I want to talk about a certain very deep, primordial intuition that we have about the structure of the world, a deep conviction with which we make our way around in it, and which in modern physical discourse is referred to as the principle of locality. Very crudely, this is the conviction that what happens at a particular point in space and time only directly affects things that are immediately adjacent to it in space and time.
The fact that I used the word directly is important. It is, of course, the case that our experience is full of instances of somebody doing something, or something happening, at a certain point in space and time and producing an effect at a remote point in space and time.
If I flip a light switch on one side of a room, a light may go on at the other corner of the room. In every such case, we have a very deep conviction that if we were to rip up the wall behind the light switch, we would find wires running continuously from the switch to the light. We expect to be able to tell a story about my flipping the switch having effects on the intensities of various electrical fields in its immediate vicinity, setting some electrons in motion in a wire, which then knock into other electrons a little farther away, and those into others farther still, and so on. The whole process amounts to a continuous row of dominoes: one thing knocking into another, without a break, stretching all the way from the location in space and time where I flipped the switch to the location where the light went on.
The same structure appears when I shout across the room to someone and say, "Raise your arm," and they do so, despite being some spatial distance away. We are confident we can tell a story of how that command traveled from me to them by way of sound waves propagating through the air between us. I change the air pressure in the immediate vicinity of my mouth; those changes cause changes in the air pressure a little farther away, and so on, continuously and without a break, all the way from my making the noise to someone raising their arm on the other side of the room.
The same expectation holds if, instead of shouting, I hold up a sign that reads "Please raise your arm," and you raise it as a consequence. Here we expect a story that begins with my raising the sign, proceeds through light bouncing off it and propagating to your eyes, entering your eyes, stimulating your retina, and eventually producing the arm movement. Once again, we take it for granted that there is a completely local story, one that proceeds continuously, without a break, from me to you.
Another expectation that locality leads to is that my capacity to signal you depends on the physical conditions in the region of space between us. If there is an opaque screen between us, then my holding up a sign will not result in your raising your arm, because the propagation of the light signal will be obstructed. The idea that effects propagate continuously through the space between the initiating event and its final result entails that the capacity of two events to be connected in this way depends on the physical conditions in the intervening region. One can imagine conditions obtaining such that the propagation gets interrupted, and the initial cause fails to produce the final effect in the usual way.
Here is another signature of locality. We expect there to be some finite time interval between the initial cause and the final effect, because the sequence of mediating causes and effects has to be something that unfolds over time. In classical physics there is no obvious upper speed limit, but there must be some finite time between the initial cause and the final effect, because we expect to be able to explain what is going on in terms of a temporal sequence. A chain of dominoes knocking into one another is a useful image to keep in mind.
Those are two signatures of a local world. In cases where something causes something else that is remote, either in space, in time, or in both, the capacity of the causal chain to function depends on what is going on in the region of space and time between cause and effect, and it must take a finite amount of time. Furthermore, by placing enough measuring instruments in the region between the initial cause and the final effect, we ought to be able to observe a physical process of propagation unfolding. In the case of my yelling to you, we ought to be able to measure sound waves propagating between us; in the case of my signaling by waving my arms, we ought to be able to measure the propagation of light waves; in the case of a chain of dominoes, we expect to be able to see the dominoes themselves.
These are supposed to be physical effects in every sense of the word. We can see them, we can measure them, and we can tell a story about how they unfold in accord with the familiar laws of physics. So we have, at this point, three signatures: the propagation should be physically measurable and something we can intervene in; its existence should depend on the physical conditions in the region of space and time through which it is taking place; and it should be directly observable.
There is one more signature, somewhat less strict than the others. Our natural expectation is that when something over here affects something over there, the strength of the effect decreases as the distance between the initial cause and the final effect grows larger. We tend to think of the initial cause as radiating effects in all directions, and those effects thin out as one moves farther away. The underlying principle is that direct, unmediated effects are always local in the sense that they act only on conditions immediately adjacent to the initial cause, both spatially and temporally. That is how direct causation works in the world.
This is not a prejudice of modern physics. It is a very deep conviction, something much older than modern science, something much older than the scientific revolution of the seventeenth century. This is something we have suspected about the world forever, indeed, something presumably hardwired into us by natural selection. It predates human existence. There is a clear sense in which dogs believe in locality, and in which mice believe in locality: the lion is not going to kill them unless the lion is where they are, both spatially and temporally. Fish believe in this. I don't know how far back it goes, but it is very deep in the way the world presents itself.
