Fear of Physics (23 page)

This is more than just a pictorial simplification. The fact that water can physically be distinguished
on all scales
only by this one effective degree of freedom, which can take on these two values,
completely determines
the nature of the phase transition in the region of this critical point. This means that the liquid-gas phase transition for water becomes
absolutely identical
to the phase transition in any other material, which at its own critical point can be described by a series of ±1’s.

For example, consider iron. Few people would confuse a block of iron with a glass of water. Now, as anyone who plays with magnets knows, iron can be magnetized in the presence of a magnet.
Microscopically what happens is that each of the iron atoms is a little magnet, with a north and south pole. Under normal conditions, with no other magnets nearby, these iron atoms are randomly aligned, so that on average their individual magnetic fields cancel out to produce no macroscopic magnetic field. Under the influence of an external magnet, however, all the atomic magnets in the iron will line up in the same direction as the external field, producing a macroscopic iron magnet. If the external magnetic field points up, so will all the atomic magnets in the iron. If it points down, they too will point downward.

Now consider an idealized piece of iron, where the atomic magnets are constrained to point only up or down, but not in any other direction. At a low temperature, if an external magnetic field is present that starts out at some value pointing up, all the atomic magnets will be aligned in this direction. But if the external field decreases to zero, it will no longer dictate in which direction the atomic magnets can point. It turns out to remain energetically favorable on average for all of them to point in the same direction as one another, but the direction they choose is random. They could point either up or down. This means that such an iron magnet can have a phase transition. As the magnetic field from outside is tuned to zero, the little atomic magnets that had been following this field could now, due to some random thermal fluctuation, instead spontaneously align pointing down over the sample.

Mathematically, this begins to resemble the case of water. Replace “points up” with “density excess” and “points down” with “density deficit.” Just as in the case of water, one finds that for such a magnet when there is no external magnetic field, there is a characteristic scale such that if one examines the sample on scales smaller than this, thermal fluctuations can still change the
direction in which the magnets point. It will thus be impossible to state that the region has any net magnetic orientation. On scales larger than this, thermal fluctuations will not be able to cause the average magnetic orientation to change and it will remain constant. Furthermore, as one increases the temperature while keeping the external magnetic field zero, the sample will have a critical point. At this point, fluctuations in direction will persist throughout the entire sample in the same way on all scales, so that it will be useless to try to characterize the object by the orientation of its atomic magnets, even in an infinite sample.

What is important here is that, at the critical point, water and such a magnet are
exactly
the same. The fact that the actual microscopic structure of the two is totally different is irrelevant. Because the variations in the material at the critical point are characterized by just two degrees of freedom—up and down, or overdense and underdense—over all scales, even those much larger than the microscopic scales, the physics is insensitive to the microscopic differences. The behavior of water as it approaches its critical point, as labeled by whether it is a liquid or gas, is completely identical to that of the magnet, as labeled by whether its field points up or down. Any measurement you can make on one system will be identical for the other.

The fact that we can use the scaling properties of different systems, in this case their scale invariance near the critical point, to find uniformity and order in what is otherwise an incredibly complex situation is one of the great successes of what has become known as condensed matter physics. This approach, which has revolutionized the way we understand the physics of materials, was pioneered in the 1960s and 1970s by Michael Fisher and Kenneth Wilson at Cornell and Leo Kadanoff at the University of Chicago. The ideas developed in this endeavor have been imported
throughout physics whenever complexity associated with scale has been an issue. Wilson was awarded the Nobel Prize in 1982 for his investigations of the applicability of these techniques to understanding the behavior not just of water but also of elementary particles, as I shall describe in the final chapter. What is important here is how they expose the underlying unity associated with the diverse and complex material world we deal with every day. It is not just the submicroscopic scales of elementary particles or the potentially infinite scales of cosmology where hidden connections can simplify reality. Think about it every time the teapot whistles, or the next time you wake up and see icicles on the window.

PART THREE

FIVE

When an artist thinks of symmetries, he or she may think of endless possibilities, of snowflakes, diamonds, or reflections in a pond. When a physicist thinks of symmetry, he or she thinks of endless impossibilities. What really drives physics is not the discovery of what happens but the discovery of what does not. The universe is a big place, and experience has taught us that whatever can happen does happen. What gives order to the universe is the fact that we can say with absolute precision that certain things never happen. Two stars may collide only once every
million years per galaxy, which seems rare. Summed over all known galaxies, however, that is several thousand such events per year in the visible universe. Nevertheless you can wait 10 billion years and you will never see a ball on Earth fall
up.
That is order. Symmetry is the most important conceptual tool in modern physics precisely because it elucidates those things that do not change or cannot happen.

Symmetries in nature are responsible for guiding physicists in two important ways: They restrict the wealth of possibilities, and they fix the proper way to describe those that remain. What do we mean when we say something possesses a certain symmetry? Take a snowflake, for example. It may possess what a mathematician might call a sixfold symmetry. What this means is that you can hold the snowflake at any of six different angles, and it will look exactly the same.
Nothing has changed.
Now, let’s take a more extreme but familiar example. Imagine a cow as a sphere! Why a sphere? Because it is the most symmetrical thing we can think of. Make any rotation, flip it in the mirror, turn it upside down, and it still looks the same.
Nothing has changed!
But what does this gain us? Well, because no rotation or flip can affect a sphere, the entire description of this object reduces to a single variable, its radius. Because of this, we were able to describe changes in its properties just by scaling this one variable. This feature is general: The more symmetrical something is, the fewer variables are needed to describe it completely.

