From Eternity to Here: The Quest for the Ultimate Theory of Time - Sean Carroll (2010)
Part III. ENTROPY AND TIME’S ARROW
Chapter 7. RUNNING TIME BACKWARD
This is what I mean when I say I would like to swim against the stream of time: I would like to erase the consequences of certain events and restore an initial condition.
—Italo Calvino, If on a Winter’s Night a Traveler
Pierre-Simon Laplace was a social climber at a time when social climbing was a risky endeavor.102 When the French Revolution broke out, Laplace had established himself as one of the greatest mathematical minds in Europe, as he would frequently remind his colleagues at the Académie des Sciences. In 1793 the Reign of Terror suppressed the Académie; Laplace proclaimed his Republican sympathies, but he also moved out of Paris just to be safe. (Not without reason; his colleague Antoine Lavoisier, the father of modern chemistry, was sent to the guillotine in 1794.) He converted to Bonapartism when Napoleon took power, and dedicated his Théorie Analytique des Probabilités to the emperor. Napoleon gave Laplace a position as minister of the interior, but he didn’t last very long—something about being too abstract-minded. After the restoration of the Bourbons, Laplace became a Royalist, and omitted the dedication to Napoleon from future editions of his book. He was named a marquis in 1817.
Social ambitions notwithstanding, Laplace could be impolitic when it came to his science. A famous anecdote concerns his meeting with Napoleon, after he had asked the emperor to accept a copy of his Méchanique Céleste—a five-volume treatise on the motions of the planets. It seems unlikely that Napoleon read the whole thing (or any of it), but someone at court did let him know that the name of God was entirely absent. Napoleon took the opportunity to mischievously ask, “M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator.” To which Laplace answered stubbornly, “I had no need of that hypothesis.”103
Figure 31: Pierre-Simon Laplace, mathematician, physicist, swerving politician, and unswerving determinist.
One of the central tenets of Laplace’s philosophy was determinism. It was Laplace who truly appreciated the implications of Newtonian mechanics for the relationship between the present and the future: Namely, if you understood everything about the present, the future would be absolutely determined. As he put it in the introduction to his essay on probability:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.104
These days we would probably say that a sufficiently powerful computer could, given all there was to know about the present universe, predict the future (and retrodict the past) with perfect accuracy. Laplace didn’t know about computers, so he imagined a vast intellect. His later biographers found this a bit dry, so they attached a label to this hypothetical intellect: Laplace’s Demon.
Laplace never called it a demon, of course; presumably he had no need to hypothesize demons any more than gods. But the idea captures some of the menace lurking within the pristine mathematics of Newtonian physics. The future is not something that has yet to be determined; our fate is encoded in the details of the current universe. Every moment of the past and future is fixed by the present. It’s just that we don’t have the resources to perform the calculation.105
There is a deep-seated impulse within all of us to resist the implications of Laplace’s Demon. We don’t want to believe that the future is determined, even if someone out there did have access to the complete state of the universe. Tom Stoppard’s Arcadia once again expresses this anxiety in vivid terms.
VALENTINE: Yes. There was someone, forget his name, 1820s, who pointed out that from Newton’s laws you could predict everything to come—I mean, you’d need a computer as big as the universe but the formula would exist.
CHLOË: But it doesn’t work, does it?
VALENTINE: No. It turns out the maths is different.
CHLOË: No, it’s all because of sex.
CHLOË: That’s what I think. The universe is deterministic all right, just like Newton said, I mean it’s trying to be, but the only thing going wrong is people fancying other people who aren’t supposed to be in that part of the plan.
VALENTINE: Ah. The attraction Newton left out. All the way back to the apple in the garden. Yes. (Pause.) Yes, I think you’re the first person to think of this.106
We won’t be exploring whether sexual attraction helps us wriggle free of the iron grip of determinism. Our concern is with why the past seems so demonstrably different from the future. But that wouldn’t be nearly the puzzle it appears to be if it weren’t for the fact that the underlying laws of physics seem perfectly reversible; as far as Laplace’s Demon is concerned, there’s no difference between reconstructing the past and predicting the future.
Reversing time turns out to be a surprisingly subtle concept for something that would appear at first glance to be relatively straightforward. (Just run the movie backward, right?) Blithely reversing the direction of time is not a symmetry of the laws of nature—we have to dress up what we really mean by “reversing time” in order to correctly pinpoint the underlying symmetry. So we’ll approach the topic somewhat circuitously, through simplified toy models. Ultimately I’ll argue that the important concept isn’t “time reversal” at all, but the similar-sounding notion of “reversibility”—our ability to reconstruct the past from the present, as Laplace’s Demon is purportedly able to do, even if it’s more complicated than simply reversing time. And the key concept that ensures reversibility is conservation of information— if the information needed to specify the state of the world is preserved as time passes, we will always be able to run the clock backward and recover any previous state. That’s where the real puzzle concerning the arrow of time will arise.
Let’s play a game. It’s called “checkerboard world,” and the rules are extremely simple. You are shown an array of squares—the checkerboard—with some filled in white, and some filled in gray. In computer-speak, each square is a “bit”—we could label the white squares with the number “0,” and the gray squares with “1.” The checkerboard stretches infinitely far in every direction, but we get to see only some finite part of it at a time.