Violations of this intuition are, in many cases, the hallmark of magic. I stick a pin into a voodoo doll and a man gets a stomachache across town. In that case, we are not thinking of a chain of dominoes. Somehow, my sticking the pin into the doll directly and in an unmediated way causes something to happen at a remote point in space. That is the signature of magic. And the fact that it is the signature of magic is a symptom of how deeply convinced we are that the natural world works in what I am calling a local way.
I want to say something about the fate of this intuition, about its relationship to the development of modern physics from the scientific revolution to the present day, because there is a really interesting story there.
This story gets going the minute the modern scientific project gets going in earnest, with Newton. Newton writes down his laws of motion and his universal theory of gravitation, and it is part and parcel of that universal theory that if you have two material bodies at a certain distance from one another in space, those two bodies exert a gravitational force on one another. The way Newton originally writes this law down, it is a blatant violation of the intuition of locality. There is no account, and no attempt to give an account, of these two bodies exerting a force on one another by means of some kind of effect on the space between them that propagates from one body to the other.
It is a consequence of Newton's Law of Gravitation, if you take it seriously, that if you were to move one of these bodies, the force on the other body would immediately change, without any physical effect occurring in the space between them. Newton writes down this universal law of gravitation, and this law combined with his Laws of Motion, F = ma, is a fantastic success. In one swoop, it shows that the explanation of the motions of the planets is exactly the same as the explanation of why rocks fall near the Earth and why projectiles carve out the trajectories they do.
But more or less immediately after it was written down, everyone noticed that this violated what we are calling the intuition of locality, and people were puzzled and disturbed by this violation. In those days, the way people described this was to say that Newton's Law of Gravitation described action at a distance. What they meant by that phrase was precisely this violation of locality: the presence of one mass is somehow directly and in an unmediated way affecting another mass, producing an acceleration in it by exerting a force on it, without any accompanying account of how that happens in a local way.
What is going on in the space between the two bodies that makes it the case that the presence of one mass can produce an acceleration in the other? Newton's formulation provides no such account. Indeed, the very idea of such an account is ruled out by the way Newton writes the law down. It stipulates a direct effect of one mass upon the other, even though they are separated from one another in space.
This is absolutely unlike the case of two billiard balls colliding with one another. It was widely noticed, even in the 17th century, that this constituted a case of what people then called action at a distance, by which they simply meant a violation of what we are calling locality. People were puzzled and disturbed by it, and no one was more puzzled and disturbed than Newton himself. Newton saw this clearly, and he was perplexed by it. The law had a great deal to recommend it in terms of empirical success, but Newton was convinced that its non-locality meant it could not be the final form of the law of gravitation.
Newton believed that if the scientific project was going to succeed in the way he hoped, the law would eventually have to be replaced by a local account of how gravity worked. It is worth pausing to note that Newton himself engaged in some speculation about what such a local version of a gravitational law might look like. He entertained one particular story for a short time before quickly dismissing it, for reasons we will describe. But the very fact that he entertained it at all shows how compelling he found the intuition of locality.
Newton speculated as follows. What if the space between the Earth and the Moon, or the Earth and the Sun, space we normally think of as empty, is actually swarming with tiny, invisible material particles moving in every direction? Suppose you have some large material body floating in this soup of tiny particles. Because the particles have random velocities, some moving this way, some moving that way, the impacts on an isolated body will statistically cancel out. The net effect on the motion of that large body will be zero.
But then Newton reasoned: what if you have two material bodies at some moderate distance from one another? Each body will cast a shadow on the other, that is, the presence of one body will shield the other from some of the tiny particles coming from that direction. The result is that each body receives more particle impacts from the far side than from the near side, and so each gets pushed toward the other. It will appear as if they are attracting one another, but in fact no non-local interaction is taking place at all. The apparent attraction would be entirely explained by perfectly local, billiard-ball-type collisions.
That would be an example, if it succeeded, of a genuinely local account of gravitational phenomena. The two bodies would not be doing anything directly to one another; they would merely be shielding one another from particle impacts on one side, producing a net push in each other's direction. Newton quickly realized, however, that there are several decisive reasons why this account cannot work. First, if it were correct, the gravitational force between two bodies would depend not only on their masses but on their shapes, since different shapes would shield more or fewer of the tiny particles.
