Three Ways to Save Reality

Before the interlude about non-locality, we were talking about the initial reactions of people like von Neumann, Wigner, and others who were, in some sense, trying to be realistic, trying to tell the kind of story that Bohr had forbidden people to tell about what is going on in between measurements, and trying to use pieces of the quantum mechanical algorithm, mathematical components like the wave function, as realistic representations of what particles are actually doing between experiments.

What we were calling the measurement problem arises from the way von Neumann tries to formulate things: it seems there is a necessity of positing two completely different sorts of fundamental laws governing the evolution of the wave function, in order to account for the kinds of behaviors we see in two-path experiments. There is a way that the wave function of a given system evolves when the system is not undergoing a measurement or observation, given by von Neumann's rule one, which refers to the differential equation called the Schrödinger equation, in accord with which the wave function is supposed to evolve. And then there is another rule, von Neumann's rule two, which applies when the particle the wave function describes is measured.

In the two-path experiments, for example, if we stop the experiment in the middle and measure which path the electron is actually taking, that forces the electron to choose one path or another in a way that the Schrödinger equation will never do. In order to account for that, von Neumann believed a separate law was needed, one that applies only when measurements are taking place. The obvious objection to this, and this is all ground we have already covered, is that words like measurement do not belong in a proposed fundamental physical theory of the world. Measurement is a vague term in ordinary language, and that is not the kind of precision we expect of a fundamental physical theory.

What followed was fifty or so completely wasted years of people proposing other ways to draw the boundary between that set of physical situations in which von Neumann's rule one applies and those in which von Neumann's rule two applies. Von Neumann's rule two is sometimes referred to as the collapse of the wave function on measurement. Rather than using the word measurement, people reached for an equally useless collection of terms: macroscopic, irreversible, indelible, conscious, and so on. It is strange and remarkable that it was not obvious to all of these people that every one of those words is exactly as useless as the original word measurement in this context.

That is where things stood. Wigner hoped that references to consciousness would turn out to offer a sharper delineation between those physical circumstances in which rule one applied and those in which rule two, the rule of the collapse of the wave function, applied. That doesn't work out either. In retrospect, it feels as though people should have known better.

This state of affairs persisted for a very long time, essentially from the early 1930s until sometime in the 1980s. All anyone had to offer on the question of drawing the boundary were useless gestures toward notions like macroscopicness, indelibility, measurement, recording, and consciousness. That was the entirety of the conversation until the 1980s.

In the 1980s, for the first time, a scientifically serious theory of the collapse of the wave function was proposed by three Italian physicists named Ghirardi, Rimini, and Weber, and it has subsequently become known as the GRW theory. The GRW theory begins by giving up entirely on the project of locating a boundary between those physical circumstances in which Von Neumann's Rule One applies and those in which Von Neumann's Rule Two applies. Rather than drawing such a boundary, they very cleverly write down a single law, which involves a small stochastic modification of Von Neumann's Rule One, that is, a small stochastic modification of the Schrödinger equation.

Let me describe how the theory works for the case of a single particle. The rule governing the evolution of the wave function for a single isolated particle is that the wave function evolves almost all of the time in perfect accord with the Schrödinger equation, that is, with Von Neumann's Rule One. But every now and then, very rarely, a stochastic event interrupts this smooth, continuous, deterministic evolution. In the way GRW originally formulated the theory, this event happens on average once in 10¹² years for a single particle.

This event is genuinely stochastic: it has a certain fixed probability of occurring per unit time, and there is no deterministic rule that will allow you to predict when it will occur. When it does occur, it transforms the wave function of the particle from whatever it was, evolving along in accord with the Schrödinger equation, into a wave function that represents a particle about whose location in space there is a determinate fact of the matter.