The point of the game is to guess the pattern. Given the array of squares before you, your job is to discern patterns or rules in the arrangements of whites and grays. Your guesses are then judged by revealing more checkerboard than was originally shown, and comparing the predictions implied by your guess to the actual checkerboard. That last step is known in the parlance of the game as “testing the hypothesis.”
Of course, there is another name to this game: It’s called “science.” All we’ve done is describe what real scientists do to understand nature, albeit in a highly idealized context. In the case of physics, a good theory has three ingredients: a specification of the stuff that makes up the universe, the arena through which the stuff is distributed, and a set of rules that the stuff obeys. For example, the stuff might be elementary particles or quantum fields, the arena might be four-dimensional spacetime, and the rules might be the laws of physics. Checkerboard world is a model for exactly that: The stuff is a set of bits (0’s and 1’s, white and gray squares), the arena through which they are distributed is the checkerboard itself, and the rules—the laws of nature in this toy world—are the patterns we discern in the behavior of the squares. When we play this game, we’re putting ourselves in the position of imaginary physicists who live in one of these imaginary checkerboard worlds, and who spend their time looking for patterns in the appearance of the squares as they attempt to formulate laws of nature.107
Figure 32: An example of checkerboard world, featuring a simple pattern within each vertical column.
In Figure 32 we have a particularly simple example of the game, labeled “checkerboard A.” Clearly there is some pattern going on here, which should be pretty evident. One way of putting it would be, “every square in any particular column is in the same state.” We should be careful to check that there aren’t any other patterns lurking around—if someone else finds more patterns than we do, we lose the game, and they go collect the checkerboard Nobel Prize. From the looks of checkerboard A, there doesn’t seem to be any obvious pattern as we go across a row that would allow us to make further simplifications, so it looks like we’re done.
As simple as it is, there are some features of checkerboard A that are extremely relevant for the real world. For one thing, notice that the pattern distinguishes between “time,” running vertically up the columns, and “space,” running horizontally across the rows. The difference is that anything can happen within a row—as far as we can tell, knowing the state of one particular square tells us nothing about the state of nearby squares. In the real world, analogously, we believe that we can start with any configuration of matter in space that we like. But once that configuration is chosen, the “laws of physics” tell us exactly what must happen through time. If there is a cat sitting on our lap, we can be fairly sure that the same cat will be nearby a moment later; but knowing there’s a cat doesn’t tell us very much about what else is in the room.
Starting completely from scratch in inventing the universe, it’s not at all obvious that there must be a distinction of this form between time and space. We could imagine a world in which things changed just as sharply and unpredictably from moment to moment in time as they do from place to place in space. But the real universe in which we live does seem to feature such a distinction. The idea of “time,” through which things in the universe evolve, isn’t a logically necessary part of the world; it’s an idea that happens to be extremely useful when thinking about the reality in which we actually find ourselves.
We characterized the rule exhibited by checkerboard A as “every square in a column is in the same state.” That’s a global description, referring to the entire column all at once. But we could rephrase the rule in another way, more local in character, which works by starting from some particular row (a “moment in time”) and working up or down. We can express the rule as “given the state of any particular square, the square immediately above it must be in the same state.” In other words, we can express the patterns we see in terms of evolution through time—starting with whatever state we find at some particular moment, we can march forward (or backward), one row at a time. That’s a standard way of thinking about the laws of physics, as illustrated schematically in Figure 33. You tell me what is going on in the world (say, the position and velocity of every single particle in the universe) at one moment of time, and the laws of physics are a black box that tells us what the world will evolve into just one moment later.108 By repeating the process, we can build up the entire future. What about the past?
Figure 33: The laws of physics can be thought of as a machine that tells us, given what the world is like right now, what it will evolve into a moment later.
FLIPPING TIME UPSIDE DOWN
A checkerboard is a bit sterile and limiting, as imaginary worlds go. It would be hard to imagine these little squares throwing a party, or writing epic poetry. Nevertheless, if there were physicists living in the checkerboards, they would find interesting things to talk about once they were finished formulating the laws of time evolution.
For example, the physics of checkerboard A seems to have a certain degree of symmetry. One such symmetry is time-translation invariance—the simple idea that the laws of physics don’t change from moment to moment. We can shift our point of view forward or backward in time (up or down the columns) and the rule “the square immediately above this one must be in the same state” remains true.109 Symmetries are always like that: If you do a certain thing, it doesn’t matter; the rules still work in the same way. As we’ve discussed, the real world is also invariant under time shifts; the laws of physics don’t seem to be changing as time passes.
Another kind of symmetry is lurking in checkerboard A: time-reversal invariance . The idea behind time reversal is relatively straightforward—just make time run backward. If the result “looks the same”—that is, looks like it’s obeying the same laws of physics as the original setup—then we say that the rules are time-reversal invariant. To apply this to a checkerboard, just pick some particular row of the board, and reflect the squares vertically around that row. As long as the rules of the checkerboard are also invariant under time shifts, it doesn’t matter which row we choose, since all rows are created equal. If the rules that described the original pattern also describe the new pattern, the checkerboard is said to be time-reversal invariant. Example A, featuring straight vertical columns of the same color squares, is clearly invariant under time reversal—not only does the reflected pattern satisfy the same rules; it is precisely the same as the original pattern.
