Bohr is referring to something more or less like this: there can be no question of any unambiguous interpretation of the symbols of quantum mechanics other than that embodied in the well-known rules which allow us to predict results to be obtained by a given experimental arrangement described in a totally classical way.
What was developed, and we will talk more about this later, was a fairly straightforward and explicit algorithm for predicting the outcomes of experiments, given the arrangement and the outcomes of the initial color measurements. You had an algorithm into which you plug whatever arrangement you have set up, plug in the outcomes of the measurements you performed on these particles before feeding them into the hardness box, and use it to predict the outcomes of the color measurements on the other side.
This algorithm works, spectacularly well. The name of this algorithm is quantum mechanics. It makes predictions that are staggeringly more accurate than any previous physical theory. Bohr, however, is adamant, for reasons we will examine further, that this algorithm is all quantum mechanics can or should aspire to be.
Any attempt to offer yourself a picture, to tell yourself an intelligible story about what is going on in between the time you put the electron in here and the time you measured its color out there, any attempt to do anything like that is going to fail. It is going to collapse into paradox and contradiction. This is a profound moment in the history of science, and in the history of the aspirations of the scientific enterprise. There is a great deal one could say about this.
A good history of this period has yet to be written. What was happening in physics was occurring simultaneously with various other intellectual movements, the rise of modernism in literature and painting, among others. There is a sense in which the original aspirations of scientific inquiry were, in the zeitgeist of that era, akin to the aspirations of the nineteenth-century naturalistic novel: the belief that one could simply and straightforwardly say what happened, say who did what. Both sets of aspirations were thought to have collapsed under pressure from multiple directions, the development of psychoanalysis, and much else besides.
There was supposed to be a crisis of representation affecting the novel, affecting painting, and so forth. It seems to me that at least part of what must have been going on with Bohr was some version of the following idea: you think you have a crisis of representation? You have trouble getting to the bottom of people's motivations. I have trouble with rocks. The crisis was arising at a far more elementary level, in a far more shocking and radical way.
There were certainly many figures in literary modernism, people in the Bloomsbury circle, Virginia Woolf among them, who were acutely aware of what was happening in physics and who were clearly trying to find connections between developments in physics and what they took to be the crisis of representation that had overtaken the hopes espoused by the nineteenth-century European novel. This was not entirely independent of the larger historical zeitgeist of the time. In that context, what Bohr was saying sounded genuinely exciting, and there were serious, intelligent people who said: here is another attitude one could take.
Once again, we seem to have run out of logical possibilities. This was much later than Bohr, around the 1960s. People said, look, another attitude you could take is that there is a story to tell about how the electron got from here to there, but the story violates logic. This would represent the first case in history of an empirical scientific discovery about the laws of logic.
There were people at the time, people like John von Neumann, who were seriously entertaining this idea.
Mathematicians and physicists like von Neumann and Schrödinger, the discoverer of the Schrödinger equation, were not as comfortable as Bohr apparently was with simply abandoning the possibility of telling any story at all about what is going on between one measurement and the next. What I am about to describe is not explicitly what any of them said, but rather how things looked after the dust settled. The idea was this: there may be a way to escape the straightforward logical contradiction by saying that in the period between the initial color measurement and the final color measurement, while the electron is inside the apparatus, the electron is in some kind of situation, a situation never before dreamt of for a material particle, but one that is not obviously a logical contradiction in the way we were suggesting it might be.
Perhaps there are modes of being available to things like electrons, situations that electrons can be in where a question of the form "is it on the soft path or is it on the hard path?" simply does not make sense. This would be analogous to asking whether the number five is married or single, or whether Catholicism weighs more or less than five grams, or whether this table is a Republican or a Democrat. These are the kinds of questions philosophers are very familiar with; they are referred to as category mistakes.
If someone asks, "Is the number five married?" you say no. And if they then say, "I see, so the number five is a bachelor," you say no, neither of those is the case, and there is no logical contradiction in that. The number five is simply not the kind of thing that has a marital status. I am not asserting a contradiction, as if I were saying the number five is both married and unmarried. The category just does not apply. Similarly, if this table is not a Democrat, it does not follow that it must be a Republican or an Independent; questions of political affiliation simply do not apply to tables.
