Fear of Physics (8 page)

While this is an unusual and new way of thinking, the beauty of it is that with only one fundamental independent dimensional parameter left, we can approximate the results of what may be intrinsically very complicated phenomena simply in terms of a single quantity. In so doing we can perform some magic. For example, say a new elementary particle is discovered that has three times the mass of the proton or, in energy units, about 3 billion electron Volts—3 GeV (Giga electron Volts), for short. If this particle is unstable, what might we expect its lifetime to be before it decays? It may seem impossible to make such an estimate without knowing any of the detailed physical processes involved. However, we can use dimensional analysis to make a guess. The only dimensional quantity in the problem is the rest mass, or equivalently rest energy of the particle. Since the dimensions of time are equivalent to the dimensions of 1/mass in our system, a reasonable estimate of the lifetime would be k/(3 Ge V), where k is some dimensionless number that, in the absence of any other information, we might hope is not too different from 1. We can convert back to our normal units, say, seconds, using our conversion formula (1/1 eV) = 6 × 10
^–16
sec. Thus we estimate the lifetime of our new particle to be about k × 10
^–25
seconds.

Of course, there really is no magic here. We have not gotten something for nothing. What dimensional analysis has given us is the
scale
of the problem. It tells us that the “natural lifetime” of unstable particles with this kind of mass is around k × 10
^–25
seconds, just like the “natural” lifetime of human beings is of the order of k × 75 years. All the real physics (or, in the latter case, biology) is contained in the unknown quantity
k.
If it is very small, or very large, there must be something interesting to be learned in order to understand why.

Dimensional analysis has, as a result, told us something very important. If the quantity
k
differs greatly from 1, we know that the processes involved must be either very strong or very weak, to make the lifetime of such a particle deviate from its natural value as given by dimensional arguments. It would be like seeing a supercow 10 times the size of a normal cow but weighing only 10 ounces. Simple scaling arguments in that case would tell us that such a cow was made of some very exotic material. In fact, many of the most interesting results in physics are those in which naive dimensional scaling arguments break down. What is important to realize is that without these scaling arguments, we might have no idea that anything interesting was happening in the first place!

In 1974, a remarkable and dramatic event took place along these lines. During the 1950s and 1960s, with the development of new techniques to accelerate high-energy beams of particles to collide, first with fixed targets and then with other beams of particles of ever higher energy, a slew of new elementary particles was discovered. As hundreds and hundreds of new particles were found, it seemed as if any hope of simple order in this system had vanished—until the development of the “quark” model in the
early 1960s, largely by Murray Gell-Mann at Caltech, brought order out of chaos. All of the new particles that had been observed could be formed out of relatively simple combinations—fundamental objects that Gell-Mann called
quarks.
The particles that were created at accelerators could be categorized simply if they were formed from either three quarks or a single quark and its antiparticle. New combinations of the same set of quarks that make up the proton and neutron were predicted to result in unstable particles, comparable in mass to the proton. These were observed, and their lifetimes turned out to be fairly close to our dimensional estimate (that is, approximately 10
^–25
sec). Generically, the lifetimes of these particles were in the neighborhood of 10
^–24
seconds, so that the constant
k
in a dimensional estimate would be about 10, not too far from unity. Yet the interactions between quarks, which allow these particles to decay, at the same time seem to hold them so tightly bound inside particles like protons and neutrons that no single free quark had ever been observed. Such interactions seemed so strong as to defy attempts to model them in detail through any calculational scheme.

In 1973, an important theoretical discovery prepared the path. Working with theories modeled after the theory of electromagnetism and the newly established theory of the weak interactions, David Gross and Frank Wilczek at Princeton and, independently, David Politzer at Harvard discovered that an attractive candidate theory for the strong interactions between quarks had a unique and unusual property. In this theory, each quark could come in one of three different varieties, whimsically labeled “colors,” so the theory was called quantum chromodynamics, or QCD. What Gross, Wilczek, and Politzer discovered was that as quarks moved closer and closer together, the interactions between them, based on their “color,” should become
weaker and weaker!
Moreover, they proved that such a property was unique to this kind of theory—no other type of theory in nature could behave similarly.

This finally offered the hope that one might be able to perform calculations to compare the predictions of the theory with observations. For if one could find a situation where the interactions were sufficiently weak, one could perform simple successive approximations, starting with noninteracting quarks and then adding a small interaction, to make reliable approximate estimates of what their behavior should be.

While theoretical physicists were beginning to assimilate the implications of this remarkable property, dubbed “asymptotic freedom,” experimentalists at two new facilities in the United States—one in New York and one in California—were busily examining ever higher energy collisions between elementary particles. In November 1974, within weeks of each other, two different groups discovered a new particle with a mass three times that of the proton. What made this particle so noticeable was that it had a lifetime about 100 times longer than particles with only somewhat smaller masses. One physicist involved commented that it was like stumbling onto a new tribe of people in the jungle, each of whom was 10,000 years old!

Coincident with this result, Politzer and his collaborator Tom Appelquist realized that this new heavy particle had to be made up of a new type of quark—previously dubbed the charmed quark—whose existence had in fact been predicted several years earlier by theorists for unrelated reasons. Moreover, the fact that this bound state of quarks lived much longer than it seemed to have any right to could be explained as a direct consequence of asymptotic freedom in QCD. If the heavy quark and antiquark coexisted very closely together in this bound state, their interactions
would be weaker than the corresponding interactions of lighter quarks inside particles such as the proton. The weakness of these interactions would imply that it would take longer for the quark and its antiquark to “find” each other and annihilate. Rough estimates of the time it would take, based on scaling the strength of the QCD interaction from the proton size to the estimated size of this new particle, led to reasonable agreement with the observations. QCD had received its first direct confirmation.

