The bad news is that this story barely makes sense. What counts as an “observation”? Can the cat herself make an observation, or could a nonliving being? Surely we don’t want to suggest that the phenomenon of consciousness is somehow playing a crucial role in the fundamental laws of physics? (No, we don’t.) And does the purported collapse really happen instantaneously, or is it gradual but just very fast?
IRREVERSIBILITY
At heart, the thing that bugs us about the Copenhagen interpretation of quantum mechanics is that it treats “observing” as a completely distinct kind of natural phenomenon, one that requires a separate law of nature. In classical mechanics, everything that happens can be accounted for by systems evolving according to Newton’s laws. But if we take the collapse of the wave function at face value, a system described by quantum mechanics evolves according to two completely separate kinds of rules:
1. When we’re not looking at it, a wave function evolves smoothly and predictably. The role that Newton’s laws play in classical mechanics is replaced by the Schrödinger equation in quantum mechanics, which operates in a precisely analogous way. Given the state of the system at any one time, we can use the Schrödinger equation to evolve it reliably into the future and into the past. The evolution conserves information and is completely reversible.
2. When we observe it, a wave function collapses. The collapse is not smooth, or perfectly predictable, and information is not conserved. The amplitude (squared) associated with any particular outcome tells us the probability that the wave function will collapse to a state that is concentrated entirely on that outcome. Two different wave functions can very easily collapse to exactly the same state after an observation is made; therefore, wave function collapse is not reversible.
Madness! But it works. The Copenhagen interpretation takes concepts that would seem to be nothing more than useful approximations to some deeper underlying truth—distinguishing between a “system” that is truly quantum mechanical and an “observer” who is essentially classical—and imagines that these categories play a crucial role in the fundamental architecture of reality. Most physicists, even those who use quantum mechanics every day in their research, get along perfectly well speaking the language of the Copenhagen interpretation, and choosing not to worry about the puzzles it presents. Others, especially those who think carefully about the foundations of quantum mechanics, are convinced that we need to do better. Unfortunately there is no strong consensus at present about what that better understanding might be.
For many people, the breakdown of perfect predictability is a troubling feature of quantum mechanics. (Einstein is among them; that’s the origin of his complaint that “God does not play dice with the universe.”) If the Copenhagen interpretation is right, there could be no such thing as Laplace’s Demon in a quantum world; at least, not if that world contained observers. The act of observing introduces a truly random element into the evolution of the world. Not completely random—a wave function may give a very high probability to observing one thing, and a very low probability to observing something else. But irreducibly random, in the sense that there is no piece of missing information that would allow us to predict outcomes with certainty, if only we could get our hands on it.201 Part of the glory of classical mechanics had been its clockwork reliability—even if Laplace’s Demon didn’t really exist, we knew he could exist in principle. Quantum mechanics destroys that hope. It took a long while for people to get used to the idea that probability enters the laws of physics in some fundamental way, and many are still discomforted by the concept.
One of our questions about the arrow of time is how we can reconcile the irreversibility of macroscopic systems described by statistical mechanics with the apparent reversibility of the microscopic laws of physics. But now, according to quantum mechanics, it seems that the microscopic laws of physics aren’t necessarily reversible. The collapse of the wave function is a process that introduces an intrinsic arrow of time into the laws of physics: Wave functions collapse, but they don’t un-collapse. If we observe Miss Kitty and see that she is on the sofa, we know that she is an eigenstate (100 percent on the sofa) right after we’ve done the measurement. But we don’t know what state she was in before we did the measurement. That information, apparently, has been destroyed. All we know is that the wave function must have had some nonzero amplitude for the cat to be on the sofa—but we don’t know how much, or what the amplitude for any other possibilities might have been.
So the collapse of the wave function—if, indeed, that’s the right way to think about quantum mechanics—defines an intrinsic arrow of time. Can that be used to somehow explain “the” arrow of time, the thermodynamic arrow that appears in the Second Law and on which we’ve blamed all the various macroscopic differences between past and future?
Probably not. Although irreversibility is a key feature of the arrow of time, not all irreversibilities are created equal. It’s very hard to see how the fact that wave functions collapse could, by itself, account for the Past Hypothesis. Remember, it’s not hard to understand why entropy increases; what’s hard to understand is why it was ever low to begin with. The collapse of the wave function doesn’t seem to offer any direct help with that problem.
On the other hand, quantum mechanics is very likely to play some sort of role in the ultimate explanation, even if the intrinsic irreversibility of wave function collapse doesn’t directly solve the problem all by itself. After all, we believe that the laws of physics are fundamentally quantum mechanical at heart. It’s quantum mechanics that sets the rules and tells us what is and is not allowed in the world. It’s perfectly natural to expect that these rules will come into play when we finally do begin to understand why our universe had such a low entropy near the Big Bang. We don’t know exactly where this journey is taking us, but we’re savvy enough to anticipate that certain tools will prove useful along the way.
UNCERTAINTY
Our discussion of wave functions has glossed over one crucial property. We’ve said that wave functions assign an amplitude to any possible outcome of an observation we could imagine doing. In our thought experiment, we restricted ourselves to only one kind of observation—the location of the cat—and only two possible outcomes at a time. A real cat, or an elementary particle or an egg or any other object, has an infinite number of possible positions, and the relevant wave function in each case assigns an amplitude to every single possibility.
