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arXiv:1310.1489

Why "noncommuting common causes" don't explain anything

Dustin Lazarovici

Published Oct 5, 2013• 3 citations
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Abstract

In my commentary, I will argue that the conclusions drawn in the paper "Noncommutative causality in algebraic quantum field" theory by Gábor Hofer-Szabó (and similar publications) are incorrect. As proven by J.S. Bell, a local common causal explanation of correlations violating the Bell inequality is impossible.
DOI
10.48550/arxiv.1310.1489

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This paper is the philosopher-friendly version of our more technical work (Hofer-Szabó and Vecsernyés, 2014). It aims to give a clear-cut definition of Bell's notion of local causality. Having provided a framework, called local physical theory, which integrates probabilistic and spatiotemporal concepts, we formulate the notion of local causality and relate it to other locality and causality concepts. Then we compare Bell's local causality with Reichenbach's Common Cause Principle and relate both to the Bell inequalities. We find a nice parallelism: both local causality and the Common Cause Principle are more general notions than captured by the Bell inequalities. Namely, Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively.
Bell inequalities, understood as constraints between classical conditional probabilities, can be derived from a set of assumptions representing a common causal explanation of classical correlations. A similar derivation, however, is not known for Bell inequalities in algebraic quantum field theories establishing constraints for the expectation of specific linear combinations of projections in a quantum state. In the paper we address the question as to whether a 'common causal justification' of these non-classical Bell inequalities is possible. We will show that although the classical notion of common causal explanation can readily be generalized for the non-classical case, the Bell inequalities used in quantum theories cannot be derived from these non-classical common causes. Just the opposite is true: for a set of correlations there can be given a non-classical common causal explanation even if they violate the Bell inequalities. This shows that the range of common causal explanations in the non-classical case is wider than that restricted by the Bell inequalities.
Standard common causal explanations of the EPR situation assume a so-called joint common cause system that is a common cause for all correlations. However, the assumption of a joint common cause system together with some other physically motivated assumptions concerning locality and no-conspiracy results in various Bell inequalities. Since Bell inequalities are violated for appropriate measurement settings, a local, non-conspiratorial joint common causal explanation of the EPR situation is ruled out. But why do we assume that a common causal explanation of a set of correlation consists in finding a joint common cause system for all correlations and not just in finding separate common cause systems for the different correlations? What are the \nperspectives of a local, non-conspiratorial separate common causal explanation for the EPR scenario? And finally, how do Bell inequalities relate to the weaker assumption of separate common cause systems? \n
In the paper, the EPR-Bohm scenario will be reproduced in an algebraic quantum field theoretical setting with locally finite degrees of freedom. It will be shown that for a set of spatially separated correlating events (projections) maximally violating the Clauser–Horne inequality there can be given a common causal explanation if commutativity is abandoned between the common cause and the correlating events. Moreover, the noncommuting common cause will be local and supported in the common past of the correlating events.
The aim of this paper is to implement Bell’s notion of local causality into a framework, called local physical theory. This framework, based on the axioms of algebraic field theory, is broad enough to integrate both probabilistic and spatiotemporal concepts and also classical and quantum theories. Bell’s original idea of local causality will arise as the classical case of our definition. 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 to be locally causal. Finally, comparing local causality with the common cause principles and relating both to the Bell inequalities we find a nice parallelism: Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively.
According to a conventional view, there exists no common-cause model of quantum correlations satisfying locality requirements. In fact, Bell's inequality is derived from some locality conditions and the assumption that the common cause exists, and the violation of the inequality has been experimentally verified. On the other hand, some researchers argue that in the derivation of the inequality the existence of a common common-cause for multiple correlations is implicitly assumed and that the assumption is unreasonably strong. According to their idea, what is necessary for explaining the quantum correlation is a common cause for each correlation. However, in this paper, we will show that in almost all entangled states we can not construct a local model that is consistent with quantum mechanical prediction even when we require only the existence of a common cause of each correlation.