Causarum Investigatio and the Two Bell's Theorems of John Bell

Abstract: "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 suggest quite different conclusions, embraced by operationalists and realists, respectively. The key issue is whether Locality or Local Causality is the appropriate notion emanating from Relativistic Causality, and this rests on one's basic notion of causation. For operationalists the appropriate notion is what is here called the Principle of Agent-Causation, while for realists it is Reichenbach's Principle of common cause. By breaking down the latter into even more basic Postulates, it is possible to obtain a version of Bell's theorem in which each camp could reject one assumption, happy that the remaining assumptions reflect its weltanschauung. Formulating Bell's theorem in terms of causation is fruitful not just for attempting to reconcile the two camps, but also for better describing the ontology of different quantum interpretations and for more deeply understanding the implications of Bell's marvellous work.

• The paper demonstrates that Bell's 1964 theorem, based on separability and predetermination, challenges deterministic hidden variable theories.
• It shows that Bell's 1976 theorem reduces the discussion to local causality, revealing distinct philosophical implications for realists.
• By dissecting causal principles, the paper bridges operationalist and realist views, guiding future quantum mechanics and AI research.

An Analysis of John Bell's Dual Theorems on Quantum Phenomena

The paper "Causarum Investigatio and the Two Bell's Theorems of John Bell" authored by Howard M. Wiseman and Eric G. Cavalcanti offers a detailed exposition of the two different theorems by John Bell concerning the incompatibility of quantum phenomena with classical assumptions. The first theorem from 1964 discusses the incompatibility with the assumptions of separability and predetermination, while the second from 1976 addresses the notion of local causality. The authors argue that although these two theorems are logically equivalent, the differences in their assumptions result in distinct philosophical implications, leading to disparate interpretations by operationalists and realists.

Bell's 1964 theorem is grounded in the assumptions that the statistical predictions of quantum mechanics are incompatible with what Bell described as "separable predetermination." Here, the joint assumptions are twofold: separability or locality, and predetermination of individual measurement results. The paper discusses Bell's framing of these concepts and presents the theorem as:

1. Predetermination (1964): For certain quantum phenomena, no theory satisfying both separability and predetermination is consistent. This implies a stark difference from deterministic hidden variable theories that fail to concur with quantum mechanical predictions.

Bell's later theorem in 1976 refines the incompatibility to a single assumption: local causality. This theorem implies:

2. Local Causality (1976): There are quantum phenomena for which no theory can be locally causal while maintaining consistency with quantum mechanical predictions. This single-assumption theorem is crucial for realists who equate local causality with Einstein's principle of relativity and dismiss any outcomes inconsistent with this framework as evidence of nonlocality in the quantum field.

One of the paper's central themes is the exploration of these two interpretations through the lens of causation, which the authors dissect into more fundamental postulates and principles. They propose a version of Bell's theorem that could potentially reconcile these seemingly opposed views by deconstructing Reichenbach’s Principle of Common Cause into smaller postulates, such as the Principle of Free Choice and the postulates of local causality and predetermination. This nuanced approach provides a basis for objective discussion among scholars, avoiding the ambiguity associated with terms like "locality."

Wiseman and Cavalcanti also illustrate the belief that Bell's theorems, when framed as principles of causation, enrich the understanding of quantum interpretations and the ontology underlying them. This has practical implications as it allows researchers to address the fundamental nature of quantum mechanics beyond the operational perspective, exploring the inherent randomness and deterministic underpinnings of quantum theory and testing the constraints of Einstein’s relativity.

In conclusion, the chapter argues for a formulation of Bell's theorem that can be accepted by both operationalists and realists. By examining Bell's work through the prism of causation, the authors demonstrate that the divide between these camps arises from different understandings of causation itself. Through this detailed causal dissection, the paper offers a unique standpoint that encourages dialogue, minimizes sensationalist interpretations of locality, and aims for a comprehensive understanding of quantum phenomena within the scientific community. This formulated approach sets the stage for future research directions that may resolve some of the core philosophical tensions in quantum mechanics and explore the prospective developments in understanding causal structures prevalent in quantum theory and their implications for AI developments.