These are going to be some talks meant to introduce the foundations of quantum mechanics. I have some friends here to help me out, who you'll meet. Let me just jump right in.
I want to start by telling some stories about things that can happen to electrons. These are, I think, among the most unsettling and surprising stories to have emerged from the natural sciences since the scientific revolution of the 17th century. The experiments I am going to describe will be presented schematically, but they are all experiments that have actually been performed.
I will talk about what would happen if you did these experiments with electrons, though for various technical reasons some of them are easier to carry out with neutrons and others with photons. That will not affect the logical structure of the situation, which is mainly what I want to get across.
These stories will involve measurements of various measurable physical properties of electrons. Once again, exactly which physical properties those are will not be important to the logic of the situation. For those who want to look into the matter more deeply, the properties I will be discussing are different components of the intrinsic spin of electrons. Here, however, we will refer to them by deliberately facetious names, simply to keep things as clear as possible.
One of the properties whose measurements we will be discussing here, let's call the color of these electrons. Electrons don't actually have colors; I'm using the term as a stand-in for certain components of the intrinsic angular momenta of electrons. When you measure this property, there appear to be only two possible numerical values it can take on, say, plus one or minus one. Let's call one of them black and the other white. Every electron whose color you measure turns out to be either black or white; you never see green electrons, purple electrons, yellow electrons, or anything of that kind.
Another property we will be discussing is what we'll call the hardness of the electron. Hardness likewise has only two possible numerical values, which we'll call hard and soft. These are properties that physicists have known how to measure in a routine way for something on the order of a hundred years, and we are very good at measuring them in the laboratory.
The way you measure color and hardness is by arranging a certain configuration of magnetic fields. A particular configuration of magnetic fields will deflect an incoming white electron in one direction and an incoming black electron in a different direction, and similarly for hard and soft electrons.
These arrangements of magnetic fields will be referred to as boxes, or more precisely, measuring boxes. It is routine and straightforward to construct in a laboratory something one might call a hardness box. This is a box with three apertures: you feed an electron into the input, and if the electron is a hard electron, the magnetic fields inside arrange things so that it exits through the hard aperture. If the electron is a soft electron, those same fields direct it out through the soft aperture.
We can build, in a very similar way, color boxes. You feed an electron into the input aperture of a color box, and it exits through one aperture if it is a black electron, and through the other if it is a white electron. We have been remarkably good, for something on the order of a century now, at measuring these properties, manipulating electrons that possess them, and directing them into whichever boxes we choose.
Once you have boxes like this, one thing that might occur to you right away, once you have discovered these properties and learned how to measure them, is that these measurements are repeatable in the way you would expect a measurement of a bona fide physical variable to be. That is, if I measure the color of an electron and find it to be white, and immediately feed it into a second color box, 100% of the time it will come out the white aperture of that second color box. And if it came out the black aperture of the first color box and I feed it immediately into another color box, it will come out the black aperture of that second box as well.
There are things you can do to electrons, as we will see, which can affect their color values. So it is not the case that any electron that has ever come out a white aperture will necessarily come out the white aperture of any color box you feed it into at some later point in its life. But if you keep the environment free of things that could disrupt these color and hardness values, repeating two measurements in a row will always yield the same result. If that were not true, we would have reason to be puzzled about what we even mean by speaking of ourselves as measuring a physical variable when we carry out a measurement like that.
We are able to move these electrons around as we wish, and we are able to build these color boxes. Something that might immediately occur to us to do is to ask whether there is any relationship between these two physical properties of electrons that we know how to measure, whether the properties are connected with one another in some way. An easy way to begin to get a handle on that question is to look for statistical correlations between the color value of a given electron and its hardness value.
We can measure the color values and hardness values of a large collection of electrons, feed the results into a computer, and ask the computer to look for correlations between those values. These experiments are straightforward to perform, and it turns out that there are no correlations whatsoever. Of any large collection of electrons known to be white, exactly 50% turn out to be hard and 50% turn out to be soft, and similarly for all other possible combinations. There appear to be no correlations at all between the color value of a given electron and its hardness value.
