9 A Hidden Variable Theory
It was many years ago that I first encountered the Great Predictor.
I was thrilled to meet him. I’d been looking forward to the encounter for years. The Predictor was famous—world-famous. He was legendary for the number of his predictions, and for their amazing accuracy. Many people had relied on those predictions, and always with profit.
What intrigued me the most, however, was how bizarre were some of his predictions. “Tomorrow you will be in two places at once” was one. “On Wednesday an event will occur for which there is no cause” was another.
How could such things be? I was captivated by the strangeness of these prognostications. Could such weird things really come to pass? That’s why I had been so anxious to meet the Predictor. For years I had looked forward to finally getting to know him.
At long last I was getting my wish. I was twenty years old, and I was thrilled. I was sitting in a classroom, in college, on the first day of a course called Introduction to Quantum Mechanics.
That was many long years ago. And throughout my career I have maintained my early fascination with quantum mechanics. Somehow, though, I never felt that I really understood the theory. It always sat lodged in the back of my mind—enigmatic, mysterious, enticing. Over and over again, I found myself thinking that someday I really ought to go back and figure it all out, and finally put all those early juvenile confusions to rest.
Part of that project was an effort to understand Bell’s Theorem. To be honest I found myself dreading getting to work on that particular topic. While I had never felt comfortable with quantum mechanics in general, Bell’s Theorem was a topic that I felt positively unnerved by. Over and over again I had tried to master it, and over and over again I had failed.
Eventually I did come to some sort of understanding of Bell’s work. I recall feeling pretty pleased with things … until the fateful day when I looked at my reflection in the mirror—and this is literally true—and I spoke aloud. “Greenstein,” I said to my reflection, “you were just kidding yourself, weren’t you? You never really understood Bell’s Theorem at all, did you?”
“Time to get going,” I told my reflection.
And I did.
What did I do? I read some books. I read some scientific articles. Among all my readings, one stood out: an article provocatively entitled “Is the Moon There When Nobody Looks?,” whose explicit function was to make Bell’s Theorem as accessible and as comprehensible as possible. I found it wonderful: immensely readable and immensely informative.a I read that article—not just once, but over and over again. It was helpful … and yet, no matter how many times I returned to it, I still felt that something was eluding me. Some central insight, some clear understanding I still found out of reach.
Eventually I realized the error I had been making. I had been trying to adopt someone else’s thoughts and make them my own. And then I realized that it was time for me to stop heading down this other person’s path, and strike out by myself. It was time to take seriously the central mystery, and try to think about it in my own personal terms.
I decided to tackle the famous and intimidating EPR paradox.
At first glance it might be hard to see what the fuss over the EPR Paradox is all about. Why do people even call it a “paradox?” Perhaps you, the reader, feel this way. After all, is it not obvious what is going on in the EPR thought experiment? Is it not clear that the source of electrons, however it may work, is merely shooting out a pair of particles, one flying toward Alice and the other toward Bob, set spinning in opposite directions (figure 9.1)?
Naive view of the entangled state. It makes intuitive sense, but it will turn out to be wrong.
Figure 9.1 illustrates the underlying reality that we might think our Great Predictor sees, but of which he so adamantly refuses to speak. And isn’t this simple picture all we need to explain the fact that Alice’s and Bob’s results disagree?
I decided to take seriously this picture, and see how well it dealt with the EPR scenario. I decided to see if I could create the very hidden-variable theory that Einstein sought, and Bohr declared impossible—a theory that made the same predictions as quantum theory, but that went further and described in full detail the workings of the microworld. I would create my own predictor—a new predictor, a loquacious predictor, one who was willing to speak up: a “Greater Predictor.” Notice that in doing so I was doing just what quantum theory failed to do.b
How far could I go in this project? I was not so naive as to think that I was going to succeed. As you can guess, my project failed. I did not end up accomplishing what right then seemed so easy. So am I wasting your time as I lead you through this exercise? I am not—because the reason I failed will turn out to be very instructive indeed.
Before we begin, a brief word. If mathematics makes you nervous, just skip over the details of what follows and resume reading toward the end of this chapter. But perhaps you might want to resist this urge, and just read on. Maybe it won’t be so very terrible after all!
A difficulty facing me at the outset was that a detector never told the precise direction a spin pointed. All it could reveal was whether the spin pointed more or less along or against the detector’s reference direction. If Alice’s detector found, let us say, against, the only thing I knew was that the spin of her electron must have one of the configurations illustrated by the dotted arrows in figure 9.2.c
The spin axis of Alice’s electron points along one of the indicated directions.
But I knew little more than that: any one of the dotted arrows might be the spin axis of the particle she had just detected. Her detector did not say which.
But there was one more thing I did know: since within my hidden-variable theory the spin of the particle heading toward Bob was opposite to that of Alice, its spin must have one of the configurations shown in figure 9.3.
The spin axis of Bob’s electron points along one of the indicated directions.
And since Bob’s detector lay in the same direction as Alice’s, he was sure to find the spin of the particle heading toward him to lie along his detector’s reference direction. So his measurements would invariably disagree with those of Alice. The electrons were behaving just like the angry couple I described in the EPR chapter.
This was a prediction of my hidden-variable theory—and it was just what the EPR thought experiment revealed. My theory was doing well so far: its predictions were the same as those of quantum mechanics. So maybe I had succeeded in my project of developing the hidden-variable theory that underlay quantum theory.