More important, however, there are things we can observe other than positions. Remembering our experience with classical mechanics, we might imagine observing the momentum rather than the position of our cat. And that’s perfectly possible; the state of the cat is described by a wave function that assigns an amplitude to every possible momentum we could imagine measuring. When we do such a measurement and get an answer, the wave function collapses into an “eigenstate of momentum,” where the new state assigns nonzero amplitude only to the particular momentum we actually observed.
But if that’s true, you might think, what’s to stop us from putting the cat into a state where both the position and momentum are determined exactly, so it becomes just like a classical state? In other words, why can’t we take a cat with an arbitrary wave function, observe its position so that it collapses to one definite value, and then observe its momentum so that it also collapses to a definite value? We should be left with something that is completely determined, no uncertainty at all.
The answer is that there are no wave functions that are simultaneously concentrated on a single value of position and also on a single value of momentum. Indeed, the hope for such a state turns out to be maximally frustrated: If the wave function is concentrated on a single value of position, the amplitudes for different momenta will be spread out as widely as possible over all the possibilities. And vice versa: If the wave function is concentrated on a single momentum, it is spread out widely over all possible positions. So if we observe the position of an object, we lose any knowledge of wha
t its momentum is, and vice versa.202 (If we measure the position only approximately, rather than exactly, we can retain some knowledge of the momentum; this is what actually happens in real-world macroscopic measurements.)
That’s the true meaning of the Heisenberg Uncertainty Principle. In quantum mechanics, it is possible to “know exactly” what the position of a particle is—more precisely, it’s possible for the particle to be in a position eigenstate, where there is a 100 percent probability of finding it in a certain position. Likewise, it is possible to “know exactly” what the momentum is. But we can never know precisely the position and momentum at the same time. So when we go to measure the properties that classical mechanics would attribute to a system—both position and momentum—we can never say for certain what the outcomes will be. That’s the uncertainty principle.
The uncertainty principle implies that there must be some spread of the wave function over different possible values of either position or momentum, or (usually) both. No matter what kind of system we look at, there is an unavoidable quantum unpredictability when we try to measure its properties. The two observables are complementary: When the wave function is concentrated in position, it’s spread out in momentum, and vice versa. Real macroscopic systems that are well described by the classical limit of quantum mechanics find themselves in compromise states, where there is a small amount of uncertainty in both position and momentum. For large enough systems, the uncertainty is relatively small enough that we don’t notice at all.
Keep in mind that there really is no such thing as “the position of the object” or “the momentum of the object”—there is only a wave function assigning amplitudes to the possible outcomes of observations. Nevertheless, we often can’t resist falling into the language of quantum fluctuations—we say that we can’t pin the object down to a single position, because the uncertainty principle forces it to fluctuate around just a bit. That’s an irresistible linguistic formulation, and we won’t be so uptight that we completely refrain from using it, but it doesn’t accurately reflect what is really going on. It’s not that there is a position and a momentum, each of which keeps fluctuating around; it’s that there is a wave function, which can’t simultaneously be localized in position and momentum.
In later chapters we will explore applications of quantum mechanics to much grander systems than single particles, or even single cats—quantum field theory, and also quantum gravity. But the basic framework of quantum mechanics remains the same in each case. Quantum field theory is the marriage of quantum mechanics with special relativity, and explains the particles we see around us as the observable features of the deeper underlying structure—quantum fields—that make up the world. The uncertainty principle will forbid us from precisely determining the position and momentum of every particle, or even the exact number of particles. That’s the origin of “virtual particles,” which pop in and out of existence even in empty space, and ultimately it will lead to the phenomenon of Hawking radiation from black holes.
One thing we don’t understand is quantum gravity. General relativity provides an extremely successful description of gravity as we see it operate in the world, but the theory is built on a thoroughly classical foundation. Gravity is the curvature of spacetime, and in principle we can measure the spacetime curvature as precisely as we like. Almost everyone believes that this is just an approximation to a more complete theory of quantum gravity, where spacetime itself is described by a wave function that assigns different amplitudes to different amounts of curvature. It might even be the case that entire universes pop in and out of existence, just like virtual particles. But the quest to construct a complete theory of quantum gravity faces formidable hurdles, both technical and philosophical. Overcoming those obstacles is the full-time occupation of a large number of working physicists.
THE WAVE FUNCTION OF THE UNIVERSE
There is one fairly direct way of addressing the conceptual issues associated with wave function collapse: Simply deny that it ever happens, and insist that ordinary smooth evolution of the wave function suffices to explain everything we know about the world. This approach—brutal in its simplicity, and far-reaching in its consequences—goes under the name of the “many-worlds interpretation” of quantum mechanics and is the leading competitor to the Copenhagen interpretation. To understand how it works, we need to take a detour into perhaps the most profound feature of quantum mechanics of all: entanglement.
When we introduced the idea of a wave function we considered a very minimalist physical system, consisting of a single object (a cat). We would obviously like to be able to move beyond that, to consider systems with multiple parts—perhaps a cat and also a dog. In classical mechanics, that’s no problem; if the state of one object is described by its position and its momentum, the state of two objects is just the state of both objects individually—two positions and two momenta. So it would be the most natural thing in the world to guess that the correct quantum mechanical description of a cat and a dog would simply be two wave functions, one for the cat and one for the dog.
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 sim
ply 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.
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 31