Let’s look at a more interesting example to get a better feeling for this idea. In Figure 34 we show another checkerboard world, labeled “B.” Now there are two different kinds of patterns of gray squares running from bottom to top—diagonal series of squares running in either direction. (They kind of look like light cones, don’t they?) Once again, we can express this pattern in terms of evolution from moment to moment in time, with one extra thing to keep in mind: Along any single row, it’s not enough to keep track of whether a particular square is white or gray. We also need to keep track of what kinds of diagonal lines of gray squares, if any, are passing through that point. We could choose to label each square in one of four different states: “white,” “diagonal line of grays going up and to the right,” “diagonal line of grays going up and to the left,” or “diagonal lines of grays going in both directions.” If, on any particular row, we simply listed a bunch of 0’s and 1’s, that wouldn’t be enough to figure out what the next row up should look like.110 It’s as if we had discovered that there were two different kinds of “particles” in this universe, one always moving left and one always moving right, but that they didn’t interact or interfere with each other in any way.
Figure 34: Checkerboard B, on the left, has slightly more elaborate dynamics than checkerboard A, with diagonal lines of gray squares in both directions. Checkerboard B‘, on the right, is what happens when we reverse the direction of time by reflecting B about the middle row.
What happens to checkerboard B under time reversal? When we reverse the direction of time in this example, the result looks similar in form, but the actual configuration of white and black squares has certainly changed (in contrast to checkerboard A, where flipping time just gave us precisely the set of whites and grays we started with). The second panel in Figure 34, labeled B‘, shows the results of reflecting about some row in checkerboard B. In particular, the diagonal lines that were extending from lower left to upper right now extend from upper left to lower right, and vice versa.
Is the checkerboard world portrayed in example B invariant under time reversal? Yes, it is. It doesn’t matter that the individual distribution of white and gray squares is altered when we reflect time around some particular row; what matters is that the “laws of physics,” the rules obeyed by the patterns of squares, are unaltered. In the original example B, before reversing time, the rules were that there were two kinds of diagonal lines of gray squares, going in either direction; the same is true in example B‘. The fact that the two kinds of lines switched identities doesn’t change the fact that the same two kinds of lines could be found before and after. So imaginary physicists living in the world of checkerboard B would certainly proclaim that the laws of nature were time-reversal invariant.
THROUGH THE LOOKING GLASS
Well, then, what about checkerboard C, shown in Figure 35? Once again, the rules seem to be pretty simple: We see nothing but diagonal lines going from lower left to upper right. If we want to think about this rule in terms of one-step-at-a-time evolution, it could be expressed as “given the state of any particular square, the square one step above and one step to the right must be in the same state.” It is certainly invariant under shifts in time, since that rule doesn’t care about what row you start from.
Figure 35: Checkerboard world C only has diagonal lines of gray squares running from lower left to upper right. If we reverse the direction of time to obtain C‘, we only have lines running from bottom right to top left. Strictly speaking, checkerboard C is not time-reversal invariant, but it is invariant under simultaneous reflection in space and reversal in time.
If we reverse the direction of time in checkerboard C, we get something like the checkerboard C’ shown in the figure. Clearly this is a different situation than before. The rules obeyed in C’ are not those obeyed in C—diagonal lines stretching from lower left to upper right have been replaced by diagonal lines stretching the other way. Physicists who lived in one of these checkerboards would say that time reversal was not a symmetry of their observed laws of nature. We can tell the difference between “forward in time” and “backward in time”—forward is the direction in which the diagonal lines move to the right. It is completely up to us which direction we choose to label “the future,” but once we make that choice it’s unambiguous.
However, that’s surely not the end of the story. While checkerboard C might not be, strictly speaking, invariant under time reversal as we have defined it, there does seem to be something “reversible” about it. Let’s see if we can’t put our fingers on it.
In addition to time reversal, we could also consider “space reversal,” which would be obtained by flipping the checkerboard horizontally around some given column. In the real world, that’s the kind of thing we get by looking at something in a mirror; we can think of space reversal as just taking the mirror image of something. In physics, it usually goes by the name of “parity,” which (when we have three dimensions of space rather than just the one of the checkerboard) can be obtained by simultaneously inverting every spatial direction. Let’s call it parity, so that we can sound like physicists when the occasion demands it.
Our original checkerboard A clearly had parity as a symmetry—the rules of behavior we uncovered would still be respected if we flipped right with left. For checkerboard C, meanwhile, we face a situation similar to the one we encountered when considering time reversal—the rules are not parity symmetric, since a world with only up-and-to-the-right diagonals turns into one with only up-and-to-the-left diagonals once we switch right and left, just as it did when we reversed time.
Nevertheless, it looks like you could take checkerboard C and do both a reversal in time and a parity inversion in space, and you would end up with the same set of rules you started with. Reversing time takes one kind of diagonal to the other, and reflecting space takes them back again. That’s exactly right, and it illustrates an important feature of time reversal in fundamental physics: It is often the case that a certain theory of physics would not be invariant under “naïve time reversal,” which reverses the direction of time but does nothing else, and yet the theory is invariant under an appropriately generalized symmetry transformation that reverses the direction of time and also does some other things. The way this works in the real world is a tiny bit subtle and becomes enormously more confusing the way it is often discussed in physics books. So let’s leave the blocky world of checkerboards and take a look at the actual universe.
