5 Spin
The simplest exposition I know of what came next involves electron spin.
Every rotating object possesses an axis of spin, represented as an imaginary arrow. Normally you can fully specify the orientation of that arrow: the spindle of a top, for instance, might tilt off to the right by so-and-so many degrees from the vertical.
Notice that in giving this specification I have made use of two different reference directions: one running right-and-left and one up-and-down. Fewer reference directions would not be enough to fully describe the orientation of the arrow: it would not be sufficient to say that the spin made such-and-such an angle to the vertical, had I neglected to also mention that it leaned right instead of left.
But quantum mechanics speaks a language all its own, and it is a strangely limited one. That language possesses no means of giving both specifications. Quantum mechanics can describe the fact that an electron’s spin is up as opposed to down—but in such a configuration the theory is incapable of telling us whether it leans right or left. Alternatively, the language of quantum mechanics can express the fact that an electron’s spin points to the right, but it cannot then specify the vertical component. If quantum theory is a language, it is an impoverished one, incapable of expressing many things.
Perhaps we should just try harder. Maybe with a little more work we could cook up a quantum-mechanical description of an electron with its spin arrow pointing in a definite direction. Unfortunately, no matter how hard we try we find ourselves unable to find such a specification—and indeed, it can be shown that the mathematical structure of quantum theory is such as to prohibit such descriptions. It is the Heisenberg uncertainty principle all over again.
Sometimes things are even more ambiguous than that. There are quantum-mechanical states in which even the component of spin along a single reference direction is undefined. Imagine an electron gun—a device that shoots out electrons. (These guns used to be very common: they were constituents of television sets before the advent of flat-screen technology.) Suppose that such a gun produces electrons in such a strangely ambiguous state. It fires one off in the direction of an experimenter. That experimenter is equipped with a detector that measures the component of the electron’s spin along a particular reference direction. What will it find? Will the electron’s spin turn out to be this way or that? There is no way to know. Nothing in the quantum mechanics of such a state predicts the result the detector will get.
All the theory gives is the probability of a particular result. It is precisely this refusal to go further that is so frustrating about the theory. It seems to avoid all the interesting questions. Would you like to find a description of what is going on in the experiment? Do you find yourself beset by the urge to make up for quantum theory’s lack? Do you want to posit some property of the electron flying toward the detector that explains the result that it gets? If so, then you are not alone. In the old days many people agreed with you. Einstein agreed with you.
Indeed, you may be feeling that the whole thing is trivial. Perhaps you feel that you already know what this property is. Perhaps you carry in your mind’s eye an image—an image of a tiny speck hurtling away from the electron gun, aimed precisely at the detector. Perhaps in your imagination you lean forward to gaze closely at this speck—and you notice that it is spinning. Spinning about an axis that you can see in your mind’s eye. An axis that points in a perfectly definite direction.
I too find it hard to resist this image. After all, it is what figure 5.1 shows. But that image is not provided by quantum theory. It is provided by a lifetime of experience in the large-scale world, a world that does not partake of the slippery, endlessly ambiguous nature of the microworld, a world that may be entirely inappropriate to this new zone of experience.
A spinning object and its axis of spin.
Indeed, quantum mechanics has no place for the image. It lies utterly outside the theory. It belongs to that world of which my Predictor refuses to speak—a world of real objects, of actual physical situations. A world of hidden variables. A world in which Einstein believed, but Bohr did not. A world whose very existence is in doubt.
For make no mistake: that apparently trivial figure carries with it a profound assumption about the very nature of reality. It is an assumption that I find almost impossible to question, but that I am being forced to question as I replay in my mind that long-ago battle over the completeness of quantum theory. An assumption about metaphysics. An assumption that John Bell realized could be tested by experiment.