Now, if we take that approach here, it is admittedly a much stranger situation, because there are circumstances, such as before we feed the electron into the apparatus, where it seems to make perfectly good sense to ask where the electron is, and the question has a definite answer. And when the electron comes out the other side of the apparatus, it again makes perfectly good sense to ask where it is, and again the question has a definite answer. But the suggestion is that when the electron passes through a hardness box, the hardness box puts it into some condition in which asking whether it is taking the hard path or the soft path is like asking about the marital status of the number five, or, to put it in a more positive way,
What we want to say is that the electron enters some state on passing through a hardness box, that is, what a white electron does. A hard electron doesn't do this; a hard electron simply comes out on the hard path without incident. But a white electron fed into a hardness box apparently enters into some kind of radically unfamiliar physical condition in which there is simply no fact of the matter about where it is located in space, no fact of the matter, more particularly, about whether it is moving along the soft path or moving along the hard path. The effect of passing through the recombination box is then to restore there being a fact of the matter about where the electron is.
This is radically unlike, say, the case of the number five. There are no circumstances in which the number five has a marital status, and no circumstances in which this table has a political affiliation. With electrons, the case seems to be different. Electrons can sometimes be in situations where it makes sense to ask about their location in space, about which of these two paths they are on, and there seem to be other circumstances in which that question simply fails to make sense. There is, as a fundamental matter, no fact of the matter about whether the electron is on the soft path or on the hard path.
There is, of course, a long and venerable tradition in all areas of human speculation that if you don't understand something, you can at least make up a name for it. In circumstances like this, people say that the white electron, after it emerges from the hardness box, is in a superposition of traveling along the soft path and traveling along the hard path. What that means is the following. First, if you look for the electron anywhere other than on the soft path or the hard path, you will not find it. But second, there is no fact of the matter about which of those two paths it is on. Asking whether it is on the soft path or the hard path is like asking about the marital status of the number five. A mathematical apparatus was developed throughout the course of the 1920s to handle precisely this situation.
This mathematical apparatus was more or less in finished condition by the end of the 1920s. It was developed for predicting when these sorts of circumstances arise, and more generally for making predictions, sometimes deterministic predictions, as in this case, and other times probabilistic ones. A mathematical algorithm was developed into which one plugs the external circumstances: what are the initial conditions, what was the result of the color measurement on this side, and so on, in order to predict the outcomes of measurements later on.
Audience: Would it be fair to say that when you hit a wall in the development of fundamental science, instead of giving up something like logic, you give up on things like location?
There are a lot of things one might give up, and it is always a delicate balancing act. Logic, I would think, is one of the last things you want to abandon. Indeed, I am not even sure I know what it would mean to give up on logic.
According to this algorithm, an electron in a hard state is in a superposition of being black and being white. Earlier, we discussed the fact that we could never put ourselves in a position to say of a given electron, "This electron is both black and soft." That observation acquired the name uncertainty relation. The word "uncertainty" makes it sound as though this is a question of knowledge, a question of our not knowing something.
In light of these experiments, however, the reasoning goes like this. We know it is a property of a hard electron that when you put it into the box, it comes out one way. We know it is a property of a soft electron that when you put it into the box, it comes out the other way. What we are learning here is that if you put a white electron into a hardness box, it comes out in a superposition of both ways.
If you put a white electron into a hardness box, there is no fact of the matter about which aperture it came out of. If this is a good hardness box, what that suggests is that a white electron is itself already an electron for which there is no fact of the matter about whether it is hard or soft. So when we said earlier, correctly, that there is no way to point at a given electron at a given moment and say, "This electron is now both soft and black," that is not a limitation of ours.
What these experiments suggest is that when an electron is known to be white, it is not the case that we simply do not know what its hardness is. It is the case that asking about its hardness is like asking about the marital status of the number five. The character of what we are missing here is metaphysical rather than epistemic.