THE STATE-OF-THE-SYSTEM ADDRESS
The theories that physicists often use to describe the real world share the underlying framework of a “state” that “evolves with time.” That’s true for classical mechanics as put together by Newton, or for general relativity, or for quantum mechanics, all the way up to quantum field theory and the Standard Model of particle physics. On one of our checkerboards, a state is a horizontal row of squares, each of which is either white or gray (with perhaps some additional information). In different approaches to real-world physics, what counts as a “state” will be different. But in each case we can ask an analogous set of questions about time reversal and other possible symmetries.
A “state” of a physical system is “all of the information about the system, at some fixed moment in time, that you need to specify its future evolution,111 given the laws of physics.” In particular, we have in mind isolated systems—those that aren’t subject to unpredictable external forces. (If there are predictable external forces, we can simply count those as part of the “laws of physics” relevant to that system.) So we might be thinking of the whole universe, which is isolated by hypothesis, or some spaceship far away from any planets or stars.
First consider classical mechanics—the world of Sir Isaac Newton.112 What information do we need to predict the future evolution of a system in Newtonian mechanics? I’ve already alluded to the answer: the position and velocity of every component of the system. But let’s creep up on it gradually.
When someone brings up “Newtonian mechanics,” you know sooner or later you’ll be playing with billiard balls.113 But let’s imagine a game that is not precisely like conventional eight ball; it’s a unique, hypothetical setup, which we might call “physicist’s billiards.” In our eagerness to strip away complications and get to the essence of a thing, physicists imagine games of billiards in which there is no noise or friction, so that perfectly round spheres roll along the table and bounce off one another without losing any energy. Real billiard balls don’t quite behave this way—there is some dissipation and sound as they knock into one another and roll along the felt. That’s the arrow of time at work, as noise and friction create entropy—so we’re putting those complications aside for the moment.
Start by considering a single billiard ball moving alone on a table. (Generalization to many billiard balls is not very hard.) We imagine that it never loses energy, and bounces cleanly off of the bumper any time it hits. For purposes of this problem, “bounces cleanly off the bumper” is part of the “laws of physics” of our closed system, the billiard ball. So what counts as the state of that single ball?
You might guess that the state of the ball at any one moment of time is simply its position on the table. That is, after all, what would show up if we took a picture of the table—you would see where the ball was. But we defined the state to consist of all the information you would need to predict the future evolution of the system, and just specifying the position clearly isn’t enough. If I tell you that the ball is precisely in the middle of the table (and nothing else), and ask you to predict where it will be one second later, you would be at a loss, since you wouldn’t know whether the ball is moving.
Of course, to predict the motion of the ball from information defined at a single moment in time, you need to know both the position and the velocity of the ball. When we say “the state of the ball,” we mean the position and the velocity, and—crucially—nothing else. We don’t need to know (for example) the acceleration of the ball, the time of day, what the ball had for breakfast that morning, or any other pieces of information.
We often characterize the motion of particles in classical mechanics in terms of momentum rather than velocity. The concept of “momentum” goes all the way back to the great Persian thinker Ibn Sina (often Latinized as Avicenna) around the year 1000. He proposed a theory of motion in which “inclination”—weight times velocity—remained constant in the absence of outside influences. The momentum tells us how much oomph an object has, and the direction in which it is moving114; in Newtonian mechanics it is equal to mass times velocity, and in relativity the formula is slightly modified so that the momentum goes to infinity as the velocity approaches the speed of light. For any object with a fixed mass, when you know the momentum you know the velocity, and vice versa. We can therefore specify the state of a single particle by giving its position and its momentum.
Figure 36: A lone billiard ball, moving on a table, without friction. Three different moments in time are shown. The arrows denote the momentum of the ball; it remains constant until the ball rebounds off a wall.
Once you know the position and momentum of the billiard ball, you can predict the entire trajectory as it rattles around the table. When the ball is moving freely without hitting any walls, the momentum stays the same, while the position changes with a constant velocity along a straight line. When the ball does hit a wall, the momentum is suddenly reflected with respect to the line defined by the wall, after which the ball continues on at constant velocity. That is to say, it bounces. I’m making simple things sound complicated, but there’s a method behind the madness.
All of Newtonian mechanics is like that. If you have many billiard balls on the same table, the complete state of the system is simply a list of the positions and momenta of each ball. If it’s the Solar System you are interested in, the state is the position and momentum of each planet, as well as of the Sun. Or, if we want to be even more comprehensive and realistic, we can admit that the state is really the position and momentum of every single particle constituting these objects. If it’s your boyfriend or girlfriend you are interested in, all you need to do is precisely specify the position and momentum of every atom in his or her body. The rules of classical mechanics give unambiguous predictions for how the system will involve, using only the information of its current state. Once you specify that list, Laplace’s Demon takes over, and the rest of history is determined. You are not as smart as Laplace’s Demon, nor do you have access to the same amount of information, so boyfriends and girlfriends are going to remain mysterious. Besides, they are open systems, so you would have to know about the rest of the world as well.
It will often be convenient to think about “every possible state the system could conceivably be in.” That is known as the space of states of the system. Note that space is being used in two somewhat different senses. We have “space,” the physical arena through which actual objects move in the universe, and a more abstract notion of “a space” as any kind of mathematical collection of objects (almost the same as “set,” but with the possibility of added structure). The space of states is a space, which will take different forms depending on the laws of physics under consideration.