Consider, for example, what would happen if one of these events were to occur while an electron is passing through a two-path apparatus. The probability of such an event occurring during the short transit time through the apparatus is extremely small, but if it did occur, the stochastic event would effect a choice for the particle between being on the hard path or being on the soft path. It would have exactly the effect that Von Neumann's Rule Two describes as a measurement of the particle's location. If the particle is located by this stochastic event on the hard path, it will subsequently behave like a hard particle, and, as we saw in the two-path discussion, the color statistics at the end will shift from 100% white to 50/50, and so on.

These stochastic events in the life of any single particle are very, very rare. If you were to modify Von Neumann's Rule One in the way that GRW suggests, this is not at all incompatible with our experimental experience of two-path experiments and so on, because the probability of any one particle being hit on its way through a two-path apparatus is absolutely minuscule, negligible to the point of being irrelevant. This proposal therefore has the advantage that it does not contradict anything we experimentally know to be true about the strange behavior of subatomic particles.

The theory is designed, however, so that if you have a macroscopic object in a superposition of being here and there, a baseball, say, or a pointer on a measuring device, then even if only a single one of its constituent particles gets hit by one of these GRW stochastic events, the object as a whole is forced to choose between the two locations. Because the number of subatomic particles in an object like a baseball is astronomical, the probability that at least one of them will be struck by a stochastic event within any given second or minute becomes very high.

The way the theory is designed, when just one particle in the baseball experiences a stochastic event, it not only forces that particle to choose between being here and there, but it drags the entire baseball along with it. The baseball ends up either here or there, in exact accord with the normal statistical predictions of quantum mechanics regarding how a measurement of the baseball's position would come out. One of the beautiful features of the theory is that this result, that a single particle's localization brings the whole object with it, falls out of the fundamental mathematical formulation of the theory in a completely straightforward way.

Suppose we perform a measurement of which path the particle is on. We set up a device with a pointer that stays in place if the particle does not pass by it, and swings to a different position if the particle does pass by it. In quantum mechanical language, what happens when such a device is turned on is that, according to von Neumann's Rule One, the whole situation enters into a superposition of the particle being on this path with the device going off, and the particle being on that path with the device not going off.

Now, a stochastic event of the kind GRW describes is very unlikely to happen to the particle itself. But among the millions of particles that make up the measuring device, one of the particles in the pointer will be forced by such a stochastic event to choose between one position and another. It falls very naturally out of the mathematical structure of the theory that this event will not only drag the whole pointer along with it; it will drag the measured particle along with it as well. The pointer being forced to choose between this direction and that direction forces the particle the pointer is measuring to choose between the hard path and the soft path.

This comes about because of a feature of the mathematical structure of quantum mechanics called entanglement. This is precisely the feature that gives rise to the non-localities we encountered in connection with Bell's theorem. The pointer being forced to choose between one position and another forces the measured particle, with which the pointer has become entangled through their measurement interaction, to choose between the hard path and the soft path, and to be correlated with the pointer in exactly the way the measuring device was designed to produce.

This is not exactly analogous to the baseball case, because the measured particle is not itself a constituent of the pointer. Nevertheless, the particle bears a relation to the pointer, the relation of entanglement, that is mathematically similar to the relation a constituent particle of the baseball bears to the baseball as a whole. Consequently, if any part of the pointer is forced to choose, the particle entangled with that pointer is forced to choose as well.

What about cases in which different results of a measurement don't get registered in different positions? Is that possible? What these stochastic events tend to do is to localize particles in space, and to localize any other particles those particles may be entangled with, in the way implied by the nature of the entanglement. More broadly, what these stochastic events tend to do is to localize material objects in space.

There is an assumption behind this theory that if we can manage to ensure that macroscopic material objects more or less always have more or less determinate positions, we will have solved the measurement problem. One can very naturally object that this doesn't seem right. There are all kinds of measuring instruments that indicate the outcomes of measurements in some way other than moving something around in space, instruments that indicate outcomes by flashing lights of different colors, or by making different kinds of noises.

