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

Wayne C. Myrvold

Submitted to “Quantum Nonlocality and Reality – 50 Years of Bell’s theorem”

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 relativistic causal structure.

Myrvold - Bell paper

### 15 Responses

1. Travis Norsen at | |

Hi Wayne. I just read through your paper and found it very clear and thought-provoking and nice. I should probably read it again before commenting here, but I'm not sure when I'll have time, so I'll just register a couple of half-baked reactions/questions.

So, first, just to double-check that I'm understanding your terminology the way you intend, "action at a distance" means nonlocality of the specific sort that gets classified as also involving a causal influence. Right? So your overall thesis is that while Bell + experiment shows we're stuck with nonlocality, it's an open question whether the nonlocality is "causal". Is that about right?

And then, second, (and assuming I'm on the right track above), I think my main overall reaction to the paper is a kind of discomfort that you end up defining "causal" in a way that hinges on some kind of (to me) dubious/artificial/nonfundamental/shaky distinction between (something like) agent-controllable variables, and non-agent-controllable variables. That, to me, is what's so nice and natural about Bell's "local causality" (as against the PI/OI decomposition) -- it treats settings and outcomes on an equal footing, as just different beables. Indeed, Bell's fundamental formulation (from the 1990 paper, and here unlike your equation (0.9)) is phrased exclusively in terms of generic beables.

And so (third?), while I find myself sympathetic to the point that the word "causal" has misleading/inappropriate connotations in the symmetric kind of case (exhibited in relativistic collapse theories), I'm still left feeling that there isn't really any fundamental difference (vis a vis what one normally means by "action at a distance") between something here depending on a distant (controllable) "setting" and something here depending on a distant (uncontrollable) "outcome". And so it seems at the end of the day that your overall thesis (that we're stuck with non-locality but may well be able to avoid action at a distance) is largely just a kind of terminological convention, based mostly on the arbitrary decision to restrict your formal definition of "causality" to the "paradigm" type of case (where some agency/controllability is involved).

Relatedly, I suspect Bell would question your usage of the phrase "relativistic causal structure": I think you take this to be synonymous with "fundamental Lorentz invariance", whereas I think Bell took "relativistic causal structure" to be captured by "local causality" (it thus being possible for a fundamentally Lorentz invariant theory to fail to respect "relativistic causal structure", i.e., "local causality", i.e., locality -- as in the relativistic collapse theories and, arguably, Bohmian theories in which the foliation is extracted, in a Lorentz invariant way, from the wave function instead of just being posited as extra spacetime structure).

I suspect you'll have something interesting and illuminating to say in response to that. =)

Best,
Travis

2. Travis Norsen at | |

I just have a minute before running to a meeting, so I'll come back later to your two questions and just respond to your responses to my comments:

Second point: Yes, I get, from your page 2, that you don't *want* things to rest on this distinction. But don't they end up that way? Some violations of Bell's "local causality" (which is manifestly neutral between setting-type-variables and outcome-type-variables) end up counting, for you, as the causal kind of nonlocality (i.e., actions-at-a-distance) and others don't. Can you explain why?

Third point: I don't think I disagree with anything you said here, yet I can't escape a queasy feeling that something funny is going on. I probably need to re-read your paper, or at very least think about it more.

Last: My sense of Bell's usage was just based on "La nouvelle cuisine". Looking at it again now, the phrase "causal structure" appears several times, once in section 4 ("What we have to do then is to add to the laws of relativity some responsible causal structure. To avoid causal chains going backward in time in some frames of reference, we require them to go slower than light in any frame of reference.") and then four more times in section 6, which I read as leading to his presentation (in section 7) of "local causality". I'm not arguing for any particular claim here -- just reporting why I had the sense that, for Bell, "local causality" was precisely an attempt to "add to the laws of relativity some responsible causal structure".

Bell of course knew that the collapse type theories he was excited about violated "local causality" -- but he was excited about them because they seemed promising in regard to preserving fundamental Lorentz invariance. That's all I had in mind in suggesting that, for him, it seemed (and, we now know better, is) possible for a theory to be really (not just FAPP) Lorentz invariant, but nevertheless nonlocal (i.e., in violation of his attempt to capture relativistic causal structure). There is no evidence that I know of suggesting that his excitement about collapse theories, as possible ways of maintaining fundamental Lorentz invariance, made him doubt or retract or reconsider or reformulate his notion of relativistic causal structure, "local causality". Which you would have expected if, for him, fundamental lorentz invariance and relativistic causal structure were synonymous. Of course, it could be that I'm just wrong to identify "relativistic causal structure" with "local causality"...

3. Travis Norsen at | |

"The setting variables can be the target of an intervention, the outcome variables can't. This isn't about what we, as agents, can do; it's about what sorts of processes are possible."

