Retrocausality

In the last chapter I discussed retrocausality at some length. I want to say still more about it, because it is a very important subject and, for many people, an unfamiliar one. Many textbooks on quantum mechanics never even mention it. My goal in this chapter to make up for that and give you a better understanding of it.

Rather than worrying about the details of specific retrocausal theories, I am going to focus on general principles that apply to lots of them. For purposes of this discussion, let me define a specific (but quite large) class of theories in which retrocausality is possible.

First, I assume the theory is governed by a set of differential equations that must be satisfied at every point in space and time. Newton’s second law, Maxwell’s equations, and Schrödinger’s equation are all examples of such equations.

Second, I assume the theory is both deterministic and time reversible. This means that in every physical process, there is a one-to-one correspondence between initial states and final states. Every initial state leads to a unique final state. Every final state can be traced back to a unique initial state. Classical mechanics and classical electrodynamics both satisfy this requirement. The Schrödinger formulation of quantum mechanics does not, because the Born rule is not deterministic. However, there are other formulations that do fit into this category, such as the two-state vector formalism.

In any theory of this sort, you can approach solving problems in a few different ways:

• As an initial value problem. In classical mechanics, for example, if you know the position and momentum of every particle at one point in time, you can calculate their behavior at all later times.

• As a final value problem. Given the position and momentum of every particle at one time, you can also trace backward to calculate their behavior at earlier times.

• As a boundary value problem. Given partial but incomplete state information at two different times, you can solve for the set of trajectories (or perhaps the unique trajectory) satisfying both boundary conditions.

Let’s consider a typical experiment of the sort used to test Bell’s inequality, and see how these considerations apply to it. An emitter produces two electrons. The physics of the emission process requires them to have opposite spin, but does not require their spins to be aligned with any particular axis. Each electron travels to a different detector that measures its spin along a different axis.

How would you go about computing the probability of the electrons having their spins aligned with a particular axis? What about the probability of each detector measuring a particular result? You might want to pause for a moment and think about these questions before reading further.

In one sense, this is a trick question. Remember that we assumed a deterministic theory. There are no probabilities in a deterministic theory. The universe is what it is. Each electron has whatever spin it has. Each detector produces whatever result it produces. There was never a chance of it producing any other result. In a deterministic theory, each time you perform a measurement you learn something about the universe you live in. It is one in which the first detector registers spin up and the second detector registers spin down. It always was. You just didn’t know it until now.

But of course, we do talk about probabilities even when dealing with deterministic theories, so there must be another sense in which the question means something. When we do this, we are invoking the formalism of statistical mechanics. We start by choosing a statistical ensemble, a set of microstates for the system that are all consistent with our macroscopic knowledge about it. We then invoke the postulate of equal a priori probabilities, which assumes that every microstate in the ensemble is equally probable. Here is what we really mean when we talk about the “probability” that something is true: “For what fraction of the members of some specified statistical ensemble is it true?”

The answer to this question depends on your choice of ensemble. Often, it is not obvious which is the “right” one to use. The choice of ensemble encodes subtle assumptions, both about the details of the experimental procedure and about the physics of the system being studied. Different ensembles can lead to very different conclusions.

In our electron spin experiment, one obvious choice is to treat it as an initial value problem and choose the ensemble solely based on the emission process. The emitter does not require the spins to have any particular axis, so we assume all axes are equally probable. This choice of ensemble leads to Bell’s inequality.

But there are other possible choices. We also could treat it as a boundary value problem. We know what result each detector produced, so we could restrict the ensemble to only those microstates consistent with those detections. This choice does not lead to Bell’s inequality. Instead, we conclude that the spins are more likely to be aligned with some directions than others, and that the distribution depends on our choice of axes for the detectors. This is retrocausality.

That is the essential thing to understand about retrocausality. It has very little to do with the equations governing time evolution of the system. It can appear in a huge class of theories, including very familiar ones like classical mechanics. The essence of retrocausality is what statistical ensemble you use to compute probabilities.

This raises an obvious question: which is the right ensemble to use? Of course, there is a trivial answer: the right ensemble is the one that leads to correct predictions. But that is not a satisfying answer. It tells us nothing about the more fundamental question we were really asking, why does one choice lead to correct predictions?

I cannot answer that question. It relates to one of the great mysteries of physics: why do we live in the particular universe we do? Of all the infinite possible solutions to the equations of motion, how did one solution get selected as our universe? Any comprehensive theory needs to provide an answer to that question. For example, maybe the universe minimizes a global action functional integrated over all of space and time. But right now we do not know what theory is correct, so we cannot answer this question.

Ultimately it must be answered based on experimental evidence. Retrocausality leads to measurable effects, so in principle it could be testable. For example, it allows Bell’s inequality to be violated. But as we saw in the last chapter, other mechanisms can also lead to that result, such as non-locality, faster than light communication, and superdeterminism. Bell’s inequality is too blunt an instrument to distinguish between them. We will need to look further for conclusive evidence.

There is one final important point I want to emphasize. You must never expect to see retrocausal effects in macroscopic systems, only in microscopic ones. Our local region of spacetime is dominated by an entropy gradient that creates an arrow of time. It is rooted in the big bang, a point of extraordinarily low entropy. As you move away from the big bang, entropy increases steadily.

That is why we intuitively believe that time is asymmetric. In the macroscopic world we are used to, processes only happen in one direction. Windows are easy to break and very hard to repair. Friction causes objects to slow down, never to speed up. Our brains consume energy and use it to grow new neural connections that encode memories. Each of these processes involves a change in entropy, and therefore can only happen in one direction.

In any macroscopic system, the subtle effects of retrocausality would almost certainly be swamped by the much larger entropic effects. It is unlikely that retrocausality could ever be detected in anything except very simple microscopic systems where entropy plays a much smaller role.