Second, even assuming both bodies were spherical, a straightforward calculation shows that the apparent attractive force in such a theory would not fall off with distance in the right way. In order to explain the observed motions of the planets, the force must be proportional to one over the square of the distance between the bodies, the famous inverse-square law. The shadow-particle model predicts a different dependence on distance entirely. For these reasons, Newton recognized that this could not be the correct theory of gravitation.
What is instructive, nonetheless, is the degree to which Newton's own behavior makes clear that even within his own theory, the violation of locality (the existence of action at a distance) was enough to convince him that he had not arrived at the final theory of gravitation. There is a very deep conviction at work here: that when you truly analyze how things work, when you look carefully behind the phenomena, you must be able to find physical processes of propagation actually taking place step by step through space. If no such processes can be found, then the account on offer cannot be the right one.
Things were left there for roughly 300 years. For those 300 years, this flagrantly non-local law sat right in the middle of physics. It is a good historical question, one to which I don't know the answer, how much people were bothered by this during that intervening period. Certainly Newton and his contemporaries were bothered by it. Newton himself believed this could not be the final theory of gravitation, and that the final theory was still to come.
Nothing very interesting, as far as I know, happened in these developments until about 200 years later, in the 19th century.
In the 19th century, people began to investigate electric and magnetic forces in a systematic way, and a similar issue arose. There is the so-called electrostatic force, the Coulomb force. This is a force that operates between charged particles. Unlike gravitation, this force can be either attractive or repulsive: repulsive if you have two positive charges or two negative charges, and attractive if the charges are of opposite sign.
The mathematics of the way this force works is very similar to gravitation. The force is proportional to the charges rather than to the masses, and its dependence on distance is such that as the particles get farther apart, it falls off as one over the square of the distance between them. But in its original form, the so-called electrostatic law of Coulomb, you have this exact same non-locality. You have a law to the effect that two charged particles produce forces on one another across a distance, that is, they produce accelerations in one another, with no suggestion that this depends in any way on a process of propagation, on any local process by means of which one charge lets the other know that it is there. This appears to be a direct effect that two charges at a finite distance from one another are supposed to have on one another, according to Coulomb's law.
Something else interesting happens in the 19th century in connection with these investigations of electrical and magnetic forces. Initially, it is a purely notational development, not treated as an interesting conceptual development at all. People begin talking about something they call electric fields. They describe the forces between two charged particles in the following way: one charged particle is said to produce an electric field around itself that extends out to infinity. "Producing" is perhaps the wrong word, since there is no account of how it does so, but the idea is that a charged particle is always associated with an electric field surrounding it.
This electric field extends outward to infinity, growing weaker and weaker with distance. Its strength decreases as one over R squared as you move farther from the charge. If you adopt this terminology, you can speak of the second charged particle not as reacting directly to the first particle at a distance, but as reacting to the electric field of that first particle at the point in space where the second particle happens to be located. The second particle, on this way of speaking, experiences a force as a result of the electric field associated with particle number one, evaluated at the point where particle number two sits.
As noted, this was initially regarded as a mere notational and terminological convenience, a mathematical shorthand for talking about these forces. It was not the kind of talk that anyone took ontologically seriously.
Nobody regarded this as some kind of solution to the puzzle about the non-locality of these forces. There is a purely formal, purely mathematical sense in which you can describe the force on particle two as if it arises from the field associated with particle one at the point in space where particle two happens to be located. But this was regarded, at least in the early stages, as just a mathematical way of describing what force particle two would experience if it were located at any particular point in any particular spatial relation to particle one.
Throughout the course of the 19th century, reasons began to accumulate why these fields should be taken more ontologically seriously than they had been when they were originally introduced as purely notational devices. A very important figure in this intellectual development was Faraday, one of the great 19th-century investigators of electricity and magnetism. I think Faraday was an especially important figure in this evolution of thinking about the status of fields.
This line of thinking achieves its final form in Maxwell's equations of the electromagnetic field, published toward the end of the 19th century. Faraday had anticipated many of the key features of these equations, but they take their final, clear, pristine mathematical form in Maxwell. When you look at Maxwell's equations, there are at least three distinct reasons to think that fields are not mere notational devices, that they are part of the fundamental physical furniture of the universe, no less than particles are.
