From Eternity to Here (46 page)

That’s not how it works. In quantum mechanics, no matter how many individual pieces make up the system you are thinking about, there is
only one wave function
. Even if we consider the entire universe and everything inside it, there is still only one wave function, sometimes redundantly known as the “wave function of the universe.” People don’t always like to talk that way, for fear of sounding excessively grandiose, but at bottom that’s simply the way quantum mechanics works. (Other people enjoy the grandiosity for its own sake.)

Let’s see how this plays out when our system consists of a cat and a dog, Miss Kitty and Mr. Dog. As before, we imagine that when we look for Miss Kitty, there are only two places we can find her: on the sofa or under the table. Let’s also imagine that there are only two places we can ever observe Mr. Dog: in the living room or out in the yard. According to the initial (but wrong) guess that each object has its own wave function, we would describe Miss Kitty’s location as a superposition of under the table and on the sofa, and separately describe Mr. Dog’s location as a superposition of in the living room or in the yard.

But instead, quantum mechanics instructs us to consider every possible alternative for the entire system—cat plus dog—and assign an amplitude to every distinct possibility. For the combined system, there are four possible answers to the question “What do we see when we look for the cat and the dog?” They can be summarized as follows:

(table, living room)

(table, yard)

(sofa, living room)

(sofa, yard)

Here, the first entry tells us where we see Miss Kitty, and the second where we see Mr. Dog. According to quantum mechanics, the wave function of the universe assigns every one of these four possibilities a distinct amplitude, which we would square to get the probability of observing that alternative.

You may wonder what the difference is between assigning amplitudes to the locations of the cat and dog separately, and assigning them to the combined locations. The answer is
entanglement
—properties of any one subset of the whole can be strongly correlated with properties of other subsets.

ENTANGLEMENT

Let’s imagine that the wave function of the cat/dog system assigns zero amplitude to the possibility (table, yard) and also zero amplitude to (sofa, living room). Schematically, that means the state of the system must be of the form

(table, living room) + (sofa, yard).

This means there is a nonzero amplitude that the cat is under the table and the dog is in the living room, and also a nonzero amplitude that the cat is on the sofa and the dog is in the yard. Those are the only two possibilities allowed by this particular state, and let’s imagine that they have equal amplitude.

Now let’s ask: What do we expect to see if we look for only Miss Kitty? An observation collapses the wave function onto one of the two possibilities, (table, living room) or (sofa, yard), with equal probability, 50 percent each. If we simply don’t care about what Mr. Dog is doing, we would say that there is an equal probability for observing Miss Kitty under the table or on the sofa. In that sense, it’s fair to say that we have no idea where Miss Kitty is going to be before we look.

Now let’s imagine that we instead look for Mr. Dog. Again, there is a 50 percent chance each for the possibilities (table, living room) or (sofa, yard), so if we don’t care what Miss Kitty is doing, it’s fair to say that we have no idea where Mr. Dog is going to be before we look.

Here is the kicker: Even though we have no idea where Mr. Dog is going to be before we look, if we first choose to look for Miss Kitty, once that observation is complete we know exactly where Mr. Dog is going to be, even without ever looking for him! That’s the magic of entanglement. Let’s say that we saw Miss Kitty on the sofa. That means that, given the form of the wave function we started with, it must have collapsed onto the possibility (sofa, yard). We therefore know with certainty (assuming we were right about the initial wave function) that Mr. Dog will be in the yard if we look for him. We have collapsed Mr. Dog’s wave function without ever observing him. Or, more correctly, we have collapsed the wave function of the universe, which has important consequences for Mr. Dog’s whereabouts, without ever interacting with Mr. Dog directly.

This may or may not seem surprising to you. Hopefully, we’ve been so clear and persuasive in explaining what wave functions are all about that the phenomenon of entanglement seems relatively natural. And it should; it’s part and parcel of the machinery of quantum mechanics, and a number of clever experiments have demonstrated its validity in the real world. Nevertheless, entanglement can lead to consequences that—taken at face value—seem inconsistent with the spirit of relativity, if not precisely with the letter of the law. Let’s stress: There is no real incompatibility between quantum mechanics and special relativity (general relativity, where gravity comes into the game, is a different story). But there is a tension between them that makes people nervous. In particular, things seem to happen faster than the speed of light. When you dig deeply into what those “things” are, and what it means to “happen,” you find that nothing really bad is going on—nothing has actually moved faster than light, and no real information can be conveyed outside anyone’s light cone. Still, it rubs people the wrong way.

THE EPR PARADOX

Let’s go back to our cat and dog, and imagine that they are in the quantum state described above, a superposition of (table, living room) and (sofa, yard). But now let’s imagine that if Mr. Dog is out in the yard, he doesn’t just sit there; he runs away. Also, he is very adventurous, and lives in the future, when we have regular rocket flights to a space colony on Mars. Mr. Dog—in the alternative where he starts in the yard, not in the living room—runs away to the spaceport, stows away on a rocket, and flies to Mars, completely unobserved the entire time. It’s only when he clambers out of the rocket into the arms of his old friend Billy, who had graduated from high school and joined the Space Corps and been sent on a mission to the Red Planet, that the state of Mr. Dog is actually observed, collapsing the wave function.

What we’re imagining, in other words, is that the wave function describing the cat/dog system has evolved smoothly according to the Schrödinger equation from

(table, living room) + (sofa, yard)

to

(table, living room) + (sofa, Mars).

There’s nothing impossible about that—implausible, maybe, but as long as nobody made any observations during the time it took the evolution to happen, we’ll end up with the wave function in this superposition.