With noises, it is not so hard to imagine what is going on. Noises involve moving masses of air molecules, so that if you force everything macroscopic to have a definite position, you force the device to choose to make a noise or not to make a noise. With different colored lights, it is a little harder. There is a general assumption in the background here that sooner or later, all measurements end up being recorded in the spatial position of something or other, even if not until one writes down an inscription in a laboratory notebook, or reports verbally what the outcome of the measurement was.

I wrote a paper with a colleague many years ago, when this theory first came out, analyzing a measurement in which the outcome is recorded by allowing a subatomic particle to strike an old-fashioned fluorescent television screen, making a dot of light emanate from either the top half or the bottom half of the screen. It turns out that if you analyze a situation like that via the GRW theory, none of these stochastic events are going to force a choice between the two outcomes until the light signal begins to be processed by the retina or the visual cortex.

That seems uncomfortably late, and it immediately raises further worries: what if dolphins' brains don't work like that? What if Martians' brains don't work like that? These are genuine worries associated with this theory.

There is a lingering worry, exactly as you say, about the way in which this theory is committed to a claim of the form that everything is going to be fine at the end of the day, as long as you can guarantee that macroscopic material objects always have more or less determinate spatial positions. There is a question about whether that is actually true, and to what extent it is true.

Audience: There's something that feels conspiratorial about it. When I'm measuring the position of a single subatomic particle, it's actually the wave function of the entire measuring apparatus that's deciding which outcome occurs, not the particle itself.

Yes, the theory will certainly deny that it is the particle's own state, prior to measurement, that determines the outcome. But the probabilities of the measurements coming out one way or another will be exactly the quantum mechanical probabilities. You are right that the theory does not present a situation in which what happens when you do what is normally called measuring the position of a particle is determined by what the particle was doing before you measured it.

Audience: And the worry with the example you raised at the end seems to be a straightforward worry about GRW making inconsistent predictions.

What actually happened is that when a colleague and I wrote our paper about the TV screens, GRW responded roughly as follows. They said, "Yes, that's embarrassing. The case you've described is embarrassing. It would have been nice to get the thing decided before it gets into your brain." One can imagine situations in the GRW theory where you have, on a TV screen, not a superimposition but a genuine quantum mechanical superposition of the screen displaying Lucy's face and the screen displaying Desi's face. According to this theory, until somebody walks into the room and looks at it, there was no fact of the matter about which face was on the screen, and the only thing that forced a choice was what was going on in the observer's visual cortex after they looked at it.

There were two stages to GRW's response. In our original paper, we said, "It looks like nothing is going to happen before the image hits an observer's eyes," and we left it at that. GRW then consulted some neuroscientists and came back with the claim that while nothing happens before the image hits the retina, if you analyze the behavior of the retina and the deeper parts of the brain, the optic nerve, the visual cortex, and so on, it turns out that the number of ions squirted across synapses in processing visual information is large enough to allow the GRW stochastic events to kick in quickly. Their conclusion was that by the time the brain gets involved, everything is likely to be settled. And after all, they asked, what evidence do you have that anything was settled before you walked into the room?

That response leaves a great many worries hanging. For one thing, the collapse is happening much later than I expected. I thought this theory was going to make the macroscopic world look the way we ordinarily think it looks, well before any observer enters the picture. For another, even granting that the brain's ion-based processing is sufficient to trigger collapse, what if we were to improve our brains surgically so that they store information using smaller numbers of ions? What if dolphin brains don't work that way, or Martian brains don't work that way? The right thing to say about this theory is that a genuine worry of this form remains. The question is whether we can do better with other theories.

There is a glass-half-empty and a glass-half-full way of looking at this theory. If you compare it to the earlier history of speculations about collapse, where people were talking about consciousness, macroscopicness, measurement, and so on, this is such a vast improvement, so much more recognizably a theory in the scientific tradition than any of those were, that people were enormously grateful for it. On the other hand, once you settle down and get over your initial excitement and subject the theory to careful scrutiny, there are certainly things to worry about, of exactly the kind you've raised.