By the time you get to Maxwell's equations, it is no exaggeration to say that over the course of the 19th century the fundamental ontology of the physical world had doubled. As of Newton, the physical furniture of the universe was supposed to consist entirely of material particles. By the end of the 19th century, there was a broad consensus among theoretical physicists that the fundamental physical furniture of the world consists of two different kinds of things, material particles and fields, and that these two kinds of things are ontologically on a par with one another.
Let me explain why people reached that conclusion. The first of three compelling reasons concerns energy and momentum. It was noticed by Faraday and others that if you have charged particles moving around and interacting with one another electrically and magnetically, the total energy and total momentum of those particles are not conserved. There are situations in which charged particles approach one another and then recede, and when you add up the total energy of the particles before the interaction and again once the particles are far apart, the two sums are not equal. The same is true of total momentum.
As a result of two centuries of studying Newtonian mechanics, physicists had developed enormous confidence in the principle that quantities like energy and momentum are conserved. It was therefore deeply perplexing that in certain kinds of electrical interactions, the energies and momenta of the particles alone appeared not to be conserved. The resolution came from a mathematician named Poynting, who discovered a simple formula by which energies and momenta could be associated with configurations of the electromagnetic field itself.
The algorithm works as follows: you supply a configuration of electric and magnetic fields, and the formula returns a total energy; a second formula returns a total momentum. What made this significant was that the algorithm was both simple and universal. It could be proved from Maxwell's equations that if you added the energy and momentum calculated by this formula for the fields to the standardly calculated energy and momentum of the moving particles, the combined total was strictly conserved.
Several factors converged to make this persuasive. First, physicists had a deep faith in some principle of conservation of energy and momentum. Second, the formula for associating energies and momenta with field configurations turned out to be remarkably simple. Third, adopting that simple formula allowed Maxwell's equations to entail the conservation of the combined total, particle energy and momentum plus field energy and momentum. This began to convince people that electromagnetic fields really can be carriers of energy and momentum.
Just as when two billiard balls collide we see an exchange of energy and momentum between them, Maxwell's equations describe processes in which charged particles exchange energy and momentum with the fields, and the fields in turn exchange energy and momentum with the particles. This all hangs together very naturally, and it suggests that associating energy and momentum with field configurations is not a purely mathematical convenience. It gives us at least some reason to take seriously the idea that fields are real, concrete physical objects, no less than material particles are. That is reason number one.
The second reason comes from Maxwell's equations. In order to avoid certain mathematical contradictions (something that Faraday had already anticipated through his own reasoning), Maxwell noticed that one cannot maintain the claim with which the Coulomb theory began: that electric and magnetic fields are entirely determined by the positions and motions of the charged particles in the world. Indeed, one of the famous consequences of Maxwell's equations is that there can be electric and magnetic fields doing all kinds of complicated things even in a universe containing zero charged particles.
A famous solution of Maxwell's equations runs as follows. An oscillating electric field gives rise to an oscillating magnetic field, which in turn gives rise to another oscillating electric field, which gives rise to another oscillating magnetic field, and so on, propagating forward through space. Maxwell was able to calculate the speed with which this disturbance propagates through empty space. The fields, it turns out, have an internal dynamics of their own; they are not merely determined by what the charged particles are doing. They can act on each other independently, and they possess a rich dynamical life entirely their own.
One of the great triumphant surprises in the history of physics is that when Maxwell performed this calculation of the speed at which electromagnetic disturbances propagate through space, he obtained a number astonishingly close to the recently measured velocity of light. Before this moment, no one in physics had the slightest suspicion that light had anything to do with electromagnetic fields. The phenomenon of light was considered completely disconnected from what physics had so far investigated, something to be explored in the far future of the discipline. All at once, through the work of Maxwell and Faraday, we suddenly knew what light was.
It is worth noting that accurate measurements of the velocity of light were themselves a recent achievement. People had long been interested in measuring it. Galileo apparently made attempts that were, in retrospect, endearingly naive. He arranged for himself and a companion to stand on separate mountains with hourglasses, one opening a lantern at a precise moment while the other recorded when he saw it. Given what we now know about the velocity of light, this method could not have come anywhere close to success. Galileo had no way of determining whether the light arrived before or after it was released. By the late nineteenth century, however, far more sophisticated methods had been developed, and the measurements were becoming accurate enough to confirm that the number Maxwell had calculated matched, within experimental error, what experimenters were observing.