Here is an experiment that one might think to perform if one has these color boxes available in the lab. Suppose we set up a sequence of three boxes: a hardness box, followed by a color box, followed by a second hardness box. The hardness box has a hard exit aperture and a soft exit aperture; the color box has a black exit aperture and a white exit aperture; and the second hardness box again has a hard exit aperture and a soft exit aperture.
We feed a stream of electrons into the first hardness box, discard the ones that emerge from the hard aperture, and keep only those that come out the soft aperture. We then feed those into the color box. Statistically, half will emerge from the black aperture and half from the white aperture. We discard the ones that come out the black aperture, keep the ones that come out the white aperture, and feed those into the intake of the second hardness box.
By the time the electrons reach that second hardness box, they have all been measured to be soft and they have all been measured to be white. It seems natural to suppose that what we are dealing with at this point are electrons that are simultaneously soft and white. Our expectation, then, is that feeding them into a second hardness box will simply confirm what we already know: they are soft, and so they should all emerge from the soft aperture.
The first surprise in this story is that this is not what happens. When you actually perform the experiment, half of the electrons emerge from the soft aperture and half emerge from the hard aperture. That, it must be said, is genuinely surprising.
One might wonder what is going on here. Whatever is going on, the effect is that the presence of this color box between the two hardness boxes constitutes some kind of physical factor that can disturb the hardness values of electrons passing through it. We established earlier, and it remains true, that if nothing is placed between the two hardness boxes, then every electron that exits the first hardness box by the soft aperture will also exit the second hardness box by the soft aperture. The presence of the color box, however, must constitute some kind of disruptive factor with respect to the hardness values of the electrons that pass through it.
The first natural reaction is that the color box is apparently defective in some way. It is doing its job of measuring color reliably: place two such boxes in sequence and they will both yield the same color result 100% of the time. So the color box seems to be measuring color perfectly well, but in the course of doing so, it appears to be disrupting the hardness values of the electrons that pass through it. As soon as one sees this, two questions naturally suggest themselves.
The first question is: can we build a better color box? Can we engineer things more carefully so that we end up with a box that not only measures color correctly, but does not, in the process of measuring color, disrupt the hardness values of electrons passing through it? The second question concerns the particular color box we already have. This box is defective not as a measurer of color, but because it disrupts hardness values. One naturally becomes curious about what determines which electrons get their hardness values flipped on the way through the color box and which do not. Apparently, half of these soft electrons are making their way through the color box with their original hardness value, soft, intact, while the other half are getting their hardness values flipped from soft to hard.
To summarize: the first question is whether we can engineer a better color box that leaves hardness values undisturbed. The second is whether, in the case of the defective box we already have, we can develop some account of what determines whether a given electron ends up with its hardness value flipped or left alone.
Consider the first question: can we build a better color box? There will be no rigorous proof one way or the other, but what we can do is try to engineer these color boxes as carefully as possible, spend more resources on them, think of different ways of measuring color, try different arrangements of magnetic fields, and experiment with many different designs involving more and more careful engineering. The striking result is that it is not merely the case that we fail to build a color box that leaves hardness values intact. It is much more than that. No matter what we do, as long as what we end up with is a box that successfully measures color, we seem to be stuck with a box that flips the hardness of exactly 50% of the electrons passing through it. No amount of redesign or reengineering, insofar as we can tell, moves those statistics even a thousandth of a percent away from exactly 50-50.
There is certainly no proof that it is in principle impossible to do better. But the fact that all of these redesigns and all of this more careful engineering fail to move the statistics even a thousandth of a percent off of 50-50 strongly suggests that some deeper principle is at work. What people took this to suggest is that something fundamental is operating behind the scenes, something this body of experimental experience is reflecting. It is not just that we cannot quite reach a perfect box; it is that we are getting nowhere at all. We keep trying, and we find ourselves exactly as far from perfection as when we started.
We are now addressing the second question. Color boxes, as we have seen, appear to flip the hardnesses of half the electrons that pass through them, while leaving the other half untouched. How does it get decided which electrons have their hardnesses flipped on the way through the color box, and which do not?