Audience: Can I ask a question about the realism project? Was that your project? GRW says, if I understand correctly, that the collapsed little hill is the particle. And before the collapse, it's a smeared particle?

That's a slightly different story, but here is what you could say while sticking very close to the standard way of talking about things. The theory says that particles can be in situations where there is no fact of the matter about their positions. That is not anti-realism. If I say there is no fact of the matter about the marital status of the number five, that doesn't mean I'm an anti-Platonist. It doesn't mean I don't believe in the existence of the number five. It just means that the number five is not the sort of object that has a marital status, and that inquiring into the marital status of the number five represents a category mistake.

The flat-footed realist approach here is to say: here is the true story of what happens to electrons when you put them through a two-slit apparatus. They go into a state, a genuine condition of being that we can describe with our equations, in which there fails to be a fact of the matter about where they are located in space. That doesn't mean we are being unrealistic about what's going on. It means we have made a surprising discovery: that particles like electrons can, under certain circumstances, be in situations we didn't previously think they could be in. Electrons have, as it were, modes of being available to them in which there is no fact of the matter about where they are.

There are plenty of other facts of the matter about them. They are in precisely this superposition of being here and here, and there is much we can say about them. This is supposed to be a perfectly realistic view of what is going on. As a flat-footed beginning, we say of this electron that it is in a superposition while moving through the apparatus.

Audience: So the wave equation describes the electron? It completely describes the electron?

Yes, it correctly and completely describes the electron. In other words, a superposition is a new physical state that particles can be in, one we simply didn't know about before. It is not that the particle isn't real, or that it isn't physical.

Audience: It's just that we didn't know about it before.

Exactly. We are not abstaining from talking about what the particle is doing on its way through the apparatus. We can talk about it. We have a language for talking about it, the wave function, and we have a rich vocabulary: we speak of superpositions and so on. This is all part of a realistic description of what is going on. This is not a case of obeying the Bohrian edict that we should not attempt to say anything about what is happening. Something strange and new is happening, and we can say exactly when it occurs, when it ceases to occur, and much more. This is supposed to be a realistic description of what is going on.

What we want out of a realistic description of the world is that whatever indeterminacy exists had better not spread to baseballs, tables, inscriptions, and results of experiments. There are certain features of the world which we are committed by our longstanding everyday experience to thinking there must be a fact of the matter about. There may frequently fail to be a fact of the matter about, say, the position of a single electron in space. There had presumably better not fail to be a fact of the matter, at least not frequently, about the positions of tables, chairs, baseballs, and things like that.

You have to design your theory in such a way that these random stochastic events accomplish at least two things. First, they have almost no effect on isolated microscopic systems, which we know from experience behave according to Rule 1. The fact that all the particles in the two-paths experiment come out white at the end indicates that they are behaving according to Rule 1 throughout; they never experience any collapses. Second, when a measurement is performed, the likelihood of one of these spontaneous events dragging the whole assemblage one way or the other must become very high.

The GRW theory accomplishes both of those things, and it does so without invoking any of the problematic vocabulary needed to locate a boundary, pick out a trigger, or anything of that kind. If we needed to supplement the stochastic events occurring at the microscopic level with additional special claims about what measuring instruments do, the theory would be far less attractive than it is. But we do not need to do that. The stochastic events at the micro level take care of everything.

No special assumptions are made here about what it is to be a measuring device. Measuring devices are modeled as ordinary physical systems in exactly the way they are modeled in classical mechanics. It is the microscopic stochastic modifications that are doing one hundred percent of the work. That, as I warned, is a very crude sketch of the situation with the GRW theory. The GRW theory is probably the best we have within a broad tradition that now includes a number of proposals falling under the description of scientifically respectable versions of a collapse theory. There is, however, another completely different tradition as well.