If you adopt this way of telling the story, you take the uncertainty seriously as a feature of reality itself. If instead you go with Bohr, you do not try to tell that story at all; you simply set it aside and adopt an instrumentalist picture of what science is.
The goal of science, as it was once naively conceived, was to give us a picture of what is actually going on in the world. According to Bohr, we have since grown up, and growing up is always painful. What science is really for, on this view, is simply to predict the outcomes of later measurements from the outcomes of earlier measurements. Most people who now work on the foundations of quantum mechanics regard this attitude as having been deeply harmful.
Bohr set himself up as a kind of prophet, issuing declarations that there must be no speculation about what is going on between the beginning of an experiment and the end of it, and going around dispensing aphorisms. One such aphorism: "Clarity tends to crowd out depth." Another: "A trivial truth is a truth whose negation is false. A deep truth is a truth whose negation is also true." These are the sorts of pronouncements that make one wonder what happened to the scientist one used to admire.
What unfolds over the course of the 1920s and early 1930s is the development of an algorithm, at least a version of one, setting aside technical questions about how to handle the special theory of relativity and related matters. This version of the algorithm takes shape across the 1920s and is more or less fully in place by the end of the 1930s. It is, within its domain, absolutely general.
You plug into this algorithm any experimental setup you like, along with any initial outcomes of any physical experiments, and the algorithm spits out predictions about how future physical measurements are going to come out. People were able to show, for example, that if what you plug into this algorithm are the kinds of questions we are accustomed to Newtonian physics answering, where a planet will be two weeks from now, or where a cannonball will land if shot at a certain angle and velocity, the algorithm produces predictions very similar to those of classical physics in all the cases where classical physics made correct predictions.
This algorithm was therefore proposed as a new, complete, fundamental physical science. It was not, however, the kind of fundamental physical science we had previously thought we wanted. Unless you were prepared to embrace the language of superposition, it was not the kind of algorithm that we had hoped physical science would give us. According to Bohr, if you wanted to hold onto those original aspirations, what you needed to do was simply grow up.
There were other people involved in developing this algorithm, however, who were thinking about it more literally, and who took it to imply very strange things about how material particles such as electrons, photons, and neutrons can behave. Part of what made the algorithm work the way it did points toward something that will later emerge as a very deep problem, one we will want to examine at length. But before getting to that, it is worth stepping back a little.
The algorithm uses a particular kind of mathematical object. There are several ways to represent it, but one common way is with a mathematical object called a wave function. Quantum mechanics uses these wave functions to say what there is to be said about the physical situation of an elementary particle, or of a collection of elementary particles that form a table, a computer, a university, or anything else. This is supposed to be a completely universal physical theory, and it uses wave functions to represent whatever can be represented about these physical objects.
If you are someone like Bohr, wave functions serve only as part of a prescription for calculating the probabilities of the results of future experiments. If you are someone like von Neumann, the wave function is something from which you can read off, even at intermediate times, what there are facts about and what there are not facts about. The wave function of a particle in the middle of passing through an apparatus is a mathematical object from which you can read off that, as it passes through, there is no fact of the matter about its hardness, no fact of the matter about whether it is on the hard path or the soft path, and yet the algorithm also predicts that after it passes through the black box, there will be a fact of the matter about its position and a fact of the matter about its color.
The name of this algorithm, as noted, is quantum mechanics. What is important to understand about it is that, notwithstanding how conceptually puzzling it is for anyone trying to be a realist about it, and notwithstanding how disappointing it is for anyone prepared to give up on the realist project altogether, this algorithm was the most powerful, accurate, and successful recipe for calculating the probabilities of experimental outcomes ever developed in the history of physics. The success was astounding: more or less overnight, it transformed our understanding of the physical world.
The entire separate science of chemistry became a homework problem in quantum mechanics. For simple atoms, where one could actually solve the equations of motion, it became possible to show exactly why hydrogen atoms behave the way they do, why they combine with the atoms they combine with, and fail to combine with those they don't. The success of this was beyond belief.
