From Eternity to Here (25 page)

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 10
²⁵
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
neutral
for simplicity, we will always be referring to neutral kaons.)