The measurement problem, once again, is this: if we let everything always evolve in accord with von Neumann's rule number one, if we let everything always evolve in accord with the Schrödinger equation, then the deterministic and unambiguous prediction is that when we carry out a measurement of, say, which path the electron is on in the two-paths apparatus, the world will go into a superposition. That superposition consists of the electron being on the hard path, the measuring device having recorded that the electron is on the hard path, and the observer's brain being in the state corresponding to believing the electron is on the hard path, all of this superimposed with the electron being on the soft path, the measuring device indicating the soft path, and the observer's brain being in the state corresponding to believing the electron is on the soft path.

The problem is supposed to be that this doesn't happen. If we read superpositions in the standard way we are taught to read them, this would mean that at the end of the measurement there is no fact of the matter about where the pointer is pointing, and no fact of the matter about where I, having looked at the pointer, take it to be pointing. Yet we know, in a more direct and empirical way than we know almost anything else, that that is not the case. There does end up being a fact of the matter about where the pointer is pointing, and a fact of the matter about where I take it to be pointing. That is precisely why we need von Neumann's rule number two.

There is a tradition, going back to the doctoral thesis of a physicist named Hugh Everett, written at Princeton in the 1950s, that hopes to show that, on closer examination, there is no need for von Neumann's rule number two at all. The idea is that the wave function we normally take to represent a superposition of two states is, in fact, simply what measurements look like when properly understood. To see why this might be plausible, we should ask: what was it that led us to think, in the case of the two-paths apparatus, that we could not tell a story in which the electron took one path or the other?

What stood in the way of telling such a story was the interference phenomena observable at the end, the fact that all the electrons come out white, a result to which both the state corresponding to the hard path and the state corresponding to the soft path must contribute. It is precisely this interference effect, this 100% white output, that prevents us from saying the electron simply took one path or the other.

Now here is a fact that the Schrödinger equation predicts. Imagine sending not an electron through the two-paths apparatus, but a slightly larger object, a silver atom. It turns out we can perform two-paths experiments with silver atoms as well, and they work in the same way, though one must take greater care to evacuate the chamber and isolate the atom from air molecules. The experiment can even be performed nowadays with relatively small multiatomic molecules. But as the object sent through the apparatus grows larger, these interference effects become, for various reasons connected with entanglement, more and more difficult to measure as a practical matter.

By the time one reaches a large molecule, and one need not get anywhere near a baseball, these entanglement effects make it essentially impossible to measure the interference effects at the end. This does not change the fact that the mathematics looks exactly the same: one still has what corresponds mathematically to a superposition of the object going one way and the object going the other way. But the empirical confirmation of the interference phenomenon, which is what seems to stand in the way of saying the object went this way or that way, becomes harder and harder to obtain. This suggests that there may be grounds for thinking about a state in which a baseball is in a superposition of being here and there very differently from the way we think about a state in which a single neutron is in a superposition of being here and there.

In the case of a neutron, there are things we can easily measure that ought to persuade us that describing that neutron as being either here or there, or in both places, or anything of that kind, is simply unworkable. When things get large enough so that, for all practical purposes, those measurements become impossible, one might say that it becomes sensible to regard the two branches of the superposition as two genuine, self-contained worlds. In one of those worlds, the particle is on the hard path, there is a measuring device that indicates it is on the hard path, and there is an observer who believes it to be on the hard path. In the other world, there is a particle on the soft path, a measuring device that indicates it is on the soft path, and an observer who believes it to be on the soft path.

The idea is that when things get sufficiently macroscopic, and this is admittedly vague, it is not, at least according to its proponents, vague in the damaging way that talk of measurement was vague. The exact fundamental description of the world is the wave function, and the wave function branches in this way: that is the whole story. There is also a coarser, emergent level of description, which becomes a more and more viable way to talk as the superposed states become more and more macroscopically different from one another, and as it becomes harder and harder to measure the interference terms between them.