In Newtonian mechanics, the space of states is called “phase space,” for reasons that are pretty mysterious. It’s just the collection of all possible positions and momenta of every object in the system. For our checkerboards, the space of states consists of all possible sequences of white and gray squares along one row, possibly with some extra information when diagonal lines ran into one another. Once we get to quantum mechanics, the space of states will consist of all possible wave functions describing the quantum system; the technical term is Hilbert space. Any good theory of physics has a space of states, and then some rule describing how a particular state evolves in time.
Figure 37: Two balls on a billiard table, and the corresponding space of states. Each ball requires two numbers to specify its position on the table, and two numbers to specify its momentum. The complete state of both particles is a point in an eight-dimensional space, on the right. We can’t draw eight dimensions, but you should imagine they are there. Every extra ball on the table adds four more dimensions to the space of states.
The space of states can have a huge number of dimensions, even when ordinary space is just three-dimensional. In this abstract context, a “dimension” is just “a number you need to specify a point in the space.” The space of states has one dimension for each component of position, and one dimension for each component of momentum, for every particle in the system. For a billiard ball confined to move on a flat two-dimensional table, we need to give two numbers to specify the position (because the table itself is two-dimensional), and also two numbers to specify the momentum, which has a magnitude and a direction. So the space of states of a single billiard ball confined to a two-dimensional table is four-dimensional: Two numbers fix the position, two fix the momentum. If we had nine balls on the table, we would have to specify two numbers for the position of each ball and two numbers for the momentum of each ball, so the total phase space would be thirty-six-dimensional. There are always an equal number of dimensions for position and momentum, since there can be momentum along every direction of real space. For a baseball flying through the air, which can be thought of as a single particle moving freely in three-dimensional space, the space of states would be six-dimensional; for 1,000 particles, it would be 6,000-dimensional.
In realistic cases, the space of states is very big indeed. An actual billiard ball consists of approximately 1025 atoms, and the space of states is a list of the position and momentum of each one of them. Instead of thinking of the evolution through time of all those atoms moving through three-dimensional space with their individual momenta, we can equally well think of the evolution of the entire system as the motion of a single point (the state) through a giant-dimensional space of states. This is a tremendous repackaging of a great amount of information; it doesn’t make the description any simpler (we’ve just traded in a large number of particles for a large number of dimensions), but it provides a different way of looking at things.
NEWTON IN REVERSE
Newtonian mechanics is invariant under time reversal. If you made a movie of our single billiard ball bouncing around on a table, nobody to whom you showed the movie would be able to discern whether it was being played forward or backward in time. In either case, you would just see the ball moving in a straight line at constant velocity until it reflected off of a wall.
But that’s not quite the whole story. Back in checkerboard world, we defined time-reversal invariance as the idea that we could reverse the time ordering of the sequence of states of the system, and the result would still obey the laws of physics. On the checkerboard, a state was a row of white and gray squares; for our billiard ball, it’s a point in the space of states—that is, the position and momentum of the ball.
Take a look at the first part of the trajectory of the ball shown in Figure 36. The ball is moving uniformly up and to the right, and the momentum is fixed at a constant value, pointing up and to the right. So the time-reverse of that would be a series of positions of the ball moving from upper right to lower left, and a series of fixed momenta pointing up and to the right. But that’s crazy. If the ball is moving along a time-reversed trajectory, from upper right to lower left, the momentum should surely be pointing in that direction, along the velocity of the ball. Clearly the simple recipe of taking the original set of states, ordered in time, and playing exactly the same states backward in time, does not give us a trajectory that obeys the laws of physics. (Or, apparently, of common sense—how can the momentum point oppositely to the velocity? It’s equal to the velocity times the mass.115)
The solution to this ginned-up dilemma is simple enough. In classical mechanics, we define the operation of time reversal to not simply play the original set of states backward, but also to reverse the momenta. And then, indeed, classical mechanics is perfectly invariant under time reversal. If you give me some evolution of a system through time, consisting of the position and momentum of each component at every moment, then I can reverse the momentum part of the state at every point, play it backward in time, and get a new trajectory that is also a perfectly good solution to the Newtonian equations of motion.
This is more or less common sense. If you think of a planet orbiting the Sun, and decide that you would like to contemplate that process in reverse, you imagine the planet reversing its course and orbiting the other way. And if you watched that for a while, you would conclude that the result still looked like perfectly reasonable behavior. But that’s because your brain automatically reversed the momenta, without even thinking about it—the planet was obviously moving in the opposite direction. We don’t make a big deal about it, because we don’t see momentum in the same way that we see position, but it is just as much part of the state as the position is.
It is, therefore, not true that Newtonian mechanics is invariant under the most naïve definition of time reversal: Take an allowed sequence of states through time, reverse their ordering, and ask whether the new sequence is allowed by the laws of physics. And nobody is bothered by that, even a little bit. Instead, they simply define a more sophisticated version of time-reversal: Take an allowed sequence of states through time, transform each individual state in some simple and specific way, then reverse their ordering. By “transform” we just mean to change each state according to a predefined rule; in the case of Newtonian mechanics, the relevant transformation is “reverse the momentum.” If we are able to find a sufficiently simple way to transform each individual state so that the time-reversed sequence of states is allowed by the laws of physics, we declare with a great sense of achievement that those laws are invariant under time reversal.