But only so far—and this was where the genius of John Bell came into play. Bell added a new twist to the EPR scenario. He asked what would happen were Bob to swing his detector about, so that it was no longer parallel with Alice’s (figure 9.4).
Bob’s detector points in a different direction than Alice’s.
Would the predictions of my hidden-variable theory still match those of quantum mechanics?
Suppose that Alice happened to find her electron’s spin to be against the direction of her detector. Then the other electron’s spin would lie along it. Of course, that electron was not heading toward her: it was heading toward Bob. When he measured it along his new direction, what result would he get? Would he continue to disagree with Alice so regularly? Or was the angry couple in our analogy starting to find at least a few areas of agreement? I needed to figure out how many times Bob’s and Alice’s results disagreed. And I needed to compare the result with that predicted by quantum theory.
There’s no need for secrecy: I found that my prediction failed to match that of quantum mechanics. So I had failed in my quest to improve on quantum theory. Let us see how this comes about.
In my hidden-variable theory there were many possible orientations the spin of Bob’s particle might have (figure 9.5). Were they along or against his detector’s direction? Well, some were the one and some were the other (figure 9.6).
The electron that reaches Bob has one of the indicated spin axes.
When do Bob’s and Alice’s measurements agree?
In figure 9.6 the possible spin directions that led to a disagreement between Bob and Alice were those that lay in the dashed arc. But not all of the possible spins lay within that arc! There was a “bite” taken out of it. So there would be some agreements—those configurations in which Bob’s arrow lay in the bite.
Just to be specific, I imagined that Bob’s detector lay at an angle of 60° to Alice’s. Then that dotted “bite” was also 60° in extent. Since the dashed arc’s extent was 180°, the part of it which contained spins which disagreed with Alice was 180°–60° = 120° in extent. And this span was 120° / 180° = 2/3 of the full span.
Whew! But that was my result: of all the possible directions the spins may point, 2/3 of them led to a disagreement between Bob’s measurement and Alice’s. So now I had my hidden variable theory’s prediction: when the detectors had been aligned, all measurements disagreed—but if they were tilted by 60°, only 2/3 of them did.d
But this was not what quantum theory predicted. Quantum theory predicted that they would disagree more often than that: 3/4 of the time, in fact.
So what had once seemed so easy had suddenly become less so. My intuitively obvious picture of the EPR situation—so simple, so clear—turned out to have failed. That picture was making different predictions than quantum mechanics. If I wanted to find the deeper theory underlying quantum mechanics, if I wanted to add to quantum mechanics by describing the hidden variables it so conspicuously fails to depict—then I had not succeeded.
Should this bother me? Maybe I could modify my picture, adding a new element here and getting rid of one there, and succeed in creating a theory that perfectly reproduced all the predictions of quantum mechanics. Could I do this?
I already knew what I had to do: I needed to find a way of increasing the number of disagreements between Alice’s and Bob’s results. To do this I needed to modify the configuration of the spins of the two particles heading toward Alice and Bob as they left the device that created them—modify it in such as a way as to reduce the number of dotted arrows that lay in the “bite” in figure 9.6. Was this so hard to arrange?
Not at all. Indeed, it was easy. I just had to rearrange the orientations of some of the electrons as they zipped off from the electron gun.
But not so fast! Bob might have oriented his detector, not 60° to the right, but 60° to the left. And in this case, that strategy would have exactly the wrong effect—not of increasing the number of disagreements, but of decreasing it.
The problem with this attempted fix was that Bob was free to orient his detector in any way at all, and whatever strategy I adopted was sure to work for only some of the orientations he might have chosen. No strategy would work for every possible choice. My theory could not be adjusted to yield the same behavior as quantum theory.
Before, I had likened the EPR state to an angry couple, intent on disagreeing with one another when asked the same question. Our new situation is more complicated: now each could be asked one of many different questions—sometimes the same, sometimes different, but questions they have no way of anticipating. In order to reproduce the predictions of quantum mechanics they would have to synchronize their replies, sometimes agreeing and sometimes disagreeing. But they can’t: neither of them knows which question is asked of the other, nor does either know the reply given. So they have no way to synchronize their replies appropriately.
But they do it anyway.
So I was glad that I had gone through my little exercise. It had not really failed at all—for I had never honestly thought that I would succeed in creating a theory with which to challenge one of the greatest achievements of twentieth-century thought. Indeed, that had never been the point. The real point of the exercise had been to make clear to me just how enigmatic quantum mechanics really was. For at first it had all seemed so simple … and now I realized that it was not so simple at all. Indeed, suddenly it was profound. For now I realized that quantum mechanics was making a prediction for which there was no sensible explanation.
a. You will find a reference to this article in the “Further Reading” appendix.
b. The theory was going to make the same “locality” assumption that Einstein, Podolsky, and Rosen made in their famous EPR paper—an assumption so obvious that they did not even bother to discuss it, but one that we will need to devote a whole chapter to later on.
c. In what follows I am going to do a simplified analysis in which the spin arrow always lies in the plane of the page.
d. I got this result based on Alice’s having gotten the result against. Had she gotten along, the result turned out to be the same. It’s not surprising, since the two situations are perfectly symmetrical.