When the various branches become sufficiently macroscopically distinct, and I confess this is a view I find deeply puzzling, it becomes permissible, or at least not improper, to describe these superpositions as cases in which the world splits into a multiplicity of different universes. In each of those universes there is an observer who is entirely unaware of the others, and the structure of the equations governing the evolution of the wave function explains why this must be so. The analogy Everett himself often used is instructive: someone asks, "If all these other worlds exist, why don't we see them?" Everett's answer is that this is exactly like people asking, during the scientific revolution, "If Newton is right that the Earth is in motion, why don't we fall off?"

In the Newtonian case, the very same theory, F = ma together with the law of gravitation, that entails the Earth is in rapid motion around the sun also entails that we would not feel that motion and would not fall off. The claim about the Schrödinger equation is precisely analogous: the very same feature of the Schrödinger equation that produces the branching also guarantees that an observer on one branch will be entirely unaware of the existence of any of the other branches. This talk of an observer being on one branch is admittedly difficult to make fully rigorous, but the core idea is that when the branches become sufficiently macroscopically distinct, it begins to make sense, approximately and in a vague way, to describe the quantum state as depicting a multiplicity of independent universes that evolve thereafter in a way that is more or less independent of one another.

Our impression that a measurement of, say, which path a particle takes in a two-path device has a single determinate outcome is, on this view, simply false: an illusion produced by the dynamics, in precisely the way that our impression that the Earth is stationary is an illusion produced by the very same dynamical laws that demand the Earth is in motion in the first place.

If this can be made to work, then all of the hysteria about how to impose a collapse, about when to impose a collapse, about where to impose a collapse, about what causes the collapse, was a complete waste of time. We don't need it. Sidney Coleman, who was a very famous physicist at Harvard until about ten years ago, used to call this way of talking "quantum mechanics in your face," precisely because it is just von Neumann's rule number one. There is a way of resisting the temptation to mutilate this rule with any kind of collapse postulate, whether triggered by things like measurement or consciousness, or even the small stochastic modification of the Schrödinger equation that we encounter in something like the GRW theory. If you simply look at what the Schrödinger equation all by itself is telling you, you find that it gives you a picture of the world which fully explains your empirical experience of it.

This is, in my judgment, for a whole host of reasons, a very difficult theory to make clean sense of. But the part of it that has received the most attention, not because it is the hardest to make sense of, but because it is at least a place where a question can be posed in a clear and crisp way, is the following. If somebody asks what our reasons for believing in anything like quantum mechanics are, those reasons have to do with its probabilistic predictions about how experiments come out. We observe that the frequencies of certain experimental outcomes are exactly the type one would take to be confirmatory of a fundamental probabilistic law. Our evidence for quantum mechanics therefore consists entirely of evidence that its probabilistic predictions, its chance predictions, are correct.

There is a prima facie puzzle about a theory like this. If the world is simply deterministically splitting all the time, and there will certainly be, at the end of the experiment, a version of you that sees the electron on the hard path and a version of you that sees the electron on the soft path, it is just not clear what sense one could make of probability talk. One might be tempted to drop all the probability talk, but if you drop the probability talk, you are dropping every story you know how to tell about what our empirical evidence for quantum mechanics is in the first place. So dropping the probability talk does not seem to be an option, and yet if it cannot be dropped, it is not clear what it refers to or what it is about, since this is, after all, a completely deterministic theory.

By way of analogy, suppose you stroll over and have a chat with an amoeba that is about to divide, and you ask it: "What do you think the chances are that you will end up as the one on the right rather than the one on the left?" That is a strange question, and the amoeba would rightly respond by saying it does not understand the question: who, first of all, is the me we are talking about? There seem to be several ways one could describe what things will be like after the split. One might say that the amoeba existing now simply does not survive the splitting process, or one might say that it is in some sense both the left and the right descendant. But to say something beyond that, to say, for instance, that the probability is two-thirds of ending up on the left and one-third of ending up on the right, is just not clearly coherent. It is not clear what room there is to make sense of probability talk in the context of something like the many-worlds interpretation of quantum mechanics.

