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arXiv:2610.41944

Locality, Probability and the Relevance of History

Sebastian Horvat

quant-phphysics.hist-phSubmitted Oct 30, 2026• Announced Nov 2, 2026
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Abstract

Relativistic theory and experiments teach us that causal influences are constrained by the speed of light. Deterministic theories incorporate this lesson in $\textit{functional}$ terms: whether an event occurs or not is functionally determined by events that occur in its immediate causal past. In this paper we suggest a natural generalization of this constraint that applies to probabilistic theories: the $\textit{probability}$ that an event occurs is to be functionally determined by the $\textit{probability}$ of the event-occurrences in its immediate causal past. This constraint differs from Bell's criterion of local causality and turns out to imply it only under an additional assumption that requires history to be, in a sense, $\textit{ir}$relevant. Bell-inequality-violating quantum phenomena are thus arguably $\textit{not}$ in tension with relativity, locality, "statistical independence", nor with a supposed "realism". What they instead show is that history is, in a sense, $\textit{relevant}$.

AI Summary

The paper asks whether Bell’s local causality is the appropriate probabilistic expression of relativistic locality, or whether Bell-inequality violations can instead be understood as indicating dependence on history. It introduces “probabilistic world-theories”: sets of spacetime event configurations equipped with probability measures indexed by initial conditions on a Cauchy surface. Its proposed principle, Probabilistic Locality, requires differences in an event’s probability distribution to be accompanied by differences in the joint distribution of events in its arbitrarily immediate causal past. This compares distributions associated with different initial conditions, whereas Bell’s local causality imposes conditional-independence constraints within each distribution. The paper establishes their logical independence through examples, including a theory with only one initial-condition class, for which the new principle holds trivially. It then introduces “Irrelevance of History,” requiring that future probabilities conditioned on a complete present configuration coincide with probabilities obtained by treating that configuration as fresh initial conditions. A formal argument claims that this assumption together with Probabilistic Locality implies Bell’s local causality. Rejecting the history assumption is consequently proposed as a way to accommodate Bell violations while retaining the new locality principle. Time symmetry additionally yields a future-directed counterpart of that principle. No concrete theory reproducing Bell correlations under these requirements is constructed. Stated limitations include unresolved continuum and measure-theoretic details, a fixed spacetime background, and deferred comparisons with existing quantum formulations, other locality assumptions, and GHZ phenomena. The conclusions concern a proposed foundational framework rather than new experimental predictions or quantitative bounds.