This episode provides a second category of reasons for believing that electromagnetic fields are not mere notational devices but part of the concrete, fundamental physical furniture of the universe. First, these fields are not simply things we associate with the presence of charged particles, or imagine particles producing and carrying along with them; they have a dynamical life of their own, capable of complex behavior even in hypothetical universes containing no particles whatsoever. Second, there is the phenomenon of light, which everyone already regarded as part of the physical furniture of the universe without knowing what it was. It turns out to be nothing other than oscillating electric and magnetic fields.
There is a third reason, which also emerges from Maxwell's equations, to take the fields ontologically seriously as parts of the furniture of the universe. In Newtonian mechanics, if you are given the positions and velocities of all the particles in the world at any one time, together with their masses and charges, you can predict their positions and velocities at any future time. In Maxwellian electrodynamics, that is not the case.
Suppose you want to concern yourself only with the particles and set the fields aside. It turns out that the initial positions and velocities of the particles are not enough information to determine even the future positions and velocities of those same particles. As we have been discussing, what the fields are doing is not completely determined by what the particles are doing at any given moment, and what the fields are doing can in turn influence what the particles will do later on. Light rays can encounter charged particles and set them jiggling, and so forth. Particulate initial conditions are simply not sufficient initial conditions for predicting the future behavior of the particles.
If you insist that particles are the only things you take ontologically seriously, you will not be able to write down a satisfactory theory of them. In order to know what the particles are going to be doing at later times, you need initial conditions that refer not only to the particles but to the electromagnetic fields as well. This produces the strong impression that a correct description of the world's initial situation must include the situation the fields are in, not merely the situation the particles are in.
Once you include the fields in your description of the world's initial state, everything snaps into place. You have a completely deterministic theory, not only of the motions of the particles, but of the evolution of the fields as well. Adding the fields gives you precisely what you need: a complete description of what the world is doing at the initial time.
For these reasons, and for others more peripheral, there was a striking evolution over the course of the nineteenth century. The attitude toward fields shifted from treating them as metaphysically uninteresting notational devices to a slow recognition that there was no scientifically viable alternative to acknowledging that the fields are genuinely part of the fundamental furniture of the universe, entirely on a metaphysical par with material particles.
Let us consider how this bears on the question of locality we have been following. If you look at Maxwell's equations, you have two charged particles, one here and one here. The fields are more or less quiet, except for the Coulomb fields of these charged particles. One particle experiences a force associated with the existence of the other, which you can calculate from the Coulomb field of that particle.
We can now ask what would happen in Maxwell's theory if we were to suddenly move one of the particles to another point. We know that eventually the other particle would feel a different force, an attractive force pulling it in a new direction rather than the old one. Maxwell's equations tell you exactly what happens to the fields in this situation.
Initially, the fields change only in the immediate vicinity of the particle that was moved. Outside a certain radius from that point, the field has not changed at all. Those changes in the fields then produce further changes a little farther out, and those in turn produce changes farther out still. What you get is an outgoing wave of field changes.
This wave is an electromagnetic wave, similar to light, something whose presence can be detected. It moves outward at a finite velocity, and that finite velocity turns out to be exactly the speed of light. It therefore takes a finite time for the second particle to feel any change in the electric force on it as a result of the motion of the first.
We now have a complete and thoroughly local account of how that change eventually propagates to the second particle, of how the second particle eventually "learns," through the force acting on it, that the first particle has been moved. Initially, the Coulomb force looked to be non-local in exactly the way Newton's gravitational force looked to be non-local. But when you sit with the problem and examine it in detail, it turns out that is not what is going on. There is a thoroughly local explanation of exactly the kind we were hoping for.
This satisfies all of the desiderata we want. Whether the effect occurs depends on the physical conditions in the region between the two particles. It takes a finite amount of time. The process of propagation is itself a detectable physical process, one we can observe unfolding by placing measuring devices in the region between the two particles. We have everything we want: a thoroughly local explanation of what is going on.