Here is one way to try to address that. We are simply doing the best we can; we are not giving logical proofs of anything. Run the experiment a large number of times. Try to hold all of the physical conditions constant between runs, except for the motions of the electrons themselves. And as the electrons are on their way in, measure everything you can think of about them: their velocities, the exact angles at which they are entering, anything at all. Keep track of the outcomes of those measurements and match them up with the question of whether or not the hardness got flipped on the way through the color box. Feed it all into a computer and look for correlations.
It turns out, and I am describing a body of empirical experience here, not a logical proof, that we can discover no correlations whatsoever between the properties of the incoming electrons and the question of whether their hardnesses get flipped. We can also narrow the range of incoming properties: ensure that all the electrons have exactly the same velocity, that they are as nearly identical as we can make them in every measurable respect. This has no effect on the statistics.
The impressive thing is not merely that we are unable to find reliable predictors. No matter how we filter the incoming electrons, it does not move the flipping statistics even a thousandth of one percent away from exactly 50-50.
Audience: Are you sure they all have exactly the same properties? What about properties you cannot measure?
Of course we can only work with the properties we know how to measure. We have to think of every property we can and take advantage of every measurement technique available to us. But keep in mind, we are not asking for the moon here. We are not asking to find some set of properties that will tell us with certainty whether the hardness is going to get flipped. All we want is to find some way of filtering the incoming electrons that moves the flipping statistics even one millionth of one percent away from exactly 50-50, and we find nothing.
What this body of experience suggests is that there is nothing about the initial conditions that determines in advance which electrons will have their hardnesses flipped on the way through the color box, and which will not. This is the first time in the history of physics that this kind of situation arose. What these experiments seem to suggest is that even at the microscopic level, there is an irreducible element of chance in how systems evolve, because nothing we are able to measure about the incoming electrons sheds any light whatsoever on whether they will have their hardnesses flipped by passing through a color box.
Audience: Can you say a little about what you mean by chance? A lot of people might associate it with something magical, some fairy going into the box that we cannot measure with physical instruments, doing its work.
No, we are looking for regularities in nature. And it looks like all we can say is that whatever kinds of electrons we have going in, if we feed in a large ensemble, half of them are going to come out the soft aperture and half of them are going to come out the hard aperture, statistically speaking. We will be relying in this discussion on the assumption that most of us have some rough and ready sense of what we mean by probabilistic claims. We think of flipping a coin, rolling a die, something like that. Of course, being fully clear and explicit about what we mean by probability claims is a highly non-trivial matter, philosophically.
Audience: People usually associate chance with unpredictability. That is why it seems important to emphasize that what is really surprising here is not so much the fact that measuring color interferes with hardness measurements, but the fact that however you vary all the properties of the electrons you put into the hardness box, there is no way to change the statistics.
That is exactly right, and that is the fact that suggests we are dealing with a probabilistic law, a genuine law of nature. But there is still predictability, just not predictability at the level of any individual electron. There is statistical predictability, and that is what distinguishes this from magic, or from mere chaos. Good.
Let us pause to note an important corollary. This is the first time in the history of physics that we encounter a suggestion of this kind. There had been plenty of talk about probability and chance in earlier periods of physics, but those occasions were always associated with situations in which we do not know the exact microstate of the world. The reason we could not make definite predictions about the future had to do with ignorance of present circumstances. What these experiments suggest is something fundamentally different: that the process of evolving through time involves transitions from one state to another that are genuinely, irreducibly chancy.
Consider what it would mean to build what we might call a super box. This would be a box with not three apertures but five. An electron is fed in, and it exits through one of four output apertures: one for electrons that are hard and black, one for hard and white, one for soft and black, and one for soft and white. A box like this would put us in a position, after observing which aperture an electron exits through, to say with confidence, "That electron is now hard and black," or "That electron is soft and white." It would allow us to announce simultaneously both the color value and the hardness value of a given electron.