What I have said so far is just the beginning of this story. It is universally acknowledged by proponents of the many-worlds interpretation that what I have described amounts to an important challenge for their picture, something they must have some answer to. There has been a tremendous amount of ingenuity poured into the project of trying to make sense of probabilities in the many-worlds interpretation. Let me recommend two books at this point. There is a very helpful conference volume from a meeting held at Oxford about ten or fifteen years ago, whose papers were collected in a book called Many Worlds?, which will give readers a useful survey of various attempts to make sense of probability talk in the context of the many-worlds interpretation.

There is also a book by David Wallace called The Emergent Multiverse, which most people would now identify as reporting on the state of the art in attempts to make sense of the many-worlds interpretation, and in particular, attempts to make sense of ordinary probability talk in quantum mechanics within that context. For whatever it is worth, my own sense of things is that these attempts have not yet succeeded, and I am not optimistic that they can succeed, but this is by no means a settled question. It is a hotly debated question at the moment.

Audience: We are trying to understand probability in the context of many worlds. A very naive thing to say is that when people talk about probabilities or chances, they say something like, "There is a world where I do and a world where I don't." Is there anything connecting that to this story? Another way of putting it is that the story of probability in GRW as you told it is that there are these fundamental stochastic chance laws, which is an unanalyzed, basic feature of what probability is and how it arises. Could one not also just say that that is equally incomprehensible, and so here is another way of looking at it: that all probability is, is a measure on possible worlds?

That is a fair point, and it was, at least in the beginning (though I think people have grown much more sophisticated), a kind of double-standard response. People would say, "Nobody knows what probability is. What makes you think it is not this?" There is a serious question there. It is indeed the case that there is no widely agreed-upon, satisfactory philosophical analysis of probability. But it does seem fair to point out the following. In every case in physics or elsewhere prior to Everett where we are tempted to say something like "the probability that such-and-such will occur five minutes from now is 0.7," it has been a feature of all those cases, even if we do not know what probability is, that there is something about the future of the universe that we do not know for certain at the moment we say it.

It would be nonsensical to make such a statement if there were nothing we were uncertain about regarding the future evolution of the universe. There may be various reasons for that uncertainty. We might not know because, although the laws of evolution are perfectly deterministic, we have only incomplete information about present conditions: that is what goes on in classical statistical mechanics. Or the reason we do not know what is going to happen may be that even with complete information about initial conditions, we would be confronted with the fact that the fundamental dynamical laws are themselves stochastic: that is what is going on in the GRW theory. In the many-worlds case, however, neither of those things is going on.

Nothing would be sharpened by a more precise microscopic description of the situation before I perform a measurement of which path the particle is on. The challenge for a many-worlder is to make it plausible that it makes sense for me to say something like, "The probability that the outcome of this experiment I am about to witness will be the soft path is 0.7," and to say that while having no ignorance whatsoever about the future physical evolution of the world. There is an immediate temptation to respond: "Of course there is nothing third-personal that I don't know about the future evolution of the world. What I don't know, and what these chances are about, is where I am going to end up." But then the referent of this "I" becomes very mysterious, and that is exactly what we were discussing in the case of the amoeba.

If people simply say, "Nobody knows what probability means, so I can make it mean anything I want," that seems a little unfair. It does seem to be a feature of every previous kind of sensible discourse that includes a statement like "the probability that this is going to happen is 0.7" that there is something about the future of the world that one is not presently certain of, and that is precisely what seems absent here, and what seems very puzzling. That having been said, if you look at Deutsch's book, or at Many Worlds?, you will find a variety of spectacularly ingenious proposals for trying to make sense of this talk in the presence of complete certainty about what is physically going to happen in the future.

For reasons I do not have a chance to go into here, I remain skeptical of these strategies. There is an essay of mine in the collection Many Worlds?, and there is also an essay forming the last chapter of a book I published a few years ago called After Physics, which goes through various of these strategies and explains why I find them unconvincing. But once again, it would misrepresent the situation to suggest there is anything like consensus about this at the moment.