Einstein was deeply impressed by these developments. This was what people were excited about when Einstein was in college, and it was one of his inspirations for the development of his theory of gravitation, the general theory of relativity. The general theory of relativity is revolutionary in all sorts of ways, many of which are not directly relevant to our story here. As people have probably heard, it is a complete reinterpretation of the phenomenon of gravitation, conceiving of gravity not as a force but as a phenomenon in which the presence of material bodies distorts the structure of space and time in their vicinities, in such a way as to make projectiles move the way they do. That is the truly profound revolution of general relativity, but it is not the aspect I want to focus on here.
What general relativity also does, it turns out, is something very similar to what Maxwell's theory does with respect to the question of locality. If you move one material body, the gravitational influence on nearby bodies does not change instantaneously, as Newton's theory demanded. Instead, moving a body creates gravitational disturbances, variations in the geometry of space-time, right in its immediate vicinity. Those variations then propagate outward, each causing the next, further and further out. The structure is exactly the same as in electromagnetism, and, interestingly, it also propagates at the speed of light.
A distant body does not immediately feel a different gravitational force when a nearby one is moved; it takes time for the disturbance to propagate outward. General relativity predicts that if you look very carefully, and in this case you do have to look very carefully, you will be able to detect these disturbances. They are called gravitational waves, in analogy to electromagnetic waves. Indeed, two or three years ago, a group of researchers won the Nobel Prize for a spectacularly beautiful and sensitive experiment in which they detected gravitational waves for the first time.
With this, physics had literally held its breath for three hundred years for the problem of non-locality in Newton's gravitational theory to be resolved along exactly the lines Newton had hoped. It is replaced by a thoroughly local account, and that deep conviction, held long before the rise of modern science, that the world must be local is vindicated after three hundred years by the development of theoretical physics. Through the development of electromagnetism, through Einstein's new theory of gravitation inspired by those developments, and finally through the direct detection of gravitational waves, the conviction that the world must ultimately be local is confirmed. It is an amazing story.
What is interesting, what is poignant, about this story is that the whole thing blows up ten years later. Einstein proposes his theory of gravitation in more or less final form around 1915, and a couple of years after that we see the beginning of the development of quantum mechanics. It happens to be a feature of the quantum mechanical algorithm we have been discussing that there are certain circumstances involving pairs of particles in which, when you carry out a measurement on particle number one, you are immediately required to change your description of particle number two. The algorithm gives you the instruction to make that change in a way that looks strikingly non-local.
The instruction is that you should change your description of particle number two instantaneously, no matter how far apart the two particles may be, and no matter what physical conditions prevail in the region between them. It does not matter if that entire region is filled with the densest conceivable matter. What is happening between the two particles is completely irrelevant, and so is the distance separating them. The moment at which you are supposed to change your description of particle number two, in order to continue your calculations, is exactly the moment at which you carry out the measurement on particle number one. This is the postulate of collapse, von Neumann's rule number two.
This looks strikingly non-local, and Einstein noticed it. I do not know the history well enough to be certain, but I believe Einstein was the first person to draw attention to the fact that this particular feature of the quantum mechanical algorithm, these instructions for predicting the outcomes of future experiments, contained a glaringly non-local prescription. What this suggested to Einstein, given everything physics had been through over the previous three hundred years, and given in particular what Einstein himself had been through over the previous twenty, was that we were obviously dealing with another incomplete theory. We had seen this before, and we knew how the story ended.
Every now and then, something arises in physics that is calculationally useful and yet formally non-local. This is what happened with Newton's universal theory of gravitation, and it is what happened with Coulomb's law of electrostatic attraction and repulsion. When you first write those theories down, you write them in a crude form, and their non-locality is a sure sign that the story is not over. It is a reliable indicator that there is more to be discovered. Einstein, of all people, was in a position to say: there is no reason to take this non-locality seriously at face value, because we have seen this happen before and we know how it comes out.
You work at it a little further and you discover that the non-locality is an artifact of an imperfect, incomplete mathematical formalism, a sign that a deeper theory remains to be found. The quantum mechanical algorithm, as it stood around 1930 when Einstein was confronting it, simply was not the final theory. Indeed, Einstein hoped that whatever the final theory turned out to be would not only restore locality to the world, but would also eliminate all the strange features of quantum mechanics, superposition and everything that goes along with it.