I haven't read Woodward yet, but I don't see why we can't talk about the causal effect on Y (one of the outcomes) due to X (the other outcome). According to the kind of theory in question, there is some (stochastic) process that sets the value of X, such that different values are possible (even holding fixed all the other stuff that's relevant to Y -- here, obviously, the settings and some "lambda"). So we can compare the probabilities for a given value of Y when X=0, to the same thing when X=1. If those are different we have a violation of "outcome independence" and of course a violation of "local causality". What I don't understand is why this violation comes out as non-causal on your account. Is it just that (in your terminology from the paper) X~Y, so that a causal influence between them is ruled out? That would seem like putting the cart before the horse...

4. Travis Norsen at | |

Hi again Wayne. I thought about this a little more and re-read some of your paper, and now I feel like I understand the situation better. I think I was groping in the right direction before. So it indeed seems to me like your general description of causation in section 2 leads naturally to Bell's "local causality", rather than just one or the other of PI/OI. That is, a violation of OI would seem to establish that one outcome is a causal influence on the other outcome, in exactly the same way that a violation of PI is said to establish that one setting is a causal influence on the distant outcome.

It then seems to me that the asymmetry (i.e., your saying that violations of PI are causal while violations of OI aren't) comes in, just in the way you relate each of these cases to spacetime structure. In the case of PI, if I understand correctly, you argue as follows: a violation of PI establishes a causal influence from one setting onto the distant outcome; but these are spacelike separated so in a relativistic spacetime this should be impossible; therefore theories that violate PI are incompatible with relativistic spacetime. Whereas then in the case of OI, you seem to argue more along these lines: in a relativistic spacetime the two outcomes are spacelike separated and hence not viable candidates for causally influencing each other; so violation of OI does not imply a causal influence and hence satisfies "no action-at-a-distance".

I'm not sure if you'll agree that that's a fair summary of your arguments. Admittedly, it's only attempting to summarize *part* of your argument: in the case of OI, you also bring in the point that, assuming a relativistic spacetime, whatever we say about outcome X influencing outcome Y, we will have to say the same thing about outcome Y influencing outcome X; this symmetry is thus also at least part of your argument against classifying violations of OI as "causal". But to me that's a separable (no pun intended) issue. I think it does come out from your basic exposition of "causation" that (e.g., in your toy example) X is causally relevant to Y and vice versa. Whether one regards that as problematic seems unrelated to the main question at issue here, so I set it aside. (But perhaps I shouldn't... I look forward to your comments on all this.)

In any case, it's probably obvious, but if my summary two paragraphs up is fair, it seems troubling that one could easily just turn the tables and have the main conclusions come out the other way (if for some reason you wanted to). Here is the same thing I wrote above but with the roles of OI and PI just swapped:

A violation of OI establishes a causal influence from one outcome onto the distant outcome; but these are spacelike separated so in a relativistic spacetime this should be impossible; therefore theories that violate OI are incompatible with relativistic spacetime. Whereas ... in a relativistic spacetime each setting is spacelike separated from the distant outcome and hence not a viable candidate for causally influencing it; so violation of PI does not imply a causal influence and hence satisfies "no action-at-a-distance".

Of course, I think the only really fair thing to say is that if "action at a distance" means causal influence between spacelike separated events, then you have that with both violations of PI and violations of OI. And so if you think that "action at a distance" is incompatible with relativistic spacetime, then no empirically viable theory is compatible with relativistic spacetime. Of course, we have some reason to think that's wrong (e.g., rGRWf, rGRWm) so apparently it's just false that "action at a distance" is incompatible with relativistic spacetime.

This of course leads us back to the questions you posed earlier (about the differences between collapse/stochastic and deterministic theories, vis a vis compatibility with relativistic causal structure, relativistic spacetime, fundamental Lorentz invariance, etc.) about which, unfortunately, I don't have much to say. I find all of this confusing, largely because (surprisingly, to me at least) these various things (relativistic causal structure, relativistic spacetime, fundamental Lorentz invariance) turn out to be really different, such that it's no longer even remotely clear (to me at least) what the heck it even means for a theory to be "relativistic" in the way that we were all trained to want and expect.

5. Travis Norsen at | |

"Consider the toy example of the two boxes, with irreducibly chancy underlying physics, making no assumption about whether it's in a relativistic spacetime. The outcome variables are not the sorts of things that can be a target of an intervention---there's no process that can set them to particular values---and so they don't fall within the paradigm cases of causal relations."

There *is* a process that can (and does) set them to particular values -- it's just not a process that some agent can control. So I agree that it wouldn't fall under the paradigm case... but I thought we agreed, upthread, that the concept of causal influence shouldn't be restricted to just this kind of paradigm case (where the cause is controllable/influencable by an agent). ??