But if you think carefully about how you would actually build such a box, you immediately run into a serious problem. The natural approach would be to first pass the electron through a color box and then through a hardness box, routing it to the appropriate output based on the outcomes of both measurements. What we have just learned, however, is that this will not work. Once we pass the electron through the color box, the hardness value is no longer reliable; it may have been scrambled. The same problem arises in the reverse order. One might also try combining the two boxes by superimposing their internal magnetic field configurations, but it turns out that does not work either.
What these experiments suggest is that, in a fundamental and principled way, it is beyond our capacities, by any means whatsoever, to ever put ourselves in a position to say of a given electron, "Its hardness value is such-and-such and its color value is such-and-such." We can measure the color of any electron as accurately as we like, and we can measure the hardness of any electron as accurately as we like. But color measurements disturb hardness values, and hardness measurements disturb color values, and this appears to make it impossible ever to know both simultaneously. We cannot say of any electron, at any given moment, "I know that it is black and I know that it is soft."
This is a simple example of what is referred to in the quantum mechanical literature as the uncertainty principle. You will usually find this discussed in situations where the two variables under consideration are not color and hardness, but rather the position of an electron and its momentum. Nevertheless, this is an example of exactly the same phenomenon.
We call observable properties like color and hardness incompatible with one another because there is no procedure by means of which we can put ourselves in a position to simultaneously know both of those values. Variables like color and hardness, or position and momentum, are referred to as quantum mechanically incompatible with one another. We say that there is an uncertainty principle relating them: if we are certain about the color value, we are completely at sea about the hardness value, and vice versa.
Audience: Does this suggest that it is about our uncertainty, that we simply cannot know both values simultaneously, or that a particle cannot actually be both hard and black at the same time?
At this point in the story, it looks like the former: it appears to be an epistemic issue. You are anticipating, quite correctly, that it will turn out to be something deeper than an epistemic issue, and we will encounter that as we continue. That is a very good question to keep in the back of one's mind. The name uncertainty principle itself suggests that this has to do with limitations on what we can know, rather than limitations on what there can simultaneously be facts about. That judgment is going to evolve as we elaborate the story further.
Consider the following, more complicated contraption. Down here I have a hardness box. Here is the hard exit aperture, and here is the soft exit aperture. Here is the route that an electron emerging from the hard aperture would follow, and here is the route that an electron exiting the box by the soft aperture would follow. And here I have placed what I will call an electron mirror.
What this mirror does, and this claim can be independently verified by removing the mirror from the apparatus and performing the appropriate experiments on it, is simply to change the direction in which the electron is moving, without altering any of its other properties. It does not alter the electron's hardness value; it does not alter its color value. The electron bounces off the mirror exactly as light bounces off an optical mirror. These devices are straightforward arrangements of magnetic fields, and we can confirm independently that every hard electron entering the mirror exits as a hard electron, every soft electron exits as a soft electron, every black electron exits as a black electron, and every white electron exits as a white electron.
There is a second mirror further along, equally innocent in the same sense. The two paths converge at what I will call a black box. This black box is innocent in a similar way: if you send a soft electron into it, a soft electron emerges; if you send a hard electron into it, a hard electron emerges. Within this setup, the black box is hardness-innocent as well.
The apparatus I have just described is essentially a neutron interferometer. This type of experiment is most commonly performed with neutrons, and detailed, explicitly physical descriptions can be found by searching for neutron interferometry. I will continue to describe the experiment as though it were performed with electrons, simply to keep everything consistent with our earlier discussion. In principle it could be done with electrons, though the fact that electrons are electrically charged makes such experiments technically more difficult than they are with neutrons, which are electrically neutral. With that said, let us proceed to perform several experiments with this contraption.
Put in white electrons here and measure their hardnesses here. You'll get 50/50. Why? Half of them will go this way, half of them will go that way. The half that go this way are hard here, hard here, hard here — you get a hard result. The half that go this way are soft here, soft here, soft here — you get a soft result.
Put in hard electrons here and measure color at the end. Everything is straightforward: they all go this way, they're hard here, hard here, hard here, and the color statistics should be 50/50 — and they are. That is exactly what you get if you feed in a bunch of soft electrons here and measure the color at the end. They'll be soft here, soft here, soft here, with random colors.