The amazing thing is that it emerges thirty years later in the work of John Bell, and this time the story does not come out the same way. This time we have an argument, a quite simple argument, which I'm going to try to present here, that this quantum mechanical non-locality cannot be eliminated. That is, one can give a straightforward mathematical proof, as Bell does, that there could not be any algorithm for correctly predicting the outcomes of experiments that matches the quantum mechanical predictions, which were, as far as anyone knew at the time, correct. There cannot be any theory which reproduces those predictions without including, at one point or another, a non-local instruction of exactly this kind.
This is an almost unbearably poignant way for the story to come out, an astonishing way for it to come out. If Bell is right, and I don't want to get ahead of myself before going through the argument, then after all of this drama about losing non-locality and regaining it and losing it again, it turns out that this deep, primordial conviction about how the world works is not merely wrong but demonstrably wrong. At the end of the day, locality simply didn't make it. Many people have said, and I don't think this is any kind of crazy exaggeration, that this is probably the single most shocking result in natural science since the scientific revolution of the seventeenth century.
So let's talk about this result. The experiments that are actually performed here are easier to carry out with photons than with electrons, but the logic is exactly the same in either case. I'm going to describe a version of Bell's reasoning that has to do with electrons. The experimental scenario Bell considers involves three different measurable properties of electrons. Let's call them property A, property B, and property C. These three properties, just like the hardness and color properties we discussed earlier, always take one of two possible values: the value of property A for any given electron is either plus one or minus one, and the same is true for property B and property C.
Here is how the argument starts. This is something that was first noticed by Einstein and two of his collaborators, Podolsky and Rosen. It turns out that if quantum mechanics is right, there is a way to prepare a pair of particles, a so-called EPR pair, such that if you follow a certain set of preparation instructions, call them P, you will get a pair of particles with the following feature. If you measure property A on particle one and property A on particle two, the results will always be opposite: if particle one turns out to have an A-value of plus one, particle two will have an A-value of minus one. The same holds for property B and for property C.
Einstein, Podolsky, and Rosen then reasoned that no matter what quantum mechanics says, even if quantum mechanics says that measuring particle one should change your description of particle two, if anything like locality is true, and they are assuming that something like locality is true, then getting the two particles far enough apart from one another means that measuring A on particle one cannot affect anything about particle two, and vice versa. So if it is a feature of this pair that measuring A always yields opposite results, and similarly for B and C, that must already have been the case as soon as the particles were prepared. They must have had definite values of A, B, and C from the moment of preparation.
If you deny that, the only alternative is that measuring A on particle one somehow produced the opposite value in particle two, and that would be a violation of locality. So if we assume locality is true, then whenever I follow these preparation instructions, I am preparing a pair of particles for each of which there is already, as soon as the preparation is complete, a perfectly determinate fact of the matter about the value of their A property, their B property, and their C property.
It is useful at this point to write down the possible values of these properties. Consider particle one and particle two, each with an A-value, a B-value, and a C-value. What EPR have argued is that once we follow these preparation instructions, there are exactly eight possibilities for what the values of A, B, and C might be for the two particles. We know that if particle one has A-value plus one, particle two must have A-value minus one, and similarly for B and C. This anti-correlation across all three properties constrains the joint state to precisely these eight combinations.
The eight possibilities, listing the A, B, C values for particle one followed by particle two, are: (+1, +1, +1, −1, −1, −1), (+1, +1, −1, −1, −1, +1), (+1, −1, +1, −1, +1, −1), (+1, −1, −1, −1, +1, +1), (−1, +1, +1, +1, −1, −1), (−1, +1, −1, +1, −1, +1), (−1, −1, +1, +1, +1, −1), and (−1, −1, −1, +1, +1, +1). Whenever I follow these preparation instructions, what I am left with is one of these eight conditions, because measuring A on both sides always gives opposite results, measuring B on both sides always gives opposite results, and measuring C on both sides always gives opposite results.
Which of the eight situations do I actually get? That is harder to say. Perhaps it depends on the fine details of how the preparation was carried out, or perhaps there is some probabilistic element such that if I prepare a large number of these pairs in the same way, I get some statistical mixture: a certain percentage in one condition, a certain percentage in another, and so forth. Something along those lines must be going on. Bell takes up this argument of EPR.
