The British Journal for the Philosophy of Science

Stochastic Outcomes in Branching Space-Time: Analysis of Bell's Theorem

Tomasz Placek

Published Sep 1, 2000• 29 citations

Abstract

The paper extends the framework of outcomes in branching space-time (Kowalski and Placek [1999]) by assigning probabilities to outcomes of events, where these probabilities are interpreted either epistemically or as weighted possibilities. In resulting models I define the notion of common cause of correlated outcomes of a single event, and investigate which setups allow for the introduction of common causes. It turns out that a deterministic common cause can always be introduced, but (surprisingly) only special setups permit the introduction of truly stochastic common causes. I analyse next the Bell-Aspect experiment and derive the Bell-CH inequalities. I observe that we postulate there not a common cause for outcomes of a single event but rather a common common cause that accounts for outcomes of many events, where 'events' mean 'measurements with (different) directions of polarization'. Since the inequalities are violated, I claim that no causal story can be told about the Bell correlations, where causality is subliminal and restricted by screening-off condition. Similarly, given certain intuitive principles, no deterministic story can be told about these correlations.

DOI: 10.1093/bjps/51.3.445

Related papers

Branching space-time analysis of the GHZ theorem

Nuel Belnap, László E. Szabó

Greenberger, Horne, Shimony and Zeilinger gave a new version of the Bell theorem without using inequalities (probabilities). Mermin summarized it concisely; but Bohm and Hiley criticized Mermin’s proof from contextualists ’ point of view. Using the Branching Space-time language, in this paper a proof will be given that is free of these difficulties. At the same time we will also clarify the limits of the validity of the theorem when it is taken as a proof that quantum mechanics is not compatible with a deterministic world nor with a world that permits correlated space-related events without a common cause.

Jan 1, 199652 citations

On propensity-frequentist models for stochastic phenomena; with applications to Bell's theorem

Tomasz Placek

The paper develops models of statistical experiments that combine propensities with frequencies, the underlying theory being the branching space-times (BST) of Belnap (1992). The models are then applied to analyze Bell's theorem. We prove the so-called Bell-CH inequality via the assumptions of a BST version of Outcome Independence and of (non-probabilistic) No Conspiracy. Notably, neither the condition of probabilistic No Conspiracy nor the condition of Parameter Independence is needed in the proof. As the Bell-CH inequality is most likely experimentally falsified, the choice is this: contrary to the appearances, experimenters cannot choose some measurement settings, or some transitions, with spacelike related initial events, are correlated; or both.

Sep 1, 200921 citations

Bell's Theorem: What it Takes

Jeremy Butterfield

I compare deterministic and stochastic hidden variable models of the Bell experiment, exphasising philosophical distinctions between the various ways of combining conditionals and probabilities. I make four main claims. (1) Under natural assumptions, locality as it occurs in these models is equivalent to causal independence, as analysed (in the spirit of Lewis) in terms of probabilities and conditionals. (2) Stochastic models are indeed more general than deterministic ones. (3) For factorizable stochastic models, relativity's lack of superluminal causation does not favour locality over completeness. (4) If we prohibit all superluminal causation, then the violation of the Bell inequality teaches us a lesson, besides quantum mechanics' familiar ones that quantities can lack precise values and that pairs of quantities can lack joint probabilities: namely, some pairs of events are not screened off by their common past.

The British Journal for the Philosophy of ScienceMar 1, 199280 citations

Stochastic Einstein Locality Revisited

Jeremy Butterfield

I discuss various formulations of stochastic Einstein locality (SEL), which is a version of the idea of relativistic causality, that is, the idea that influences propagate at most as fast as light. SEL is similar to Reichenbach's Principle of the Common Cause (PCC), and Bell's Local Causality.My main aim is to discuss formulations of SEL for a fixed background spacetime. I previously argued that SEL is violated by the outcome dependence shown by Bell correlations, both in quantum mechanics and in quantum field theory. Here I reassess those verdicts in the light of some recent literature which argues that outcome dependence does not violate the PCC. I argue that the verdicts about SEL still stand.Finally, I briefly discuss how to formulate relativistic causality if there is no fixed background spacetime. 1. Introduction2. Formulating Stochastic Einstein Locality2.1. Events and regions2.2. The idea of SEL2.3. Three formulations of SEL2.3.1. The formulations2.3.2. Comparisons2.4. Implications between the formulations2.4.1. Conditions for the equivalence of SELD1 and SELD22.4.2. Conditions for the equivalence of SELS and SELD23. Relativistic Causality in the Bell Experiment3.1. The background3.1.1. The Bell experiment reviewed3.1.2. My previous position3.2. A common common cause? The Budapest school3.2.1. Resuscitating the PCC3.2.2. Known proofs of a Bell inequality need a strong PCC3.2.3. Two distinctions3.2.4. Szabó's model3.2.5. A common common cause is plausible3.2.6. Bell inequalities from a weak PCC: the Bern school3.3. SEL in the Bell experiment3.3.1. PCC and SEL are connected by PPSI3.3.2. The need for other judgments3.3.3. Weak vs. strong SELD4. SEL in Algebraic Quantum Field Theory4.1. The story so far4.2. Questions4.2.1. Our formulations4.2.2. The Budapest and Bern schools5. SEL in Dynamical Spacetimes5.1. SEL for metric structure?5.2. SEL for causal sets?5.2.1. The causal set approach5.2.2. Labelled causal sets; general covariance5.2.3. Deducing the dynamics5.2.4. The fate of SEL

arXiv:0708.2192The British Journal for the Philosophy of ScienceNov 22, 200758 citations

Lessons of Bell's Theorem: Nonlocality, Yes; Action at a Distance, Not Necessarily

Wayne C. Myrvold

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.

Cambridge University Press eBooksAug 31, 201615 citations

Causation and Locality

Richard Healey

By moving to the context of relativistic space-time structure, this chapter completes the argument of Chapter 4 that we can use quantum theory locally to explain correlations that violate Bell inequalities with no instantaneous action at a distance. Chance here must be relativized not just to time but to a space-time point, so that an event may have more than one chance at the same time—it may even be certain relative to one space-time point but ‘at the same time’ completely uncertain relative to another. This renders Bell’s principle of Local Causality either inapplicable or intuitively unmotivated. Counterfactual dependence between the outcomes of measurements on systems assigned an entangled state is not causal since neither outcome is subject to intervention: but it may still be appealed to in a non-causal explanation of one in terms of the other.

Oxford University Press eBooksJul 20, 20170 citations