Now suppose you put a bunch of white electrons in here and measure color at the end. We should expect that half of them will take the hard route and half will take the soft route. Consider the ones that take the hard route: they'll be hard here, hard here, hard here, and the color statistics should be 50/50. Consider the ones that take the soft route: they'll be soft here, soft here, soft here, and the color statistics should again be 50/50. Since half of the white electrons take the hard route and half take the soft route, the overall color statistics at the end ought to be 50/50.
The surprise is that this is not what happens. When you do this experiment — feed in a bunch of white electrons and measure their colors at the end — the colors are 100% white. To emphasize just how surprising this is: we have taken white electrons, passed them through a hardness box, and everything else is straightforward. We know that passing through a hardness box randomizes color values. Yet these electrons, once they emerge, remember their original color. They are 100% white.
In this case, for some reason that apparently has to do with the fact that we put the two routes back together before we measured the color, passing through this hardness box causes no disruption of the color. For every white electron we feed in, if we do the experiment properly and carefully, we get a white electron out the other side.
Let me say one or two other things about this. You might think that the reason the merging of the two routes makes a difference is that some of the electrons going one way are bumping into some of the electrons going the other way and having some kind of effect on one another. That is easy to eliminate. We have very good control over these experiments, and we can easily arrange things so that electrons are fed in so slowly — one at a time — that there is never more than a single one of them anywhere in the apparatus at any moment.
For a typical apparatus, it takes a small fraction of a second for an electron to pass through and come out the other side. We can arrange it so that we feed in only one electron per minute, one electron per hour, or one electron per week. This does not decrease the effect at all. Electron by electron, even with only one electron in the apparatus at any given time, 100% of the ones that go in white come out white.
This is very odd. If you ask yourself how a given electron — focus on the fifth one you send through — got from one side to the other, you do not want to say it took the hard route, because you know that electrons taking the hard route have 50/50 color statistics, and this one does not. Equally, you know that electrons taking the soft route have 50/50 color statistics, and this one does not either.
Now suppose you modify the experiment in the following way. You insert a small measuring device somewhere along the hard route that clicks if the electron takes the hard route and does nothing if the electron takes the soft route — something that will tell you which route the electron took. The moment you insert that device, things change. To make it more dramatic: insert the device but leave it switched off. You still get 100% white electrons out the other side. Turn it on, and you get 50/50 white and black.
What always happens when the device is on is that it clicks half the time and does not click the other half. This is more and more puzzling. What difference does this device make? It does not change the hardness values of electrons that pass near it, and it does not change the color values of electrons that pass near it — we can confirm that by removing it from the apparatus and checking independently. But when it is inserted and switched on, the 100% white result vanishes immediately and switches to half white, half black.
Here is a way to make this even more acute. What I am about to describe is an effect called the Aharonov–Bohm effect.
Consider a box with two apertures — an input aperture and an exit aperture. I'm going to call a box like that a total of nothing box if the following conditions hold: all measurable properties of a particle passing through it — velocity, momentum, mass, color, hardness, everything you can think of — are left completely undisturbed. The particle comes out with exactly the same set of measurable properties it went in with, excepting only its position in space. Moreover, to make the requirement even more stringent, the time required to pass through the box must be exactly the same as the time required to traverse that same extent of empty space.
There are infinitely many different ways one could build a total of nothing box. The simplest way is just to build an empty box. But another way is to have a box that does all kinds of things to the electron while it's inside, then undoes all of them before the electron exits, and manages to accomplish this in exactly the time it would take the electron to pass through that amount of empty space. Any box satisfying those conditions we will call a total of nothing box.
What Aharonov and Bohm were able to show is that there exist boxes which satisfy all the conditions for being a total of nothing box, yet are not empty. These are boxes containing specific kinds of fields inside. All incoming properties are identical to the outgoing properties except for position in space, and the transit time is the same as it would be through empty space. We will call this specific construction an Aharonov-Bohm total of nothing box, and Aharonov and Bohm tell you exactly how to design one. These experiments have since been carried out and confirmed the predicted results.