Indeed, twenty years later, at the height of World War II and then of the Cold War, this algorithm made theoretical physicists the masters of the universe. They were determining the fate of human history; they were the people to whom everyone looked for advice and guidance.
I want to point out something curious about this algorithm. What the algorithm does for you is tell you how this wave function evolves.
From the wave function you can read off either future predictions, if you are Bohr, or you can read off all the facts there are to be had about a particle passing through this device, including which things there are facts about and which things there are not. What the algorithm gives you are equations of motion, like the Schrödinger equation, which determine how the wave function evolves in time, just as Newton's equations of motion told us how the positions of material particles evolve over time. The quantity that was central to Newtonian mechanics, the position of a particle or set of particles, is now replaced by another mathematical quantity: the wave function. What you want from physics in either case is to be told how these quantities change with time, and there is an algorithm that does exactly that for the wave function.
Bohr's constant caution was this: do not, under any circumstances, think of the wave function as a description of what the system is actually doing. The business of describing what the system is doing is, on his view, hopeless. Other thinkers, von Neumann, Schrödinger, had more scientific realist sympathies and wanted to treat the wave function as a genuine description of what the system is doing. But on that reading, the system is doing something very, very strange. It is being described as existing in circumstances where even asking about its location in space is like asking about the marital status of the number five.
That having been said, there is something deeply strange about the algorithm itself, and this is a second problem, a second reason why Bohr thought the entire project of scientific realism had not merely collapsed but, more dramatically, had committed suicide. It was not killed by philosophical critique from the outside; it was killed by itself. Among the things the algorithm had better explain is why it is that inserting a particle detector changes everything.
Consider a specific setup. We have a two-path apparatus with a total-of-nothing box, which is turned off for the moment. When the total-of-nothing box is off, every white electron fed in comes out white. The algorithm predicts that. It also predicts that when the total-of-nothing box is switched on, every white electron fed in comes out black. But there is a third phenomenon. Suppose the total-of-nothing box is on, though it will not matter whether it is on or off for what follows. If we place a detector at one of the paths, two things happen: first, if the detector fires, all future attempts to locate the particle will find it with certainty on the hard route; and second, the color statistics at the output become 50-50.
The mere act of placing a detector, the mere act of looking, of asking where the particle is, transforms the situation entirely. Recall that we have very compelling reasons to believe that when the detector is absent, there is simply no fact of the matter about whether the electron is on the hard route or on the soft route. Yet when we place the detector and turn it on, the electron immediately begins to behave like a familiar electron that is determinately on one route or the other. Something about the act of looking does something very violent.
It forces the electron, as it were, to choose. It drives the electron out of the superposition, the state in which there was no fact of the matter about which route it was on, and into one of the two definite conditions: being on the hard route or being on the soft route, where there is now a fact of the matter. This shatters a fantasy that had been very much at the heart of the physical project right up until quantum mechanics: the fantasy of what one might call passive observation.
Everybody knew, of course, that in order to observe anything, in order to get any information about any physical system, you had to interact with it physically in one way or another. After all, the embodied information needs to get from whatever you're looking at into your head, so you have to bounce light off it or smack into it or something like that. But the thought was that if you were careful enough, you could in principle reduce the degree to which you disturbed a physical system by observing it to as low a level as you wanted. It was just a matter of doing the engineering more carefully, spending enough money, trying as hard as you could.
Here is what people observed in the laboratory. You do something as a result of which you can know which path the electron took, and it snaps onto one route or another. Then it keeps going on its way as if it were always on that route, and the outcomes of future experiments are exactly as if it had always been on that route. Many people assume that when you say "looking for it," what is meant is a person looking at the results. The fact is that people didn't know whether they meant that or not.
Consider von Neumann's formulation of this algorithm, which became standard after its publication. Von Neumann wrote a very famous book, I believe in 1929, called Mathematical Foundations of Quantum Mechanics, in which he laid out the algorithm as follows. The rules about how wave functions evolve have two different components: one rule that applies when you are not measuring the system, and another that applies when you are. What exactly he meant by that distinction, he didn't know, and he left it open to interpretation.