I cannot overstress the importance of this one feature, and I shall praise it more later. For now, it is important to focus on how symmetries forbid change. One of the most striking things about the world, as Sherlock Holmes pointed out to the bewildered Watson, is that certain things do not happen. Balls do not spontaneously start bouncing up the stairs, nor do they pick up and roll
down the hallway on their own. Vats of water don’t spontaneously heat up, and a pendulum does not rise higher in the second cycle than it did in the first. All of these features arise from the symmetries of nature.

The recognition of this fact began in the work of the classical mathematical physicists of the eighteenth and nineteenth centuries, Joseph-Louis Lagrange in France and Sir William Rowan Hamilton, in England, who put Newton’s mechanics on a more general, consistent mathematical footing. Their work reached fruition in the first half of this century through the brilliant German mathematician Emmy Noether. Unfortunately, her keen intellect did not help this woman in a man’s world. Her untenured and unsalaried position at the distinguished mathematics department at Göttingen University was terminated by anti-Semitic laws in 1933—in spite of support from the greatest mathematician of the time, David Hilbert. (He argued unsuccessfully to the Göttingen faculty that they were part of a university and not a bathing establishment. Alas, university faculties have never been hotbeds of social awareness.)

In a theorem that bears her name, Noether demonstrated a mathematical result of profound importance for physics. After the fact, Noether’s theorem seems eminently reasonable. Its formulation in physics goes essentially as follows: If the equations that govern the dynamical behavior of a physical system do not change when some transformation is made on the system, then for each such transformation there must exist some physical quantity that is itself
conserved,
meaning that it does not change
with time.

This simple finding helps explain one of the most misunderstood concepts in popular science (including many undergraduate texts) because it helps show why certain things are impossible.
For example, consider perpetual motion machines, the favorite invention of crackpot scientists. As I described in chapter 1, they can be pretty sophisticated, and many reasonable people have been duped into investing in them.

Now, the standard reason why most machines of this type cannot work is the conservation of energy. Even without rigorously defining it, most people have a fairly good intuitive idea of what energy is, so that one can explain relatively easily why such a machine is impossible. Consider again the contraption illustrated on page 12. As I described there, after one complete cycle, each of its parts will have returned to its original position; if it was at rest at the beginning of the cycle, it would have to be at rest at the end. Otherwise it would have more energy at the end of the cycle than at the beginning. Energy would have had to be produced somewhere; since nothing has changed in the machine, no energy can be produced.

But the diehard inventor may say to me: “How do I know for sure that energy is conserved? What makes this law so special that it cannot be violated? All existing experiments may support this idea, but maybe there is a way around it. They thought Einstein was crazy too!”

There is some merit in this objection. We should not take anything on faith. So all these books tell undergraduates that Energy Is Conserved (they even capitalize it). And it is claimed that this is a universal law of nature, true for energy in all its forms. But while this is a very useful property of nature, the important issue is
Why?
Emmy Noether gave us the answer, and it disappoints me that many physics texts don’t bother going this far. If you don’t explain why such a wonderous quality exists, it encourages the notion that physics is based on some set of mystical rules laid
down on high, which must be memorized and to which only the initiated have access.

So why
is
energy conserved? Noether’s theorem tells us that it must be related to some symmetry of nature. And I remind you that a symmetry of nature tells us that if we make some transformation, everything still looks the same. Energy conservation is, in fact, related to the very symmetry that makes physics possible. We believe the laws of nature will be the same tomorrow as they are today. If they weren’t, we would have to have a different physics text for every day of the week.

So we believe, and this is to some extent an assumption—but, as I shall show, a
testable
one—that
all
the laws of nature are invariant, that is, they remained unchanged, under time translation. This is a fancy way of saying that they are the same no matter when you measure them. But if we accept this for the moment, then we can show rigorously (that is, mathematically) that there must exist a quantity, which we can call energy, that is constant over time. Thus, as new laws of nature are discovered, we do not have to worry at each stage whether they will lead to some violation of the law of conservation of energy. All we have to assume is that the underlying physical principles don’t change with time.

How, then, can we test our assumptions? First, we can check to see that energy is indeed conserved. This alone may not satisfy you or my inventor. There is another way, however. We can check the laws themselves over time to see that their predictions do not vary. This is sufficient to guarantee energy conservation. But aside from this new method of testing energy conservation, we have learned something much more important. We have learned what
giving up
energy conservation is tantamount to doing. If we
choose not to believe that energy is conserved, then we must also believe that the laws of nature change with time.

It is perhaps not so silly to wonder whether, at least on some cosmic time scale, various laws of nature might actually evolve with time. After all, the universe itself is expanding and changing, and perhaps somehow the formulation of microphysical laws is tied to the macroscopic state of the universe. In fact, such an idea was proposed by Dirac in the 1930s. There are various large numbers that characterize the visible universe, such as its age, its size, the number of elementary particles in it, and so on. There are also some remarkably small numbers, such as the strength of gravity. Dirac suggested that perhaps the strength of gravity might vary as the universe expanded, getting weaker with time! He thought this might naturally explain why gravity might be so weak today, compared to the other forces in nature. The universe is old!