It’s all very reminiscent (or should be, if my master plan has been successful) of the diagonal lines from checkerboard C. There we found that if you simply reversed the time ordering of states, as shown on checkerboard C‘, the result did not conform to the original pattern, so checkerboard C is not naïvely time-reversal invariant. But if we first flipped the checkerboard from right to left, and only then reversed the direction of time, the result would obey the original rules. So there does exist a well-defined procedure for transforming the individual states (rows of squares) so that checkerboard C really is time-reversal invariant, in this more sophisticated sense.
This notion of time reversal, which involves transforming states around as well as literally reversing time, might seem a little suspicious, but it is what physicists do all the time. For example, in the theory of electricity and magnetism, time-reversal leaves the electric field unchanged, but reverses the direction of the magnetic field. That’s just a part of the necessary transformation; the magnetic field and the momentum both get reversed before we run time backward.116
The lesson of all this is that the statement “this theory is invariant under time reversal” does not, in common parlance, mean “you can reverse the direction of time and the theory is just as good.” It means something like “you can transform the state at every time in some simple way, and then reverse the direction of time, and the theory is just as good.” Admittedly, it sounds a bit fishy when we start including phrases like in some simple way into the definitions of fundamental physical concepts. Who is to say what counts as sufficiently simple?
At the end of the day, it doesn’t matter. If there exists some transformation that you can do to the state of some system at every moment of time, so that the time-reversed evolution obeys the original laws of physics, you are welcome to define that as “invariance under time reversal.” Or you are welcome to call it some other symmetry, related to time reversal but not precisely the same. The names don’t matter; what matters is understanding all of the various symmetries that are respected or violated by the laws. In the Standard Model of particle physics, in fact, we are faced precisely with a situation where it’s possible to transform the states in such a way that they can be run backward in time and obey the original equations of motion, but physicists choose not to call that “time-reversal invariance.” Let’s see how that works.
RUNNING PARTICLES BACKWARD
Elementary particles don’t really obey the rules of classical mechanics; they operate according to quantum mechanics. But the basic principle is still the same: We can transform the states in a particular way, so that reversing the direction of time after that transformation gives us a perfectly good solution to the original theory. You will often hear that particle physics is not invariant under time reversal, and occasionally it will be hinted darkly that this has something to do with the arrow of time. That’s misleading; the behavior of elementary particles under time reversal has nothing whatsoever to do with the arrow of time. Which doesn’t stop it from being an interesting subject in its own right.
Let’s imagine that we wanted to do an experiment to investigate whether elementary particle physics is time-reversal invariant. You might consider some particular process involving particles, and run it backward in time. For example, two particles could interact with each other and create other particles (as in a particle accelerator), or one particle could decay into several others. If it took a different amount of time for such a process to happen going forward and backward, that would be evidence for a violation of time-reversal invariance.
Atomic nuclei are made of neutrons and protons, which are in turn made of quarks. Neutrons can be stable if they are happily surrounded by protons and other neutrons within a nucleus, but left all alone they will decay in a number of minutes. (The neutron is a bit of a drama queen.) The problem is that a neutron will decay into a combination of a proton, an electron, and a neutrino (a very light, neutral particle).117 You could imagine running that backward, by shooting a proton, an electron, and a neutrino at one another in precisely the right way as to make a neutron. But even if this interaction were likely to reveal anything interesting about time reversal, the practical difficulties would be extremely difficult to overcome; it’s too much to ask that we arrange those particles exactly right to reproduce the time reversal of a neutron decay.
But sometimes we get lucky, and there are specific contexts in particle physics where a single particle “decays” into a single other particle, which can then “decay” right back into the original. That’s not really a decay at all, since only one particle is involved—instead, such processes are known as oscillations. Clearly, oscillations can happen only under very special circumstances. A proton can’t oscillate into a neutron, for example; their electrical charges are different. Two particles can oscillate into each other only if they have the same electric charge, the same number of quarks, and the same mass, since an oscillation shouldn’t create or destroy energy. Note that a quark is different from an antiquark, so neutrons cannot oscillate into antineutrons. Basically, they have to be almost the same particle, but not quite.
Figure 38: A neutral kaon and a neutral antikaon. Since they both have zero electric charge, and the net number of quarks is also zero, the kaon and antikaon can oscillate into each other, even though they are different particles.
Nature hands us the perfect candidate for such oscillations in the form of the neutral kaon. A kaon is a type of meson, which means it consists of one quark and one antiquark. If we want the two types of quarks to be different, and for the total charge to add up to zero, the easiest such particle to make will consist of one down quark and one strange antiquark, or vice versa.118 By convention, we refer to the down/anti-strange combination as “the neutral kaon,” and the strange/anti-down combination as “the neutral antikaon.” They have precisely the same mass, about half the mass of a proton or neutron. It’s natural to look for oscillations between kaons and antikaons, and indeed it’s become something of an industry within experimental particle physics. (There are also electrically charged kaons, combinations of up quarks with strange quarks, but those aren’t useful for our purposes; even if we drop the neutralfor simplicity, we will always be referring to neutral kaons.)