He wrote this down in order to account for all the experiments, in order to get all the predictions right. Von Neumann said it seemed like there would need to be two rules about how wave functions change with time, one applying under certain circumstances and the other under different circumstances. He called them Rule 1 and Rule 2. Rule 1 is given by the differential equations of the Schrödinger equation, which tells you smoothly how the wave function describing the system evolves with time. That is what tells you, for instance, that if you don't have a detector in place and you feed in a white particle, it will definitely come out white.
But then, said von Neumann, we have to account for what happens when we put detectors in. When a measurement is performed (and I believe that is the precise phrase he uses in the book), the usual Schrödinger rule gets suspended and is replaced by Rule 2. Rule 2 says that when you look, the particle is forced to choose a value of whatever it is you measured and then proceeds from there. This is coupled with a rule about the probability that the particle will end up choosing one outcome or another. In a case like the one we've been discussing, this rule, called the Born rule, tells you that the particle has a 50% chance of choosing the hard route and a 50% chance of choosing the soft route, and whichever it chooses, you go on from there.
Here are the facts on the ground. These two rules taken together, coupled with some rough-and-ready idea that everybody knows when a measurement is being made and when it isn't, turn out to constitute the most spectacularly successful algorithm for predicting the outcomes of experiments in the history of the world. And as it stands, this is clearly madness.
Even if you are Bohr, even if all you are looking for is an algorithm that is completely agnostic about the story of how the electron gets from one place to another, if you want an algorithm that will predict in detail and correctly the probabilities of outcomes of any measurement at all, you have to know exactly under which circumstances you are supposed to apply rule one and exactly under which circumstances you are supposed to apply rule two. Different claims about when to apply rule one and when to apply rule two will lead to different predictions about how later experiments come out. Relatively reasonable differences will lead to differences in predictions that are very hard to detect with currently available technologies, but this cannot be the way a fundamental theory of physics looks, even if you are only after an instrumentalist theory.
Audience: Given some arbitrary wave function, are there cases where there is nothing one can measure about it with certainty without changing it and having to apply rule two? Or is there always something one can measure and determine?
It turns out there is always something you can measure whose outcome the Schrödinger equation by itself will predict with certainty. That is, it will always be the case that there is something you can measure and get the correct prediction about without having to apply rule two. That is a technical point that is difficult to justify quickly off the cuff, but it is a very good question and the answer is yes.
There are two major topics I want to discuss in these sessions. The first concerns a problem with this algorithm that has now been put on the table. As it stands, the distinction between the situations in which we are supposed to apply von Neumann's rule one and those in which we are supposed to apply von Neumann's rule two is supposed to be drawn by whether or not a measurement is taking place. It is as clear as it can be that the English word measurement does not have anything like the requisite precision to play a fundamental role in what is supposed to be our most fundamental physical theory.
We will need a new theory that takes care of measurement situations and non-measurement situations in a more precise, unified way. There are several genuinely interesting and profoundly different proposals that have evolved over the course of the last fifty years or so, each attempting to come to grips with this problem: to imagine what an algorithm, or a description involving superpositions of what particles are doing, would look like if it handled these distinctions in the way we expect a fundamental physical theory to handle them. In future lectures we will be more precise about exactly what the problem is; we have alluded to it here, but we need to sharpen it up before we are in a position to discuss the attempts to resolve it.
The other topic I want to discuss is the second great shock that quantum mechanics delivers, but that will come in turn.
Another profound challenge that emerged from the foundations of quantum mechanics concerns the intuition that physical events can only affect other physical events that are immediately contiguous to them in both space and time. The idea is that things only directly affect other things right next to them, and that if something done here affects something over there, there must be a row of dominoes in between, each one knocking the next over, however difficult those intermediate steps may be to see.
It turns out that quantum mechanics poses a radical challenge to this intuition, the intuition that the world must work, at a fundamental level, in a so-called local way. This challenge is almost as radical as the challenge to realism itself, which we discussed today. These two large topics, the challenge to realism and the challenge to locality, are what I am planning to take up in future lectures.