Special Relativity and the Bell Theorems

The Bell Theorems, together with a collection of experimental results (such as those of Aspect et al.), provide good statistical evidence that quantum theory is "non-local." Roughly, this means that the interaction between two bodies in quantum theory doesn't necessarily get weaker as those bodies become spatially separated.

Is this a problem for Special Relativity? That depends on what you think Special Relativity means. Here's a simple flow-chart illustrating some of what's at stake.

For a very accessible view on how we should navigate many of these options, I highly recommend Tim Maudlin's excellent book on the subject. But here are a few thoughts on each of the steps.

• Bell-inequality violation. There is a sect of conspiracy theorists who aren't convinced that the Bell-inequalities are violated by experiment. If that's you, then there's no reason to worry about Special Relativity.
  
• Minkowski Geometry. If Special Relativity requires only that the background spacetime be Minkowski spacetime, then there is no problem for non-local quantum effects. After all, we have plenty of matter theories (quantum field theories) that take place on such a background, and even respect its symmetries to a certain extent. Non-locality is not a problem here.
• Upper limit on the speed of mass-energy transfer. We would normally like to add that matter-energy cannot be transferred faster than the speed of light. But this is not a problem for quantum non-locality, either -- unless you adopt a pretty unusual view of matter-energy transfer. Then what matters is statistical correlation -- see below.
• Signal/Information Transfer. These terms are a bit vague, and people disagree about how to explicate them. However, as the chart suggests, I think that what's really important is whether or not you think there are consequences for the statistical behavior of distant regions.
• Statistical Correlation. This seems to be the heart of the problem. If you think that Special Relativity implies an upper limit on the "speed" at which statistical correlation can occur, then you'll think the Bell-type results violate this. What I mean by that is: interactions in one region can have near-immediate consequences for the statistical behavior of another region, no matter how far apart the two regions are.

But why would someone answer "yes" to the last choice in the chart? Why should we think that Special Relativity implies anything at all about the statistical behavior of matter?

There is no probability measure in SR. Of course, matter satisfying the assumption of local realism appears consistent with Special Relativity, and the Bell inequalities hold for such matter. But I see no reason to think that such matter is required by Special Relativity. If it isn't, then Special Relativity isn't enough to derive the Bell inequalities, and doesn't contradict non-locality.

And that's exactly how we all like it. Right?

How Special Relativity Thwarts Eternalism (And More)

Image Credit: John D. Norton
INTRODUCTION. The image above illustrates a well-known argument due to Rietdijk and Putnam, which says that Special Relativity implies Eternalism (also called the "Block Universe" view). I recommend John Norton's exposition if you're not already familiar with the argument.

Norton has pointed out that the Rietdijk-Putnam argument requires assumptions that are not implied by Special Relativity (I review this in the Part I.) But the plight of the Eternalist is worse than that. After observing that observer-dependence is an essential part of the Eternalist claim (Part II), I'll show that Special Relativity actually implies that there is no fact of the matter about Eternalism (or the "determinate present" itself, for that matter), since observers disagree about the Eternalist claim (Part III).

PART I: SR Does Not Imply Eternalism. Eternalism is the view that all events in the past and future have a kind of Platonic or "determinate" existence. But Norton points out there are two hidden assumptions in the Rietdijk-Putnam argument for this view. They are:

1. IF two events lie on the same hypersurface of simultaneity, THEN they are equally determinate; and


2. for all events e₁, e₂, e₃: IF e₁ and e₂ are equally determinate and e₂ and e₃ are equally determinate, THEN e₁ and e₃ are equally determinate (transitivity of determinateness).

These assumptions seem to be independent of the theory of Special Relativity (SR). Therefore: SR does not imply eternalism.

PART II: Determinateness is Observer-Dependent. Notice that assumptions (1) and (2) only allow one to say that two events are equally determinate. There's not yet a way to say that any event actually is determinate. So if the case for Eternalism is to be made, a third assumption needed, that

3. There exists at least one event that is determinate.

How might one establish (3)? The only way to do it is through an observer. If a property of spacetime (like "determinateness") isn't in principle accessible by some observer, then we have good reason to suspect that it's meaningless.

Fortunately, a precise kind of "observer-dependence" is already built into the Eternalist view, and into the view of (her nemesis) the Presentist. Both agree that an observer's experience of the event "the present now" is required to first establish that an event is determinate. Therefore: both the Eternalist and the Presentist have only established the ability to say that an event is determinate for some observer.

In order to establish the Eternalist claim (that Minkowski spacetime is factually determinate), one would have to claim that spacetime in Special Relativity is determinate for all observers.

PART III: SR Thwarts Eternalism. Minkowski spacetime (the spacetime of Special Relativity) is not determinate for every observer. In fact, for every event e in Minkowski spacetime, there are observers who disagree about whether or not e is determinate. For example, consider Alice:


(In this spacetime diagram, the vertical axis is time, the horizontal axis is space, and c = 1.) Alice accelerates uniformly from v = -c to +c over the course of her lifetime. Her hypersurfaces of simultaneity are indicated in red. But none of Alice's hypersurfaces of simultaneity intersect any event in regions I or II. So Alice will claim that regions I and II are indeterminate.

On the other hand, suppose Bob travels with constant velocity for all of time. Bob's hypersurfaces of simultaneity will collectively cover all of Minkowski spacetime. So Bob will conclude (by way of assumptions 1-3 above) that all events (and thus also the in regions I and II) are determinate.

CONCLUSION. Determinateness cannot be an objective property of spacetime, because observers disagree whether or not a given event is determinate. As a result, there is no fact of the matter about Eternalism, either.

This conclusion falls straight out of the assumptions (1-3) above, which were adopted by the Eternalist in the Rietdijk-Putnam argument. So insofar as one accepts these assumptions, Special Relativity not only thwarts Eternalism: it implies that the very notion of a determinate event is bogus.