So you’d like to make a collection of kaons and antikaons, and keep track as they oscillate back and forth into each other. If time-reversal invariance is violated, we would expect one process to take just a bit longer sthan the other; as a result, on average your collection would have a bit more kaons than antikaons, or vice versa. Unfortunately, the particles themselves don’t come with little labels telling us which kind they are. They do, however, eventually decay into other particles entirely—the kaon decays into a negatively charged pion, an antielectron, and a neutrino, while the antikaon decays into a positively charged pion, an electron, and an antineutrino. If you measure how often one kind of decay happens compared to the other kind, you can figure out whether the original particle spends more time as a kaon than an antikaon.
Even though the theoretical predictions had been established for a while, this experiment wasn’t actually carried out until 1998, by the CPLEAR experiment at the CERN laboratory in Geneva, Switzerland.119 They found that their beam of particles, after oscillating back and forth between kaons and antikaons, decayed slightly more frequently (about 2/3 of 1 percent) like a kaon than like an antikaon; the oscillating beam was spending slightly more time as kaons than as antikaons. In other words, the process of going from a kaon to an antikaon took slightly longer than the time-reversed process of going from an antikaon to a kaon. Time reversal is not a symmetry of elementary particle physics in the real world.
At least, not “naïve” time reversal, as I defined it above. Is it possible to include some additional transformations that preserve some kind of time-reversal invariance in the world of elementary particles? Indeed it is, and that’s worth discussing.
THREE REFLECTIONS OF NATURE
When you dig deeply into the guts of how particle physics works, it turns out that there are three different kinds of possible symmetries that involve “inverting” a physical property, each of which is denoted by a capital letter. We have time reversal T, which exchanges past and future. We also have parity P, which exchanges right and left. We discussed parity in the context of our checkerboard worlds, but it’s just as relevant to three-dimensional space in the real world. Finally, we have “charge conjugation” C, which is a fancy name for the process of exchanging particles with their antiparticles. The transformations C, P, and T all have the property that when you repeat them twice in a row you simply return to the state you started with.
In principle, we could imagine a set of laws of physics that were invariant under each of these three transformations separately. Indeed, the real world superficially looks that way, as long as you don’t probe it too carefully (for example, by studying decays of neutral kaons). If we made an anti-hydrogen atom by combining an anti-proton with an antielectron, it would have almost exactly the same properties as an ordinary hydrogen atom—except that, if it were to touch an ordinary hydrogen atom, they would mutually annihilate into radiation. So C seems at first blush like a good symmetry, and likewise for P and T.
It therefore came as quite a surprise in the 1950s when one of these transformations—parity—was shown not to be a symmetry of nature, largely through the efforts of three Chinese-born American physicists: Tsung-Dao Lee, Chen Ning Yang, and Chien-Shiung Wu. The idea of parity violation had been floating around for a while, suggested by various people but never really taken seriously. In physics, credit accrues not just to someone who makes an offhand suggestion, but to someone who takes that suggestion seriously enough to put in the work and turn it into a respectable theory or a decisive experiment. In the case of parity violation, it was Lee and Yang who sat down and performed a careful analysis of the problem. They discovered that there was ample experimental evidence that electromagnetism and the strong nuclear force both were invariant under P, but that the question was open as far as the weak nuclear force was concerned.
Lee and Yang also suggested a number of ways that one could search for parity violation in the weak interactions. They finally convinced Wu, who was an exper imentalist specializing in the weak interactions and Lee’s colleague at Columbia, that this was a project worth tackling. She recruited physicists at the National Bureau of Standards to join her in performing an experiment on cobalt-60 atoms in magnetic fields at very low temperatures.
As they designed the experiment, Wu became convinced of the project’s fundamental importance. In a later recollection, she explained vividly what it is like to be caught up in the excitement of a crucial moment in science:
Following Professor Lee’s visit, I began to think things through. This was a golden opportunity for a beta-decay physicist to perform a crucial test, and how could I let it pass?—That Spring, my husband, Chia-Liu Yuan, and I had planned to attend a conference in Geneva and then proceed to the Far East. Both of us had left China in 1936, exactly twenty years earlier. Our passages were booked on the Queen Elizabeth before I suddenly realized that I had to do the experiment immediately, before the rest of the Physics Community recognized the importance of this experiment and did it first. So I asked Chia-Liu to let me stay and go without me.
As soon as the Spring semester ended in the last part of May, I started work in earnest in preparing for the experiment. In the middle of September, I finally went to Washington, D.C., for my first meeting with Dr. Ambler. . . . Between experimental runs in Washington, I had to dash back to Columbia for teaching and other research activities. On Christmas eve, I returned to New York on the last train; the airport was closed because of heavy snow. There I told Professor Lee that the observed asymmetry was reproducible and huge. The asymmetry parameter was nearly -1. Professor Lee said that this was very good. This result is just what one should expect for a two-component theory of the neutrino. 120
Your spouse and a return to your childhood home will have to wait—Science is calling! Lee and Yang were awarded the Nobel Prize in Physics in 1957; Wu should have been included among the winners, but she wasn’t.
Once it was established that the weak interactions violated parity, people soon noticed that the experiments seemed to be invariant if you combined a parity transformation with charge conjugation C, exchanging particles with antiparticles. Moreover, this seemed to be a prediction of the theoretical models that were popular at the time. Therefore, people who were surprised that P is violated in nature took some solace in the idea that combining C and Pappeared to yield a good symmetry.