Audience: I have a perhaps naive question. Suppose I set up a contraption that measures an electron passing through a measurement device, and if it comes out one way, the contraption shoots me in the head, and if it comes out the other way, it does not. I do the measurement, it shoots me in the head, and as I bleed to death, I know for certain that in another branch of the universe, that person lived. So it seems I do know a great deal, even if the chances are fifty-fifty. What sense can talk about probabilities make in that situation?

That was exactly the point: everything branches, and you know everything. There is a nice story in this vein that is worth taking a moment to tell. David Lewis, toward the end of his life, became interested in the many-worlds interpretation of quantum mechanics, and was drawn to cases exactly like the one you are describing. What is poignant about this is that Lewis wrote this paper at a time when he knew he was about to die of diabetes, his condition having grown much worse. By coincidence, when he first delivered this paper at a conference, I was asked to comment on it.

Lewis's thought was this: suppose you perform a measurement where, if the result is on the soft path, you are immediately killed, and if it is on the hard path, you observe that result. His thought, which I never really understood, and which my comment was largely about, was that in that case, the probability you should assign to seeing the hard-path outcome is one. You will definitely see it on the hard path, because you go entirely into whichever branch of you survives. What Lewis quickly realized was that this leads to an ultimate fate for human beings that is worse than any nightmare anyone has ever constructed.

Suppose you step in front of a train and the train runs you over. There will be a very small but non-zero quantum mechanical probability of your surviving, but you will be surviving in a kind of agony that no human being has ever experienced, and that is with probability one what is going to happen to you if you do this. So you have this man who is about to die, and who is inventing stories about how, if many worlds is true, there are things much, much worse than death. Lewis had a nice line somewhere in the talk where he said that you would not be able to report it to anyone else, but if you would like to find out whether this theory is true, you can do so this very day: just go out and dance in traffic, and you will find out. It is a poignant anecdote, given that he knew he was dying. Rather than inventing comforting stories about what was to come, he found himself inventing stories about how things could be incomparably worse. If there are no other questions, that is what I have to say about the many-worlds tradition.

There is a third live tradition of attempts to solve the measurement problem which, unlike GRW, involves no modification of the Schrödinger equation, but which, unlike many-worlds, denies that the wave function is a complete description of physical reality. Bell, who was a wonderful writer, put the lesson of the measurement problem succinctly: either the Schrödinger equation is not right, or it is not everything. The option that the Schrödinger equation is not right is the collapse-theory option, the GRW option, according to which the Schrödinger equation alone cannot be the right story of what is going on, and must be supplemented either by von Neumann's rule two or by the stochastic modifications found in GRW.

The third tradition is the so-called hidden variable tradition, or extra variable tradition. The most successful such theory we have was first formulated by David Bohm in the 1950s, and was completely ignored at the time. Part of what facilitated that neglect was the House Un-American Activities Committee, which hounded Bohm out of the country. A book that tells this historical story nicely is Adam Becker's What Is Real, which covers Bohm's story and many other fascinating episodes in the history of these discussions over the course of the twentieth century.

What is going on in Bohm's theory is, from a metaphysical standpoint, radically and thrillingly conservative. According to the theory, there are two kinds of concrete physical objects in the world. In a world we would normally describe as consisting of a single particle, there are two kinds of concrete fundamental physical objects: a particle, which always has a perfectly determinate position in space, and a wave function field. The wave function field pushes the particle around, not exactly like a force field, but better understood by analogy with a fluid, like a river. The particle is like an infinitely light cork floating in that river, dragged along by whatever currents are present.

Consider the two-paths experiment. When you feed the particle into the hardness box at the beginning, you are feeding both the particle and its accompanying wave function fluid into the box. The particle comes out one aperture or the other — there is always a perfectly definite fact about its location in space. It either takes the hard route or it takes the soft route. The fluid, however, splits and goes in both directions, as a river might when it divides.