But the implications are somewhat surprising. When Billy unexpectedly sees Mr. Dog bounding out of the spaceship on Mars, he makes an observation and collapses the wave function. If he knew what the wave function was to begin with, featuring an entangled state of cat and dog, Billy
immediately
knows that Miss Kitty is on the sofa, not under the table. The wave function has collapsed to the possibility (sofa, Mars). Not only is Miss Kitty’s state now known even without anyone interacting with her; it seems to have been determined instantaneously, despite the fact that it takes at least several minutes to travel between Mars and Earth even if you were moving at the speed of light.

This feature of entanglement—the fact that the state of the universe, as described by its quantum wave function, seems to change “instantaneously” throughout space, even though the lesson of special relativity was supposed to be that there’s no unique definition of what “instantaneously” means—bugs the heck out of people. It certainly bugged Albert Einstein, who teamed up with Boris Podolsky and Nathan Rosen in 1935 to write a paper pointing out this weird possibility, now known as the “EPR paradox.”
²⁰³
But it’s not really a “paradox” at all; it might fly in the face of our intuition, but not of any experimental or theoretical requirements.

The important feature of the apparently instantaneous collapse of a wave function that is spread across immense distances is that it cannot be used to actually transmit any information faster than light. The thing that bothers us is that, before Billy observed the dog, Miss Kitty back here on Earth was not in any definite location—we had a 50/50 chance to observe her on the sofa or under the table. Once Billy observes Mr. Dog, we now have a 100 percent chance of observing her to be on the sofa. But so what? We don’t actually know that Billy did any such observation—for all we know, if we looked for Mr. Dog we would find him in the living room. For Billy’s discovery to make any difference to us, he would have to come tell us about it, or send us a radio transmission—one way or another, he would have to communicate with us by conventional slower-than-light means.

Entanglement between two far-apart subsystems seems mysterious to us because it violates our intuitive notions of “locality”—things should only be able to directly affect other nearby things, not things arbitrarily far away. Wave functions just don’t work like that; there is one wave function that describes the entire universe all at once, and that’s the end of it. The world we observe, meanwhile, still respects a kind of locality—even if wave functions collapse instantaneously all over space, we can’t actually take advantage of that feature to send signals faster than light. In other words: As far as things actually bumping into you and affecting your life, it’s still true that they have to be right next to you, not far away.

On the other hand, we shouldn’t expect that even this weaker notion of locality is truly a sacred principle. In the next chapter we’ll talk a little bit about quantum gravity, where the wave function applies to different configurations of spacetime itself. In that context, an idea like “objects can affect each other only when they are nearby” ceases to have any absolute meaning. Spacetime itself is not absolute, but only has different amplitudes for being in different configurations—so the notion of “the distance between two objects” becomes a little fuzzy. These are ideas that have yet to be fully understood, but the final theory of everything is likely to exhibit non-locality in some very dramatic ways.

MANY WORLDS, MANY MINDS

The leading contender for an alternative to the Copenhagen view of quantum mechanics is the so-called
many-worlds interpretation
. “Many worlds” is a scary and misleading name for what is really a very straightforward idea. That idea is this: There is no such thing as “collapse of the wave function.” The evolution of states in quantum mechanics works just like it does in classical mechanics; it obeys a deterministic rule—the Schrödinger equation—that allows us to predict the future and past of any specific state with perfect fidelity. And that’s all there is to it.

The problem with this claim is that we appear to
see
wave functions collapsing all the time, or at least to observe the effects of the collapse. We can imagine arranging Miss Kitty in a quantum state that has equal amplitudes for finding her on the sofa or under the table; then we look for her, and see her under the table. If we look again immediately thereafter, we’re going to see her under the table 100 percent of the time; the original observation (in the usual way of talking about these things) collapsed the wave function to a table-eigenstate. And that way of thinking has empirical consequences, all of which have been successfully tested in real experiments.

The response of the many-worlds advocate is simply that you are thinking about it wrong. In particular, you have misidentified
yourself
in the wave function of the universe. After all, you are part of the physical world, and therefore you are also subject to the rules of quantum mechanics. It’s not right to set yourself off as some objective classical observing apparatus; we need to take your own state into account in the wave function.

So, this new story goes, we shouldn’t just start with a wave function describing Miss Kitty as a superposition of (sofa) and (table); we should include your own configuration in the description. In particular, the relevant feature of your description is what you have observed about Miss Kitty’s position. There are three possible states you could be in: You could have seen her on the sofa, you could have seen her under the table, and you might not have looked yet. To start with, the wave function of the universe (or at least the bit of it we’re describing here) gives Miss Kitty equal amplitude to be on the sofa or under the table, while you are uniquely in the state of not having looked yet. This can be schematically portrayed like this:

(sofa, you haven’t yet looked) + (table, you haven’t yet looked).

Now you observe where she is. In the Copenhagen interpretation, we would say that the wave function collapses. But in the many-worlds interpretation, we say that your own state becomes entangled with that of Miss Kitty, and the combined system evolves into a superposition:

(sofa, you see her on the sofa) + (table, you see her under the table).

There is no collapse; the wave function evolves smoothly, and there is nothing special about the process of “observation.” What is more, the entire procedure is reversible—given the final state, we could use the Schrödinger equation to uniquely recover the original state. There is no intrinsically quantum mechanical arrow of time in this interpretation. For many reasons, this is an altogether more elegant and satisfying picture of the world than that provided by the Copenhagen picture.