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Central claim may not hold: Principle 1 permits two worlds that agree everywhere through t=0 and disagree everywhere afterward, contradicting Proposition 1’s assertion that locality implies determinism. The proof of the central Proposition 3 unjustifiably propagates arbitrarily nearby probability differences backward to a specified initial surface, without assumptions that would validate that propagation. Principle 4 does not require a probability-preserving correspondence between future histories, so it does not generally imply the conditional-probability identities used to prove Proposition 3. The paper does not establish that its proposed Probabilistic Locality is the constraint supported by relativity, and the criterion is vacuous for theories with only one initial-condition probability distribution.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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Relativistic theory and experiments teach us that causal influences are constrained by the speed of light. Deterministic theories incorporate this lesson in functional\textit{functional}functional terms: whether an event occurs or not is functionally determined by events that occur in its immediate causal past. In this paper we suggest a natural generalization of this constraint that applies to probabilistic theories: the probability\textit{probability}probability that an event occurs is to be functionally determined by the probability\textit{probability}probability of the event-occurrences in its immediate causal past. This constraint differs from Bell's criterion of local causality and turns out to imply it only under an additional assumption that requires history to be, in a sense, ir\textit{ir}irrelevant. Bell-inequality-violating quantum phenomena are thus arguably not\textit{not}not in tension with relativity, locality, "statistical independence", nor with a supposed "realism". What they instead show is that history is, in a sense, relevant\textit{relevant}relevant.
The paper asks whether Bell’s local causality is the appropriate probabilistic expression of relativistic locality, or whether Bell-inequality violations can instead be understood as indicating dependence on history. It introduces “probabilistic world-theories”: sets of spacetime event configurations equipped with probability measures indexed by initial conditions on a Cauchy surface.
Its proposed principle, Probabilistic Locality, requires differences in an event’s probability distribution to be accompanied by differences in the joint distribution of events in its arbitrarily immediate causal past. This compares distributions associated with different initial conditions, whereas Bell’s local causality imposes conditional-independence constraints within each distribution. The paper establishes their logical independence through examples, including a theory with only one initial-condition class, for which the new principle holds trivially. It then introduces “Irrelevance of History,” requiring that future probabilities conditioned on a complete present configuration coincide with probabilities obtained by treating that configuration as fresh initial conditions. A formal argument claims that this assumption together with Probabilistic Locality implies Bell’s local causality. Rejecting the history assumption is consequently proposed as a way to accommodate Bell violations while retaining the new locality principle. Time symmetry additionally yields a future-directed counterpart of that principle.
No concrete theory reproducing Bell correlations under these requirements is constructed. Stated limitations include unresolved continuum and measure-theoretic details, a fixed spacetime background, and deferred comparisons with existing quantum formulations, other locality assumptions, and GHZ phenomena. The conclusions concern a proposed foundational framework rather than new experimental predictions or quantitative bounds.
Relativistic theory and experiments teach us that causal influences are constrained by the speed of light. Deterministic theories incorporate this lesson in functional terms: whether an event occurs or not is functionally determined by events that occur in its immediate causal past. In this paper we suggest a natural generalization of this constraint that applies to probabilistic theories: the probability that an event occurs is to be functionally determined by the probability of the event-occurrences in its immediate causal past. This constraint differs from Bell’s criterion of local causality and turns out to imply it only under an additional assumption that requires history to be, in a sense, irrelevant. Bell-inequality-violating quantum phenomena are thus arguably not in tension with relativity, locality, “statistical independence”, nor with a supposed “realism”. What they instead show is that history is, in a sense, relevant.
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 is in tension with the requirement that the direct causes and effects of events be nearby. Introduction I never met John Bell, but his writings have supplied me with a continual source of new insights as I read and reread them over 40 years. As I worked toward a rather different understanding of quantum mechanics he was foremost in my mind as a severe but honest critic of such attempts. We all would love to know what Einstein would have made of Bell's theorem. I confess that the deep regret I feel that Bell cannot respond to this paper is sometimes assuaged by a sense of relief. Locality and Local Causality In his seminal 1964 paper [1], John Bell expressed locality as the requirement that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past. [2, p. 14] This seems to require that the result of a measurement would have been the same, no matter what operations had been performed on such a distant system. But suppose the result of a measurement were the outcome of an indeterministic process. Then the result of the measurement might have been different even if exactly the same operations (if any) had been performed on that distant system. So can no indeterministic theory satisfy the locality requirement? Bell felt no need to address that awkward question in his 1964 paper [1], since he took the EPR argument to establish that any additional variables needed to restore locality and causalitywould have to determine a unique result of ameasurement.
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 is in tension with the requirement that the direct causes and effects of events are nearby.
Introduction Fifty years after the publication of Bell's theorem, there remains some controversy regarding what the theorem is telling us about quantum mechanics, and what the experimental violations of Bell inequalities are telling us about the world. This chapter represents my best attempt to be clear about what I think the lessons are. In brief: There is some sort of nonlocality inherent in any quantum theory, and, moreover, in any theory that reproduces, even approximately, the quantum probabilities for the outcomes of experiments. But not all forms of nonlocality are the same; there is a distinction to be made between action at a distance and other forms of nonlocality, and I will argue that the nonlocality needed to violate the Bell inequalities need not involve action at a distance. Furthermore, the distinction between forms of nonlocality makes a difference when it comes to compatibility with relativistic causal structure. The Bell locality condition is a condition which, if satisfied, renders possible a completely local account of the correlations between outcomes of spatially separated experiments. Bell's theorem tells us that the probabilities that we compute from quantum mechanics do not admit of such an account. Modulo auxiliary assumptions that, in my opinion, should be noncontroversial, this yields the conclusion that there must be something nonlocal about any theory that yields the quantum mechanical probabilities or anything close to them. One way to violate the Bell locality condition – and this is the way that any deterministic theory must do it – is via straightforward action at a distance. However, we should also consider the possibility of a stochastic theorywith probabilistic laws that involve irreducible correlations. The sort of nonlocality inherent in such a theory involves relations between events that are significantly different from causal relations as usually conceived. Perhaps surprisingly, such a theory, unlike its deterministic cousins, need not invoke a distinguished relation of distant simultaneity. This makes a difference for the situation of the two sorts of theories vis-a-vis relativity. Though a deterministic theory can achieve the appearance of being relativistic, at the level of observable phenomena, it can do so only by introducing a preferred foliation and with it a causal structure at odds with relativistic causal structures. Things are otherwise with dynamical collapse theories.
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 conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom. Locality is the principle that the direction of causality matches the direction of time, and that causal influences need time to propagate spatially. Realism and freedom are directly connected to statistical thinking on causality: they relate to counterfactual reasoning, and to randomisation, respectively. Experimental loopholes in state-of-the-art Bell type experiments are related to statistical issues of post-selection in observational studies, and the missing at random assumption. They can be avoided by properly matching the statistical analysis to the actual experimental design, instead of by making untestable assumptions of independence between observed and unobserved variables. Methodological and statistical issues in the design of quantum Randi challenges (QRC) are discussed. The paper argues that Bell’s theorem (and its experimental confirmation) should lead us to relinquish not locality, but realism.
In a recent note, Hance and Hossenfelder (arXiv:2211.01331) recall that "locally causal completions of quantum mechanics are possible, if they violate the assumption [called statistical independence or measurement independence] that the hidden variables do not in any way depend on measurement settings" and that, consequently, the experimental violations of Bell inequalities "show that maintaining local causality requires violating statistical independence". However, Hance and Hossenfelder also argue that "we should (...) look for independent experimental evidence that can distinguish the two different options: non-locality and statistical independence, or locality and violations of statistical independence" and that "the unwillingness to consider theories without statistical independence may be the reason we do not yet have a locally causal theory for the foundations of physics that is consistent with general relativity". Here, we recall that there is a third option, namely, rejecting that measurement outcomes are governed in any way by hidden variables. Moreover, we argue that some recent results in the search for principles singling out the sets of quantum correlations for Bell and Kochen-Specker contextuality scenarios point out that this third option is scientifically more plausible and answers the question of why and how nature produces quantum nonlocality.