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arXiv:1201.5848 · Journal of Mathematical Physics

Noncommuting local common causes for correlations violating the Clauser–Horne inequality

Gábor Hofer-Szabó, Péter Vecsernyés

Published Nov 6, 2012• 19 citations
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Abstract

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.

DOI
10.1063/1.4763468

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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.
States in algebraic quantum field theory “typically” establish correlation between spacelike separated events. Reichenbach's Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions VA and VB, respectively, there is a local projection C not necessarily commuting with A and B such that C is supported within the union of the backward light cones of VA and VB and the set {C, C⊥} screens off the correlation between A and B.
States in algebraic quantum field theory ”typically” establish correlation between spacelike separated events. Reichenbach’s Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions VA and VB , respectively, there is a local projection C not necessarily commut- ing with A and B such that C is supported within the union of the backward light cones of VA and VB and the set {C, C⊥ } screens off the correlation between A and B.
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
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.
In this paper, we revisit the concept of noncommuting common causes; refute two objections raised against them, the triviality objection and the lack of causal explanatory force; and explore how their existence modifies the EPR argument. More specifically, we show that 1) product states screening off all quantum correlations do not compromise noncommuting common causal explanations; 2) noncommuting common causes can satisfy the law of total probability; 3) perfect correlations can have indeterministic noncommuting common causes; and, as a combination of the above claims, 4) perfect correlations can have noncommuting common causes which are both nontrivial and satisfy the law of total probability.