It doesn’t. In 1964, James Cronin and Val Fitch led a collaboration that studied our friend the neutral kaon. They found that the kaon decayed in a way that violated parity, and that the antikaon decayed in a way that violated parity slightly differently. In other words, the combined transformation of reversing parity and trading particles for antiparticles is not a symmetry of nature.121 Cronin and Fitch were awarded the Nobel Prize in 1980.
At the end of the day, all of the would-be symmetries C, P, and T are violated in Nature, as well as any combination of two of them together. The obvious next step is to inquire about the combination of all three: CPT. In other words, if we take some process observed in nature, switch all the particles with their antiparticles, flip right with left, and run it backward in time, do we get a process that obeys the laws of physics? At this point, with everything else being violated, we might conclude that a stance of suspicion toward symmetries of this form is a healthy attitude, and guess that even CPT is violated.
Wrong again! (It’s good to be the one both asking and answering the questions.) As far as any experiment yet performed can tell, CPT is a perfectly good symmetry of Nature. And it’s more than that; under certain fairly reasonable assumptions about the laws of physics, you can prove that CPT must be a good symmetry—this result is known imaginatively as the “CPT Theorem.” Of course, even reasonable assumptions might be wrong, and neither experimentalists nor theorists have shied away from exploring the possibility of CPT violation. But as far as we can tell, this particular symmetry is holding up.
I argued previously that it was often necessary to fix up the operation of time reversal to obtain a transformation that was respected by nature. In the case of the Standard Model of particle physics, the requisite fixing-up involves adding charge conjugation and parity inversion to our time reversal. Most physicists find it more convenient to distinguish between the hypothetical world in which C, P, and T were all individually invariant, and the real world, in which only the combination CPT is invariant, and therefore proclaim that the real world is not invariant under time reversal. But it’s important to appreciate that there is a way to fix up time reversal so that it does appear to be a symmetry of Nature.
CONSERVATION OF INFORMATION
We’ve seen that “time reversal” involves not just reversing the evolution of a system, playing each state in the opposite order in time, but also doing some sort of transformation on the states at each time—maybe just reversing the momentum or flipping a row on our checkerboards, or maybe something more sophisticated like exchanging particles with antiparticles.
In that case, is every sensible set of laws of physics invariant under some form of “sophisticated time reversal”? Is it always possible to find some transformation on the states so that the time-reversed evolution obeys the laws of physics?
No. Our ability to successfully define “time reversal” so that some laws of physics are invariant under it depends on one other crucial assumption: conservation of information. This is simply the idea that two different states in the past always evolve into two distinct states in the future—they never evolve into the same state. If that’s true, we say that “information is conserved,” because knowledge of the future state is sufficient to figure out what the appropriate state in the past must have been. If that feature is respected by some laws of physics, the laws are reversible , and there will exist some (possibly complicated) transformations we can do to the states so that time-reversal invariance is respected.122
To see this idea in action, let’s return to checkerboard world. Checkerboard D, portrayed in Figure 39, looks fairly simple. There are some diagonal lines, and one vertical column of gray squares. But something interesting happens here that didn’t happen in any of our previous examples: The different lines of gray squares are “interacting” with one another. In particular, it would appear that diagonal lines can approach the vertical column from either the right or the left, but when they get there they simply come to an end.
Figure 39: A checkerboard with irreversible dynamics. Information about the past is not preserved into the future.
That is a fairly simple rule and makes for a perfectly respectable set of “laws of physics.” But there is a radical difference between checkerboard D and our previous ones: This one is not reversible. The space of states is, as usual, just a list of white and gray squares along any one row, with the additional information that the square is part of a right-moving diagonal, a left-moving diagonal, or a vertical column. And given that information, we have no problem at all in evolving the state forward in time—we know exactly what the next row up will look like, and the row after that, and so on.
But if we are told the state along one row, we cannot evolve it backward in time. The diagonal lines would keep going, but from the time-reversed point of view, the vertical column could spit out diagonal lines at completely random intervals (corresponding, from the point of view portrayed in the figure, to a diagonal hitting the vertical column of grays and being absorbed). When we say that a physical process is irreversible, we mean that we cannot construct the past from knowledge of the current state, and this checkerboard is a perfect example of that.
In a situation like this, information is lost. Knowing the state at one time, we can’t be completely sure what the earlier states were. We have a space of states—a specification of a row of white and gray squares, with labels on the gray squares indicating whether they move up and to the right, up and to the left, or vertically. That space of states doesn’t change with time; every row is a member of the same space of states, and any possible state is allowed on any particular row. But the unusual feature of checkerboard D is that two different rows can evolve into the same row in the future. Once we get to that future state, the information of which past configurations got us there is irrevocably lost; the evolution is irreversible.
In the real world, apparent loss of information happens all the time. Consider two different states of a glass of water. In one state, the water is uniform and at the
Figure 40: Apparent loss of information in a glass of water. A future state of a glass of cool water could have come either from the same state of cool water, or from warm water with an ice cube.
same cool temperature; in the other, we have warm water but also an ice cube. These two states can evolve into the future into what appears to be the same state: a glass of cool water.
We’ve encountered this phenomenon before: It’s the arrow of time. Entropy increases as the ice melts into the warm water; that’s a process that can happen but will never un-happen. The puzzle is that the motion of the individual molecules making up the water is perfectly invariant under time reversal, while the macroscopic description in terms of ice and liquid is not. To understand how reversible underlying laws give rise to macroscopic irreversibility, we must return to Boltzmann and his ideas about entropy.