On Cartwright's models for EPR

Abstract

We assess Cartwright's models for probabilistic causality and, in particular, her models for EPR-like experiments of quantum mechanics. Our first objection is that, contrary to econometric linear models, her quasi-linear models do not allow for the unique estimation of parameters. We next argue that although, as Cartwright proves, Reichenbach's screening-off condition has only limited validity, her generalized condition is not empirically applicable. Finally, we show that her models for the EPR are mathematically incorrect and physically implausible.

Introduction

This paper deals with two claims of Nancy Cartwright: (1) that the so-called screening-off condition of (Reichenbach, 1956, pp. 157–160) is overly restrictive and, in particular, should not be imposed on models of the EPR, and (2) that the EPR can be correctly described by her causal models with a generalized screening-off condition.¹ To appreciate the weight of these claims, and relate them to the foundational debate of quantum mechanics (henceforth abbreviated as QM), let us recall some facts.

QM predicts the violation of some testable inequalities (the Bell inequalities, henceforth BI). Leaving aside a few dissenters,² the consensus has it that the inequalities are also experimentally violated.³ A typical test of BI involves the measurement of spin components on far separated particles, each pair of particles being prepared in the singlet state and there being two alternative polarization directions to be selected in each, say ‘left’ and ‘right’, apparatus. Moreover, in each apparatus a polarization direction is selected just an instant before an incoming particle enters the apparatus, and the selection is governed by a semi-random process. Importantly, results at two wings are correlated, which is the fact (Einstein, Podolsky, & Rosen, 1935) exploited in the EPR argument, and then (Bell, 1964) used to prove that local factorizable hidden variable models cannot account for empirical predictions of QM.

Now, BI are derivable from a set of rather intuitive or physically motivated premises, but, since BI are most likely violated, at least one of the premises must be false. Although all suspects have been tried, the condition known as ‘factorizability’⁴ has perhaps most often been implicated, and factorizability is in turn equivalent to the conjunction of two more familiar conditions, Hidden Locality and Screening-off.⁵ Hidden Locality says that a result at one apparatus and a polarization direction set in the other (remote) apparatus are probabilistically independent, if conditioned on any admissible value of a hidden variable. Screening-off identifies a value of a hidden variable (or, more physically, a hidden state that this value parameterizes) with a screener-off: conditioned on any particular value of the hidden variable and polarization settings on the left and right, a result on the left and a result on the right are probabilistically independent. Although this is a somewhat controversial issue (see Jones & Clifton, 1993), the two conditions appear to be disparate: Hidden Locality rests on the requirement of no faster than light causal signalling, whereas Screening-off stems from an epistemological ideal that ‘correlations should be explained’.⁶

Be that as it may, the logic of Bell's theorem opens the possibility of retaining Hidden Locality, while rejecting the Screening-off condition. Moreover, if you think (as we do) that there are grounds to prefer the former over the latter, this is a natural option to take. Then Cartwright's project appears attractive, since it aims at implementing this possibility. She first offers a uniform account as to under what conditions Screening-off is applicable. Then she produces an empirically adequate model for the EPR, in which Screening-off is not satisfied, the explanation being that some conditions of the model make it inapplicable. Still, correlations between remote results are explained in terms of a common cause, which technically means that a generalized screening-off condition is satisfied. And, since the usual Screening-off condition fails, and accordingly one premise of Bell's theorem is missing, it seems that the model is not committed to BI.

In short, we have here the best bargain ever offered in the debate over the EPR and BI: an empirically adequate local and causal model for the EPR, with correlations traced back to workings of a common cause, and an account of why Reichenbach's Screening-off condition should give way to a more general principle. Unfortunately, there is a catch in this bargain, which we are going to expose. Our main point is that Cartwright's models are both mathematically wrong and physically implausible. In addition to this, her models for studying causal relations lose the power of causal linear modeling of econometrics. The latter method can be used to test for causal relations, whereas Cartwright's models simply postulate them.

We said above ‘unfortunately’ since we as well opt for retaining Hidden Locality, while rejecting Screening-off: the branching models of Kowalski and Placek (1999) and Placek (2000) that one of us helped to develop, precisely show how local models without Screening-off should be built. The failure of Cartwright's models is not troublesome from our perspective, though. Seeing how frugal her models are, one could expect the failure, indeed. To compare, her framework merely allows for temporal ordering, whereas Bell's theorem requires a rather subtle mixture of modal, probabilistic, and spatiotemporal notions.

Our paper splits into two large parts. The first is concerned with Cartwright's models and her proofs of Screening-off and a generalized version of it, whereas the second assesses her models for the EPR. To set the stage, Section 2.1 shows how ancestors of Cartwright's models, econometric linear models, can be used to reveal causal relations behind statistical data. Section 2.2 exhibits two kinds of Cartwright's models, quasi-linear and propositional, as we call them, and shows that they lost the mentioned virtue of econometric modeling. Section 2.3 comments on Cartwright's proofs of Reichenbach's screening-off condition and a generalized screening-off condition, our complaint being that the latter is not empirically applicable. Turning next to the EPR, we begin in Section 3.1 with criticizing a quasi-linear models for the EPR, and then (Section 3.2) we move to Cartwright's main (propositional) model. Section 3.2.1 shows than that the model is mathematically wrong and the final Section 3.2.2 argues that any attempt to locate the events of the model in space-time makes the model either conspiratorial, or commits it to BI, or implements a physical scenario that most likely will be experimentally rejected. The paper ends with Conclusions, and the appendix contains technical material from Section 3.2.1.