"But if we do extend our terminology in that way, we've chosen to give up the idea that causation is an inherently asymmetric notion. Whether or not we call that relation between outcomes a causal relation, it's a symmetric relation. Being a symmetric relation, it can't be the case that it requires temporal precedence. So, unlike a notion of causal relation that requires a cause to temporally precede its effect, there's no reason to claim that this relation can't hold between spacelike separated events in a relativistic spacetime."

This is where I see the circularity I was trying to point out before. The claim that the relation is symmetric *presupposes* relativistic spacetime.

6. Travis Norsen at | |

Hi Wayne, I don't understand your position here at all. Of course I understand the "free will" or "no conspiracies" assumption in Bell's theorem (I biasedly think the very best discussion of this anywhere is in the scholarpedia article I wrote with Daniel, Nino, and Shelly) but I don't see that as being relevant at all to what's at issue here. Yes, the setting variables are controllable in a way that the outcome variables aren't. But -- unless you are arbitrarily restricting the notion of "causality" to the paradigm kind of case in which the purported cause is controllable, i.e., in which the purported cause "can be the target of an intervention" -- this simply doesn't matter. I thought we agreed about this. You wrote earlier: "I don't want the notion of causality to rest on some kind of distinction between variables that are agent-controllable and ones that aren't." So you'll have to explain the distinction you clearly *do* want the notion of causality to rest on, if it's not this one. All I can understand from what you've said is that you, after all, do want to reserve the word "causality" for "paradigm type cases" in which the variable that influences something else "can be the target of an intervention", i.e., is agent-controllable. If the "i.e." there is improper -- if, for you, something being able to be the target of an intervention, and its being agent-controllable, are not at all the same thing -- then you'll have to explain the difference because I don't get it.

Maybe the following will help? You said: "Control by an agent isn't the issue; unlike the instrument settings, the outcome can't be set by any other process, external to the system, whether controlled by an agent or not." I don't understand what you mean by "other" or by "external" or by "system". Let's leave aside the (for proof of Bell's *theorem*, quite important, but here, I think, totally irrelevant) point that there is good reason to expect the physical facts which control the setting to be independent of lambda, whereas obviously this assumption would not be appropriate for the outcome (since we obviously expect that lambda, whatever that consists of for some particular candidate theory, is among the causal influences on the outcome). What remaining difference is there between the setting and the outcome? I don't see any. Both can be understood as statements about the macroscopic spatial arrangement of some physical stuff ("the system" = a set of SG magnets, say, in the one case, and an instrument pointer in the other), and in both cases there is some complicated chain of physical causation (from "other" stuff, "external" to the "system") that produces the arrangement in question. I see no grounds whatsoever for thinking that one of these macroscopic spatial arrangments of physical stuff can, and the other cannot, be treated as a possible causal influence on other things. The only difference between them is that one of them is (at least partially) controllable by human experimentalists, and the other isn't (in the same way)... but of course, as you discuss in the paper, this doesn't (have to) mean anything metaphysical about literal human free will... at the end of the day *all* it means is that (we have good reason to think) the upstream causal factors controlling the setting are uncorrelated with lambda, while (as noted/explained above) this is not the case with the upstream causal factors controlling the outcome (because lambda is *among* those upstream causal factors for the outcome!). This independence assumption is crucial for Bell's theorem, but I think it has no relevance whatsoever to a discussion about whether both, or just one, of the two "systems" here can be understood as a possible cause of other things. Concretely, physically, they just aren't any different. Which is exactly why I think it's proper to treat them on the same footing (as Bell's "local causality" does explicitly, and as your account also seems to do ... modulo terminological games).

My bottom line here is as follows. I think you give a really clear and nice account of causation in the paper. And I think it's clear that (unless you arbitrarily restrict the terminology to the "paradigm case", something that I think there are simply no grounds for if one thinks *concretely* about the kinds of physical facts that these "setting" and "outcome" variables actually stand for) a violation of OI counts as a causal influence. I mean, it just plainly does, according to how you formulated "causality". But then it seems like you start playing word games to escape this conclusion, in particular, arbitrarily demanding a weird/anthropocentric interpretation of the word "set" or "intervention".

But maybe I'm just missing something.