In the years since this discovery, experiments performed at still higher energies, where it turns out that the approximations one uses in calculations are more reliable, have confirmed beautifully and repeatedly the predictions of QCD and asymptotic freedom. Even though no one has yet been able to perform a complete calculation in the regime where QCD gets strong, the experimental evidence in the high-energy regime is so overwhelming that no one doubts that we now have the correct theory of the interactions between quarks. In fact, Gross, Wilczek, and Politzer were awarded the Nobel Prize in 2004 for their discovery of asymptotic freedom, and with it, the ability to verify QCD as the theory of the Strong Interaction. And without some dimensional guide for our thinking, the key discoveries that helped put the theory on a firm empirical foundation would not have been appreciated at all. This generalizes well beyond the story of the discovery of QCD. Dimensional analysis provides a framework against which we can test our picture of reality.

 

 

If our worldview begins with the numbers we use to describe nature, it doesn’t stop there. Physicists insist on also using mathematical relations between these quantities to describe physical processes—a practice that may make you question why we don’t
use a more accessible language. But we have no choice. Even Galileo appreciated this fact, some 400 years ago, when he wrote: “Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.”
³

Now saying that mathematics is the “language” of physics may appear as trite as saying that French is the “language” of love. It still doesn’t explain why we cannot translate mathematics as well as we might translate the poems of Baudelaire. And in matters of love, while those of us whose mother tongue is not French may labor under a disadvantage, most of us manage to make do when it counts! No, there is more to it than language alone. To begin to describe how much more, I will borrow an argument from Richard Feynman. Besides being a charismatic personality, Feynman was among the greatest theoretical physics minds in this century. He had a rare gift for explanation, I think due in part to the fact that he had his own way of understanding and deriving almost all the classical results in physics, and also in part to his New York accent.

When Feynman tried to explain the necessity of mathematics,
⁴
he turned to none other than Newton for a precedent. Newton’s greatest discovery, of course, was the Universal Law of Gravity. By showing that the same force that binds us to this sphere we call Earth is responsible for the motions of all the heavenly objects, Newton made physics a universal science. He showed that we have the potential to understand not merely the mechanics of
our human condition and our place in the universe but the universe itself. We tend to take it for granted, but surely one of the most remarkable things about the universe is that the same force that guides a baseball out of the park governs the sublimely elegant motion of our Earth around the sun, our sun around the galaxy, our galaxy around its neighbors, and the whole bunch as the universe itself evolves. It didn’t have to be that way (or perhaps it did—that issue is still open).

Now Newton’s law can be stated in words as follows: The attractive force that gravity exerts between two objects is directed along a line joining them, and depends on the product of their masses and inversely as the square of the distance between them. The verbal explanation is already somewhat cumbersome, but no matter. Combining this with Newton’s other law—that bodies react to forces by changing their velocity in the direction of the force, in a manner proportional to the force and inversely proportional to their masses—you have it all. Every consequence of gravity follows from this result. But how? I could give this description to the world’s foremost linguist and ask him or her to deduce from this the age of the universe by semantic arguments, but it would probably take longer than this time to get an answer.

The point is that mathematics is also a system of
connections,
created by the tools of logic. For example, to continue with this famous example, Johannes Kepler made history in the early seventeenth century by discovering after a lifetime of data analysis that the planets move around the sun in a special way. If one draws a line between the planet and the sun, then the area swept out by this line as the planet moves in its orbit is always the same in any fixed time interval. This is equivalent (using mathematics!) to saying that when the planet is closer to this sun in its orbit it moves faster, and when it is farther it moves more slowly. But
Newton showed that this result is also mathematically identical to the statement that there must be a force directed along a line from the planet to the sun! This was the beginning of the Law of Gravity.

Try as you might, you will never be able to prove, on linguistic grounds alone, that these two statements are identical. But with mathematics, in this case simple geometry, you can prove it to yourself quite directly. (Read Newton’s
Principia
or, for an easier translation, read Feynman.)

The point of bringing all this up is not just that Newton might never have been able to derive his Law of Gravity if he hadn’t been able to make the mathematical connection between Kepler’s observation and the fact that the sun exerted a force on the planets—although this alone was of crucial importance for the advancement of science. Nor is it the fact that without appreciating the mathematical basis of physics, one cannot derive other important connections. The real point is that the connections induced by mathematics are completely fundamental to determining our whole picture of reality.

I think a literary analogy is in order. When I wrote this chapter, I had been reading a novel by the Canadian author Robertson Davies. In a few sentences, he summarized something that hit very close to home: “What really astonished me was the surprise of the men that I could do such a thing.... They could hardly conceive that anybody who read . . . could have another, seemingly completely opposite side to his character. I cannot remember a time when I did not take it as understood that everybody has at least two, if not twenty-two, sides to him.”
⁵

Let me make it a little more personal. One of the many things my wife has done for me has been to open up new ways of seeing the world. We come from vastly different backgrounds. She hails
from a small town, and I come from a big city. Now, people who grow up in a big city as I did tend to view other people very differently than people who grow up in a small town. The vast majority of people you meet each day in a big city are one-dimensional. You see the butcher as a butcher, the mailman as a mailman, the doctor as a doctor, and so on. But in a small town, you cannot help but meet people in more than one guise. They are your neighbors. The doctor may be a drunk, and the womanizer next door may be the inspirational English teacher in the local high school. I have come to learn, as did the protagonist in Davies’s novel (from a small town!), that people cannot be easily categorized on the basis of a single trait or activity. Only when one realizes this does it become possible truly to understand the human condition.