Bell, unlike EPR, focuses on cases where we measure one variable for particle one and a different variable for particle two. What happens if we measure A for particle one and B for particle two, or A for particle one and C for particle two, or B for particle one and C for particle two? It turns out that the quantum mechanical algorithm gives definite answers to all of those questions, and the calculation is straightforward.
The variables A, B, and C are chosen in a sufficiently symmetrical way so that quantum mechanics predicts the following: if you measure any one of these variables on particle one and any other on particle two, the probability that the results will be opposite is exactly one quarter. So if you measure A on particle one and B on particle two, the probability of opposite results is one quarter. Likewise for A and C, and likewise for B and C.
Now Bell makes the following observation. Consider all possible assignments of definite values to A, B, and C for each particle, the kind of pre-existing hidden facts that EPR argued must be there. For each such assignment, ask whether at least one of the following three conditions holds: the A-value of particle one is opposite to the B-value of particle two, or the A-value of particle one is opposite to the C-value of particle two, or the B-value of particle one is opposite to the C-value of particle two. When Bell goes through every possible assignment of values, every single one of them satisfies at least one of those three conditions. That means any statistical mixture of such assignments, whatever the preparation procedure produces, will yield a situation where at least one of those conditions holds 100% of the time.
But here is the contradiction. If the quantum mechanical predictions are correct, that each of those three pairs of measurements yields opposite results only one quarter of the time, then the probability that at least one of the three conditions holds cannot exceed three quarters. Yet the hidden-variable picture, in which the values are settled facts from the moment of preparation, forces that probability to be 100%. Those two conclusions are flatly incompatible. We have a contradiction.
We have a straightforward contradiction here, and only two assumptions went into it. The first is that the quantum mechanical predictions, specifically these one-quarter predictions, are correct. It was already known experimentally that if you measure the same observable on both particles, you get opposite results one hundred percent of the time. The question is whether that condition and the condition that measurements of different observables on the two sides come out opposite one quarter of the time are mathematically compatible with one another, whether there is any assignment of definite properties that simultaneously satisfies both statistical conditions. This argument shows that the answer is definitively no.
The second assumption is locality. It is only by assuming locality that we can conclude that, once the preparation is complete, all the relevant variables already have definite values. Locality is what rules out the possibility that measuring particle one alters what is happening with particle two in some way that could explain the statistics. So we have two assumptions that jointly lead to a contradiction: assumption one is locality, and assumption two is the correctness of the quantum mechanical predictions.
At that point it became extremely urgent to go out and experimentally test those quantum mechanical predictions, and that is exactly what people did. These are, in fact, the experiments that were awarded the Nobel Prize a few years ago. Physicists ran the experiments to confirm whether the fraction of cases in which measurements of different observables on the two sides come out opposite is indeed one quarter, as quantum mechanics predicts. The quantum mechanical predictions turned out to be correct. Therefore, locality must be false. There is no local theory that can account for these results.
This brings a story that has grown increasingly intense over the last three hundred years to a stark conclusion: locality is false. One can, of course, tell many stories about why we never noticed this before, why the relevant circumstances are physically exotic, and so on, and those stories are easy to tell. But the situation is unambiguous. No amount of theoretical ingenuity of the kind that rescued locality in the cases of gravitation and electromagnetism can help here.
Note that quantum mechanics itself does not have to be correct in every detail for this argument to go through. It only has to be correct about these specific, easily testable predictions. You go and test them, quantum mechanics turns out to be right about them, and the logical conclusion is inescapable: locality is false. The piece of the quantum mechanical algorithm that instructs you to update your description of particle two the moment you measure particle one was not a disposable mathematical artifact of the formalism. There is no formalism that will make the empirically correct predictions without including a non-local instruction of exactly that kind at one point or another.
A number of serious thinkers have called this the single most shocking result of natural science since the seventeenth century. What is at stake is a very deep intuition about how the world works, the intuition that the world is not magic, not voodoo, that there must be a story about how an effect gets from one place to another. Here, there is no such story. Quantum mechanics predicts, and experiment confirms, that if you look in the region between the two particles and search for any process of propagation, you will not find one. The outcome does not depend on what is happening in the intervening region. It is not that an influence travels across the gap; that is simply the wrong way to think about it. Doing something to particle one does something to particle two, and the effect does not get there by traveling. It jumps, with nothing to do with whatever lies in between.