And then the other issue, about whether the causal influences involved in the two distant settings are "symmetric" and how this relates to relativistic spacetime. I said that your claim that they are symmetric presupposes relativistic spacetime. What I meant is really simple. I think we agree that it makes sense to require "temporal precedence" for causality -- causes must preceed their effects. Now, as I read your account of causation, a violation of OI clearly establishes that one outcome (X) influences the other (Y). Your worry is: but it *also* establishes that Y influences X. It's symmetric! But if you assume a non- (or, uh, super-) relativistic spacetime -- with some preferred foliation understood as defining a notion of "absolute time" -- then one or the other of the outcomes will be "really first", temporally. So we'd say that the application of (something like) your equation from page 2 -- both as written and with the roles of X and Y swapped -- establishes that X and Y are "causally relevant" to each other, but we wouldn't say *both* that "X causally influences Y" and *Y causally influences X". It would just be one or the other. The "really first" one (as defined by the absolute time) causally influences the "really later" one.

So: there simply is nothing like the worrisome symmetry (that you want to argue should make one reconsider, and ultimately retract, the applicability of "causality"-talk to the outcomes) unless you assume that X and Y being spacelike separated implies X~Y... the very thing that you suggest has to be given up if, instead, PI is violated. But, to me, the two cases are completely parallel. If one (violation of PI) establishes a causal influence across spacelike separation, then so does the other (violation of OI). So if we're assuming that causality requires temporal precedence, it seems to me that we have to conclude (whether it's PI or OI that's violated!) that there is some true temporal sequence, unknown to relativistic spacetime structure.

Please remember: I don't think all of this can be exactly *right*, since there are theories that seem to marry violations of local causality with fundamental Lorentz invariance (and relativistic spacetime). I'm just not yet seeing how your analysis helps me understand how this can be possible.

7. Travis Norsen at | |

Yes, that does just kind of repeat the stuff I don't understand/buy, so I guess this is a good place to stop. It's become clearer, though, that for you the ability of a variable "to be the target of an intervention" is crucial/central, and clear that you mean something by this *other* than just that the variable in question is agent-controllable. Maybe if you revise you can try to explain what you mean (by this ability to be the target of an intervention). I'd like to understand.

And then I'm still concerned about the other point as well. You wrote: "if you extend talk of cause an[d] effect to things like the outcome variables in the toy example, you thereby lose any reason to think that ... a cause must precede its effect". I think that's true only if one also demands relativistic spacetime. You could easily reconcile the extended (??) talk of cause and effect to things like outcome variables, with the idea that causes must precede effects in time, if you drop relativistic spacetime. Fundamentally, I remain confused about why (other than the agent-controllability of the one as against the other kind of variable) there's any difference between PI and OI here. If you demand relativistic spacetime, it seems you could make peace with violations of PI in the same way I understand you to be trying to make peace with violations of OI -- just tweak the wording of what counts as "causal" so you regard the violation in question as nonlocality but not action-at-a-distance. That would of course mean excluding precisely the paradigm kind of case, so it won't strike anyone as really plausible. But -- absent some explanation of what it means to be a (potential) "target of an intervention" (or some other way to really distinguish Ps from Os) -- I continue to see the two as completely parallel in all the ways that matter. So to me your denying "action-at-a-distance" status to violations of OI strikes me as just exactly as implausible as it would strike you to deny that status to violations of PI.

OK, maybe that helps or maybe it's again just repeating. In any case, thanks for the interesting discussion. Take the last word if you want it -- I won't post anymore!

8. Eric G. Cavalcanti at | |

Hello Wayne, Travis,

I'm sorry if you are already tired of this discussion, but having read your paper and the discussion above, I'd like to add a couple of thoughts, and would be very interested in your opinion, Wayne.

The main puzzle for me is still whether or not a notion of agency is at least implicitly required in order to make sense of causation. I still can't understand how we can reduce this to something about physical processes without any reference (at least implicit) to what an agent can in principle control.

In your paper you argue that interventions really don't assume anything about agency, but rather that

"When talking about causation, I will, however, assume that it makes sense to consider some process that fixes the value of X without changing the other parameters relevant to Y, and to consider the result of such an intervention on the probability of Y. This is what distinguishes a causal relation between X and Y and situations in which X is merely informative about Y".

Now, Travis suggested, and I would like to keep pursuing that line of thought, that

"There *is* a process that can (and does) set them to particular values -- it's just not a process that some agent can control."

Wayne replied that this "isn't what *I* mean by setting it to a particular value. But I don't want to get into a dispute about terminology". It is a matter of terminology, but not *just* a matter of terminology. The whole question is whether the notion can be formulated precisely *without* (implicit) reference, or without being at least motivated by, what agents can or cannot do in the world. The process described by Travis does fit the definition given by Wayne in the paper, so the question is, how can the definition be refined to make it clear that it has nothing at all to do with agents?

By the way, my working opinion on all this is that (unlike Travis) thinking about interventionist causation can indeed help conciliate quantum correlations and relativistic causal structure, but that (unlike Wayne), the notion of intervention doesn't make sense without (implicit) reference to an agent who can or cannot control things.

Best,
Eric