`10 Bell’s Theorem`

Over the decades that saw the creation of quantum theory, the questions I have been discussing were central. People argued over them endlessly. I think it is fair to say, though, that the argument never reached a conclusion. Rather, as I have described earlier, it just petered out. People gave up talking about the subject.

But then John Bell came along.

In the previous chapter I tried to create my own hidden-variable theory, and I failed. I could not make it work. How about some other theory? Maybe I should abandon figure 9.1’s simple picture, so intuitively obvious, of what is going on in the spin experiments, and try something else.

Bell’s genius was to prove that the task is impossible. His theorem shows that no matter how hard I try, I will not be able to create a hidden-variable theory that agrees with all of the predictions of quantum theory. Yes, some of its predictions will agree—but there are sure to be others that don’t. And the amazing thing about his theorem is that it doesn’t deal with any particular hidden-variable theory at all. It deals with hidden-variable theories in general.

Bell’s Theorem is concerned with an EPR configuration in which the two detectors are no longer parallel—just as I had done in my attempt. In such a configuration it no longer remains the case that the two detectors always get opposite results. Sometimes they do and sometimes they don’t. Bell considered three possible configurations, in which the detectors were oriented in three possible ways. As I had done, for each of the three configurations he considered the fraction of times they disagreed. He was able to construct a specific mathematical expression involving these fractions. His thinking had nothing to do with quantum mechanics. As a matter of fact, it had nothing to do with physics. It was pure logic: a matter of analyzing all the ways a random variable can be distributed.^a

`Figure 10.1`

John Bell lectures. His famous theorem showed that no local description of submicroscopic reality could make the same predictions as quantum mechanics. On the blackboard behind him can be seen a segment of this famous theorem (at the top). Photo courtesy of CERN.

As an analogy, when you flip a fair coin it lands heads 50 percent of the time. An unfair coin, though, might be more likely to land heads than tails. The degree to which the coin has been altered is a hidden variable, which you do not know. But you do know that if you take the fraction of times the coin lands heads, and add to it the fraction of times it lands tails, you will always get the number one. In an analogous way, Bell constructed a specific combination of the three fractions from the three experiments of his scenario. And he found a definite restriction involving this combination, which each and every hidden-variable theory must obey.

That was his result. But he then went on to do something more. Bell showed that quantum theory violated this restriction.

The conclusion? That quantum theory is different from every possible local hidden-variable theory. That quantum theory is something else.

John Bell has an almost mythical status among his peers. A colleague has written that he “was called the Oracle … there was a certain aura about him.”¹ Another has written of how “Bell’s presence in a gathering raised the collective level of thinking, speaking and listening.”² A third—a friend of mine—once recounted to me how he had happened to meet Bell by accident one day and how the encounter, brief as it was, had left my friend positively breathless. The praise of one’s colleagues is the finest praise.

Bell was born in Belfast in 1928 to an impoverished family: his father was regularly in and out of work. He was one of four children. Schooling was an expense his family was able to meet only with difficulty. He began his university career not as a student but as a technician in a physics department. Members of the faculty, recognizing his talent and commitment, went out of their way to help, lending him books and allowing him to sit in on lectures. Ultimately he got an education, found employment as a physicist, and eventually moved to Switzerland and the giant particle accelerator at CERN, the European center for high-energy physics. There he worked on the quantum theory of fields and on accelerator design—and on the foundations of quantum mechanics. No one has probed those foundations more deeply.

In person Bell was graceful yet intense, and suffused with a quiet humor. These qualities shine through in his writings. These writings are graceful, passionate, powerful—and delightful. A few quotations will make this evident.

From an article of his on the mysteries of quantum theory: “The concept ‘velocity of an electron’ is now unproblematic only when not thought about.”³ And another: “The typical physicist feels that [these questions] have long been answered and that he will fully understand just how if ever he can spare twenty minutes to think about it.”⁴ From a review Bell had written—and he is referring to himself here: “Like all authors of noncommissioned reviews, he thinks that he can restate the position with such clarity and simplicity that all previous discussions will be eclipsed.”⁵ And finally a quote from a letter to a colleague who had sent him a paper proposing a revised quantum theory: “I read with very great interest and admiration your paper. … Of course I will be happy to receive any reply from you. But I hold the right not to reply to letters to be the most fundamental of human freedoms.”⁶

`Figure 10.2`

John Bell and his wife Mary at dinner with friends. Mary Bell is also a physicist: indeed, the two Bells often collaborated. Photo © Renate Bertlmann.

It was my good fortune to have met Bell. How that came about makes for an interesting tale.

In chapter 1 I referred to “a colleague who was as fascinated—and as confused—by the theory as I.” That colleague’s name is Arthur Zajonc (rhymes with “science”): his office was just down the hall from mine. As I wrote in chapter 1, we talked over the years, first casually and then more seriously. Ultimately these conversations led to a book. And at another point they led to a conference that Arthur and I organized. That was where I met Bell.

Here is what we wrote in our book about that conference.

Conferences are the stuff of life to the working scientist. The lectures provided at conferences provide an in-depth view of the latest advances in the field: often these lectures are collected and published as a book, which stands as an invaluable summary of the current state of the art. Paradoxically, however, what participants often find most valuable in a conference is not these lectures. Rather, it is what happens in the nooks and crannies lying between the formal presentations: the brief conversation over coffee, the chance encounter in the hallway, the scientific argument that erupts over dinner. We decided to organize a conference that would consist of nothing but these informal chats.

The meeting we envisaged was to be a week-long conversation, tightly focused on [the mysteries of quantum mechanics]. Attendance was to be kept low: in addition to the local physicists from the host institutions, only a limited number of the world’s foremost workers in the field would be invited. Total immersion, we decided, was an important consideration: the conferees would sleep under the same roof and eat all their meals together. It was essential that the accommodations be comfortable and the meals tasty. We decided to hold it at Amherst College, our host institution.

