Lessons of Bell’s Theorem: Nonlocality, yes; Action at a distance, not necessarily

Contents 1 Introduction. page 1 2 Does relativity preclude action at a distance? 2 3 Locally explicable correlations. 5 4 Correlations that are not locally explicable 9 5 Bell and Local Causality 11 6 Quantum state evolution 13 7 Local beables for relativistic collapse theories 17 8 A comment on Everettian theories 19 9 Conclusion 20 10 Acknowledgments 20 11 Appendix 20 References 25 1 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 required 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 relativis-tic 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 noncontrover-sial, 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 theory with probabilistic laws that involve irreducible correlations. The sort of non-locality 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 visa -vis relativity. Though a deterministic theory can achieve an appearance …