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
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.
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Alice’s detector
Figure 9.2
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.
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).
A Hidden Variable Theory 49
Alice’s detector
Figure 9.3
The spin axis of Bob’s electron points along one of the indicated directions.
Alice’s detector
Bob’s detector
Figure 9.4
Bob’s detector points in a different direction than Alice’s.
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Chapter 9
Alices’s detector
Bob’s detector
Dotted arrows show
the possible orientation
of the electron heading
toward Bob
Figure 9.5
The electron that reaches Bob has one of the indicated spin axes.
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).
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.
A Hidden Variable Theory 51
Alice’s detector
Bob’s detector
The “bite,”
which yields
agreements
The orientations
that yield
disagreements
Figure 9.6
When do Bob’s and Alice’s measurements agree?
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 60o to Alice’s. Then that dotted “bite” was also 60o in extent. Since the dashed arc’s extent was 180o, the part of it which contained spins which disagreed with Alice was 180o– 60o = 120o in extent. And this span was 120o / 180o = 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 60o, only 2/3 of them did.d
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.
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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 60o to the
right, but 60o to the left. And in th
is 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
A Hidden Variable Theory 53
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.
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
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Chapter 10
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.
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 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.
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.
Bell’s Theorem 57
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
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Chapter 10
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.
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
It was my good fortune to have met Bell. How that came about makes for
an interesting tale.
Bell’s Theorem 59
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.
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