If you insert the Aharonov-Bohm total of nothing box into, say, the soft path of an interferometer and feed in white electrons, every single one of them comes out black. When the box is switched off, every white electron fed in comes out white. When it is switched on, every white electron fed in comes out black.
This is very odd. Let's step back and investigate the possibility of telling ourselves a story about what happens in between this moment and this moment — about what the electron does. Let's go through the logical possibilities. Could a given electron have taken the soft path? No, because total-of-nothing boxes are confirmed to have no effect on the properties of electrons that pass through them, and yet the presence of this box changed the electron from white to black. So the soft route seems to be ruled out.
Could it have taken the hard route? No, for the same reason. Total-of-nothing boxes are explicitly confirmed to have no effect on the properties of electrons that pass outside of them either, and yet, once again, the presence of this box changed the electron from white to black. So the hard route is ruled out as well.
Could it, in some sense, have taken both routes? Not in any familiar sense. If you stop the experiment in the middle and look for the particle, you will either find an electron on the soft path with nothing on the hard path, or an electron on the hard path with nothing on the soft path. You will never find half an electron on one path and half an electron on the other. When a detector is placed in one of the paths, it goes off only statistically half the time — not every time. So the both-routes option appears to be excluded as well.
What about neither route? That seems easy to eliminate too. If you block both routes, nothing gets through. You can fill the entire intervening region with lead so dense that even Superman couldn't see through it, and it makes no difference — nothing gets through. So the electron is not taking some other way around. The neither-route option is ruled out, and those are the four logically available options. We are left with something enormously puzzling.
Bohr — the father of quantum mechanics, or at least the universally acknowledged guiding spirit of its development — thought that the lesson of examples like this is that any attempt to tell an intelligible story, any attempt to develop a mental picture of what is going on between the time we measured the color here and the time we measured the color there, will collapse into paradox. It collapses, apparently, into a flat-out logical contradiction. Any attempt to tell yourself a story about what is going on between the measurement at the beginning and the color measurement at the end reduces immediately to babbling gibberish.
This was among the reasons — and Bohr had several, others of which we will get to later — for what he concluded about the aspirations of science itself. You can already begin to see it here quite vividly. Bohr thought that in the light of examples like this, the traditional aspiration of scientific inquiry — to give us an understandable picture of what is going on, to give us an understandable picture of why things happen the way they do — had collapsed. A certain idea known as scientific realism, as an attitude toward the scientific enterprise and its aspirations, was collapsing right before your eyes. There is no story to tell about what happened between this color measurement and that color measurement. Any attempt to tell such a story collapses into gibberish, paradox, or self-contradiction the moment you try.
This begins to offer a taste of why something profoundly philosophically interesting is going on in quantum mechanics. It begins to offer a taste of what is at stake in the business of trying to make sense of it. It is not simply that the world contains surprises, that we are constantly astonished by new discoveries. It is that, in a certain sense, the founding aspiration of the scientific enterprise has, according to Bohr, hit a wall — a wall it is not going to be able to go through. Our attitude toward the entire scientific enterprise is going to have to be different from here on.
It is worth pausing to press this point a little further. There is, of course, a long history — a three-thousand-year history in Western thought — of philosophical critiques of what you might call naive scientific realism. There is skepticism of all kinds. There is transcendental idealism. There are all manner of external philosophical challenges to the idea that scientific realism is anything more than naïve. But none of that is what is going on here. This is the first occasion on which the obstacle to scientific realism does not come from the outside, from some external philosophical critique. It is as if the scientific enterprise itself is committing suicide. This is the moment in the horror movie when the operator tells you the call is coming from inside the house.
This has no precedent in the long tradition of Western philosophical worries about realism. Something utterly different is happening. Realism is not being exposed as resting on this or that questionable assumption, or as involving some internal tension. The claim, rather, is that certain experiments came out in certain ways — this pointer ended up here, that pointer ended up there — and that a certain set of outcomes of straightforward physical experiments made it logically impossible to tell a story about what was going on in between them. What is at stake here is not this or that surprising discovery about the way the world is, but a surprise about the very possibility of our being able to tell ourselves the kind of story that science had seemed to promise us about the way the world is.