From Eternity to Here

by Sean Carroll

The world is made of things, characterized by positions and momenta, pushed about by certain sets of forces; the job of physics was to classify the kinds of things and figure out what the forces were, and we’d be done.

Quantum mechanics offers an image of the world that is radically different from that of classical mechanics, one that scientists never would have seriously contemplated if the experimental data had left them with any other choice.

But despite its triumphs, quantum mechanics remains somewhat mysterious. Physicists are completely confident in how they use quantum mechanics—they can build theories, make predictions, and test against experiments, and there is never any ambiguity along the way. Nevertheless, we’re not completely sure we know what quantum mechanics really is.

This interpretational anxiety stems from the single basic difference between quantum mechanics and classical mechanics, which is both simple and world shattering in its implications:

According to quantum mechanics, what we can observe about the world is only a tiny subset of what actually exists.

Alone among all of the well-accepted laws of physics, quantum measurement is a process that defines an arrow of time: Once you do it, you can’t undo it. And that’s a mystery.

The one sure thing is that we have to confront the measurement problem head-on if we’re interested in the arrow of time.

THE QUANTUM CAT

In quantum mechanics, there is no fact of the matter about where Miss Kitty (or anything else) is located. The space of states in quantum mechanics just doesn’t work that way. Instead, the states are specified by something called a wave function

We see cats and planets and even electrons in particular positions when we look at them, not in superpositions of different possibilities described by wave functions. But that’s the true magic of quantum mechanics: What we see is not what there is. The wave function really exists, but we don’t see it when we look; we see things as if they were in particular ordinary classical configurations.

For objects such as cats that are macroscopic in size, we never find them in superpositions of the form “75 percent here, 25 percent there”; it’s always “99.9999999 percent (or more) here, 0.0000001 percent (or much less) there.” Classical mechanics is an approximation to how the macroscopic world operates, but a very good one.

HOW WAVE FUNCTIONS WORK

INTERFERENCE

COLLAPSE OF THE WAVE FUNCTION

There is no consensus within the physics community about what really constitutes an observation (or “measurement”) in quantum mechanics, nor about what happens when an observation occurs.

IRREVERSIBILITY

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.

UNCERTAINTY

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.

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.

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 WAVE FUNCTION OF THE UNIVERSE

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.”

ENTANGLEMENT

THE EPR PARADOX

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 actuall...

MANY WORLDS, MANY MINDS

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.

Before we made an observation, the universe was described by a single wave function, which assigned a particular amplitude to every possible observational outcome; after the observation, the universe is described by a single wave function, which assigns a particular amplitude to every possible observational outcome. Before and after, the wave function of the universe is just a particular point in the space of states describing the universe, and that space of states didn’t get any bigger or smaller. No new “worlds” have really been created; the wave function still contains the same amount of information (after all, in this interpretation its evolution is reversible). It has simply evolved in such a way that there are now a greater number of distinct subsets of the wave function describing individual conscious beings such as ourselves. The many-worlds interpretation of quantum mechanics may or may not be right, but to object to it on the grounds that “Gee, that’s a lot of worlds,” is wrong-headed.

DECOHERENCE

Decoherence occurs when the state of some small piece of the universe—your brain, for example—becomes so entangled with parts in the wider environment that it is no longer subject to interference, the phenomenon that truly makes something “quantum.”

The miracle of quantum mechanics was that there is no longer any such thing as “where the object is”; it’s in a true simultaneous superposition of the possible alternatives, which we know must be true via experiments that demonstrate the reality of interference. But if the quantum state describing the object is entangled with something in the outside world, interference becomes impossible, and we’re back to the traditional classical way of looking at things. As far as we are concerned, the object is in one state or another, even if the best we can do is assign a probability to the different alternatives—the probabilities are expressing our ignorance, not the underlying reality.

If the quantum state of some particular subset of the universe represents a true superposition that is un-entangled with the rest of the world, we say it is “coherent”; if the superposition has been ruined by becoming entangled with something outside, we say that it has become “decoherent.”

WAVE FUNCTION COLLAPSE AND THE ARROW OF TIME

PART FOUR

FROM THE KITCHEN TO THE MULTIVERSE

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BLACK HOLES: THE ENDS OF TIME

BLACK HOLES ARE FOR REAL

BLACK HOLES HAVE NO HAIR

Once it has settled, there are three things that we can measure about a black hole: its total mass, how fast it is spinning, and how much electric charge it has.

Making a black hole seems to be an irreversible process, even though Einstein’s equation would appear to be perfectly reversible.

LAWS OF BLACK-HOLE MECHANICS

While we can extract useful work from a black hole up to a point, there is some quantity (the area of the event horizon) that keeps going up during the process and reaches its maximum value when all the useful work has been extracted. Interesting. This really does sound eerily like thermodynamics.

Hawking showed that the area of the event horizon of a black hole never decreases; it either increases or stays constant. That’s much like the behavior of entropy, according to the Second Law of Thermodynamics. The First Law of Thermodynamics is usually summarized as “energy is conserved,” but it actually tells us how different forms of energy combine to make the total energy. There is clearly an analogous rule for black holes: The total mass is given by a formula that includes contributions from the spin and charge.

There is also a Third Law of Thermodynamics: There is a minimum possible temperature, absolute zero, at which the entropy is also a minimum. What, in the case of black holes, is supposed to play the role of “temperature” in this analogy? The answer is the surface gravity of a black hole—how strong the gravitational pull of the hole is near the event horizon, as measured by an observer very far away.

And there is a minimum value for the surface gravity of a black hole—zero!—which is achieved when all of the black-hole energy comes from charge or spin, none from “mass all by itself.”

Finally, there is a Zeroth Law of Thermodynamics: If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other. The analogous statement for black holes is simply “the surface gravity has the same value everywhere on the event horizon of a stationary black hole.”

How seriously should we take this analogy? Is it just an amusing coincidence, or does it reflect some deep underlying truth?