After eight months of planning and preparations, the conference began with a cocktail party on the back porch of what had, until recently, been a college residence hall. The school year had just ended: hard upon the heels of the departing students, a small army had descended on the building—cleaning up the mess, moving in new furniture, and putting fresh sheets on the beds. The sun shone brightly down, puffy white clouds marched across a sky of perfect blue, and the meeting’s participants enthusiastically pumped one another’s hands. Some had flown in from Europe or across the United States, others had driven for hours, and yet others had walked over from their offices. Although some collaborated regularly, others had not seen one another for years: indeed, a few had known each other only as names on scientific publications and were meeting for the first time. Suitcases stood around unattended as their owners, not bothering to carry them up to their rooms, fell deep into conversation.

The next morning, after a gourmet breakfast, we gathered around a large table downstairs. One of the participants [it happened to be John Bell that first morning] stood up to deliver a brief talk. He had not spoken for five minutes before someone interrupted him with a comment. Someone else then chimed in with a comment on the comment—and we were off.

For the rest of the week, participants gathered around the table to discuss some of the most fascinating and profound problems of modern physics. … But the conference did not take place only around that table. It also took place out in the garden, and on the streets of Amherst as two or three participants would break away from the group and wander off for a stroll. It took place over meals, which were invariably huge and invariably delicious. The conversations would zip from topic to topic with astonishing rapidity—from the implications of a recent experiment to the standings of the New York Mets, from culinary delights to a possible new theorem four of the participants thought they had just discovered (those four skipped going out to a movie with the rest of us one evening, stayed up to all hours, and eventually decided that their promising new approach was probably no more than a dead end …).

At one point [during a picnic] Greenstein spotted John Bell and [another participant] off in a corner … that afternoon they had squared off. For years Bell had probed with astonishing brilliance and depth the foundations of quantum theory, and he has argued that the theory is plagued by fundamental inadequacies. [The other participant], in turn, has argued with great subtlety that the mysteries of quantum mechanics have been widely exaggerated, and that in reality the theory poses difficulties no deeper than those raised by many other branches of physics. That afternoon their debate had risen to a passionate intensity.

Greenstein grew perturbed. [The two] stood with heads together, isolated from the rest of the throng. Were they at it again? Had emotions risen so high that they had grown furious with one another? Greenstein sidled unobtrusively over to eavesdrop—and found them quietly comparing their cameras.⁷

Tragically, Bell died, entirely unexpectedly, shortly after the conference.

As perhaps you can tell, Bell made an extraordinary impression on me. His presence was both gigantic and gentle. I felt that I was in the presence of someone who thought more deeply, more intensely, and more honestly than most—and who was at the same time among the most considerate people I have ever met.

Never have I encountered a person more committed to getting to the heart of a matter, and to a ruthless clarity and honesty. He was passionate in his argumentation. This was so even when he was an undergraduate. During his very first course in quantum mechanics its mysteries had disquieted him—as did what he regarded as too cavalier an attitude toward these mysteries on the part of the teacher. He remonstrated with that teacher. According to Bell’s own testimony years later, he got into a heated argument, essentially accusing the professor—a man who had gone out of his way to assist him—of dishonesty.

And yet John Bell was one of the gentlest people I have ever met. He was considerate, respectful, gracious. He would fight over ideas, but the fights were never personal. You could disagree with him without angering him. And in some remarkable way, he could attack your ideas without attacking you. John Bell was a gentle battler.

Bell’s work was pioneering. He showed the way to a new insight. His was the first step—and the first step is always the hardest one. You may have noticed that I have made no attempt to show the actual theorem Bell had proved. Rather, I have talked about it only in general terms. The reason is that his combination of mathematical quantities is something of a complicated mess, and the proof that they obey his restriction is more complicated still. But in the years following on Bell’s work other people have proved theorems analogous to his, going beyond his work in various ways. In particular, a generalization of his work has been found that is so simple and straightforward that it is actually possible to give the proof in a nontechnical book. This “Bell-like theorem” is described in the appendix.

Bell’s combination of mathematical quantities is, to me at least, quite strange. Nothing about it has any clear and simple interpretation, any immediately obvious intuitive significance. It just happens to be true that this particular combination of fractions happens to have the property he found. Indeed, my guess is that nobody would have found his result even slightly interesting—were it not for the amazing fact that quantum theory makes a different prediction.

How on earth did Bell do it? What led him to formulate this particular mathematical expression, and to ask whether it agreed with the predictions of quantum theory? I know of no writings in which Bell explained the train of thought that lead him to his theorem. I know of no one who asked him.

But I can hazard a guess as to what Bell might have replied to such a question. He might have said the same thing that every creative person would say: that it involved work—lots and lots of work, months and years of immersing himself in the problem, of wrestling with it day in and day out, living with it morning, noon, and night. He might have said that he used one approach and then another, trying this and trying that, groping about in the darkness—but a darkness that, as the effort wore on, seemed to be slowly lifting. He might have said that he sought to identify what was good about an approach that almost worked, and what was bad about one that came nowhere near working, and that he modified his various attempts accordingly. He might have said that the final result seemed to have come to him in a flash (but only after endless slogging). Or he might have said that it came to him gradually, incrementally, with agonizing slowness.

But most of all, I imagine, he might have leaned back in his chair and smiled, and said that in the last analysis he really did not know how he had done it.

a. In this work, Bell made the same locality assumption as had Einstein, Podolsky, and Rosen 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. So his theorem applied to local hidden variable theories only.

Notes