From Eternity to Here (24 page)

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 The rules obeyed in C’ are not those obeyed in C—diagonal lines stretching from lower left to upper right have been replaced by diagonal lines stretching the other way.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 moving
¹¹⁴
; 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.