Bell experiments, and natural classical statistical causality considerations, lead to Bell's notion of local hidden variables, and thence to the CHSH inequalities. The word "local" applies to the way that the…
We introduce a new interpretation of quantum mechanics by examining the Einstein, Podolsky and Rosen's (EPR) paradox and Bell's inequality experiments under the assumption that the vacuum has an inhomogeneous texture for energy levels below the Heisenberg time-energy uncertainty relation. In this article, selected results from the most reliable Bell's inequality experiments will be quantitatively analyzed to show that our interpretation of quantum mechanics creates a new loophole in Bell's inequality, and that the past experimental findings do not contradict our new interpretation. Under the v…
Assuming perfect detection efficiency, we present an (indeterministic) model for an EPR-Bohm experiment which reproduces the singlet correlations, without contradicting Bell's original locality condition. In this model we allow the probability distribution $ρ_λ$ of the state $λ$ at the source to depend parametrically on the orientation $ξ$ of one of the measuring devices: $ρ_λ(λ,ξ)$. In a Bell experiment, no-signaling between the source and each one of the devices would seem clearly sufficient to rule such an influence; however, not even schemes where the choice of observables takes place duri…
The relationship between the Bell article in 1964 and its well known inequalities with the Einstein Podolsky Rosen article in 1935 is revisited. Einstein views on quantum mechanics as stated in many circumstances up to his death in 1955 are recalled. The role of David Bohm is shown to be essential to understand the link between Bell and Einstein's views. This link appears not to be as usually recounted.
On the basis of local realism theory, nonlocal information is necessary for violation of Bell's inequality. From a theoretical point of view, nonlocal information is essentially the mutual information on distant outcome and measurement setting. In this work we prove that if the measurement is free and unbiased, the mutual information about the distant outcome and setting is both necessary for the violation of Bell's inequality in the case with unbiased marginal probabilities. In the case with biased marginal probabilities, we point out that the mutual information about distant outcome cease to…
Bell's theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as ``fine-tuning'' or a ``conspiracy''. We here point out that the common interpretation is at best ph…
A counterexample to Bell's theorem is presented which uses a pair of photons instead of spin-1/2 particles used in our previous counterexamples. A locally causal protocol is provided for Alice and Bob, which allows them to simulate observing photon polarizations at various angles, and record their results as A=+/-1 in S^3 and B=+/-1 in S^3, respectively. When these results are compared, the correlations are seen to be exactly those predicted by quantum mechanics; namely cos 2(alpha - beta), where alpha and beta are the angles of polarizers. The key ingredient in our counterexample is the topol…
I present what might seem to be a local, deterministic model of the EPR-Bohm experiment, inspired by recent work by Joy Christian, that appears at first blush to be in tension with Bell-type theorems. I argue that the model ultimately fails to do what a hidden variable theory needs to do, but that it is interesting nonetheless because the way it fails helps clarify the scope and generality of Bell-type theorems. I formulate and prove a minor proposition that makes explicit how Bell-type theorems rule out models of the sort I describe here.
We consider the problem of representation of quantum states and observables in the framework of classical probability theory (Kolmogorov's measure-theoretic axiomatics, 1933). Our aim is to show that, in spite of the common opinion, correlations of observables $A_1, A_2$ and $B_1,B_2$ involved in the experiments of the Bohm-Bell type can be expressed as correlations of classical random variables $a_1, a_2$ and $b_1, b_2.$ The crucial point is that correlations $\langle A_i, B_j \rangle$ should be treated as conditional on the selection of the pairs $(i, j).$ The setting selection procedure is …
Explaining observations in terms of causes and effects is central to all of empirical science. Correlations between entangled quantum particles, however, seem to defy such an explanation. To recover a causal picture in this case, some of the fundamental assumptions of causal explanations have to give way. Here we consider a broad class of models where one of these assumptions, Bell's local causality, is relaxed by allowing a direct influence from one measurement outcome to the other. We use interventional and observational data from a photonic experiment to bound the strength of this causal in…
A recent paper [P. Martin-Dussaud, C. Rovelli, F. Zalamea, arXiv:1806.08150] has given a lucid treatment of Bell's notion of local causality within the framework of the relational interpretation of quantum mechanics. However, the authors went on to conclude that the quantum violation of Bell's notion of local causality is no more surprising than a common cause. Here, I argue that this conclusion is unwarranted by the authors' own analysis. On the contrary, within the framework outlined by the authors, I argue that the implications of the relational interpretation are much more radical.
J.S. Bell's work has convinced many that correlations in violation of CHSH inequalities show that the world itself is non-local, and that there is an apparently essential conflict between any sharp formulation of quantum theory and relativity. Against this consensus, this paper argues that there is no conflict between quantum theory and relativity. Quantum theory itself helps us explain such (otherwise) puzzling correlations in a way that contradicts neither Bell's intuitive locality principle nor his local causality condition. The argument depends on understanding quantum theory along pragmat…
I argue that quantum mechanics is a realistic theory, but it violates either strong locality (no superluminal influences) or strict causality (diiferent effects cannot follow from the same cause).
We derive Bohm's trajectories from Bell's beables for arbitrary bipartite systems composed by dissipative noninteracting harmonic oscillators at finite temperature. As an application of our result, we calculate the Bohmian trajectories of particles described by a generalized Werner state, comparing the trajectories when the sate is either separable or entangled. We show that qualitative differences appear in the trajectories for entangled states as compared with those for separable states.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and are key resources for quantum advantage in information-processing tasks. Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure. Contextuality, on the other hand, is on a more controversial foundation. In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality. First, I show that Bell inequalities can be derived solely from …
