## Arnold Sommerfeld

With respect to [the fine structure constant] we are in the rather humiliating position of people who have to wrap a piece of string around a cylinder to determine pi.

—Edward M. Purcell

There are many constants in physics, but there are only a few fundamental constants and they are special because they have universal significance. Although the fine-structure constant is the subject of this chapter, let's begin with six other basic constants of nature. These six, with their current best values, are:

the gravitational constant, G = 6.67259 X 10-11 meter3/kilo-

gram second (m3/kg s); the speed of light, c = 2.99 792 458 X 108 meter/second (m/s); Planck's constant, h = 6.6260755 X 10-34 joule second (J s); the electron's charge, e = 1.60217733 X 10-19 coulomb (C); the electron's mass, me = 9.1093897 X 10-31 kilogram (kg); the proton's mass, mp = 1.6726231 X 10-27 kilogram (kg).

Immediately, one thing stands out: these constants are not approximate numbers. They are known with extreme precision. There is an uncertainty (not shown explicitly above) in each of the last digits, for example the last digit in Planck's constant or the last digit in the electron's charge. In each case, however, the digits up to the last are real. The precision of these constants is not an accident. There are reasons to know these numbers with the greatest precision possible; thus, ingenious experiments have been and are being designed to measure these constants with ever greater precision.

The importance of these constants is manifold. For example, are they really constants? Or have they changed over the life of the universe? Are they changing even now? If they are changing, the change over the course of human history would be minuscule and our only chance of detecting any change is by pushing our knowledge of these constants to ever greater precision.

Enormous significance derives from the universality of these constants. For every galaxy in the universe, in whatever way the stars are distributed throughout it, the unseen force of gravity extends across the light years, binds star with star, and gives the galaxy its shape. No current evidence challenges the conclusion that the gravitational force is the same everywhere. The gravitational constant, G = 6.67259 X 10-11 m3/kg s, determines the strength of the attractive force that binds the Earth to the Sun and holds the Milky Way together. For each star in the universe, in whatever galaxy it resides, the light emanating from it travels at the same speed, c = 2.99 792 458 X 108 m/s. For each atom in the universe, in whatever material environment it exists, its size and its behavior is determined by Planck's constant, h = 6.6260755 X 10-34 Js. Every electron in the universe carries the same charge and has the same mass. The same holds true for the proton.

The actual numerical values of these constants depend on the units used to express them. If, for example, we used the foot rather than the meter for the unit of length, the gravitational constant, G, would have the value 2.35640 X 10-9 ft3/kg s. So the value of these constants depends on the units employed. However, if we had a method of comparing units with an intelligent being from Galaxy X, we would find their numerical values for these constants agree with ours.

The first three constants listed above, G, c, and h, have their origins in four great theories: Newton's theory of gravitation, Maxwell's electromagnetic theory, Einstein's theory of relativity, and quantum mechanics. The actual values of these constants have meaning in terms of these theories. The second three constants have their origin in the nature of matter itself. These constants determine the nature of our material world. Suppose, for example, the charge on the electron were two times larger. The hydrogen atom would then be one-quarter its present size. All other atoms would be likewise scaled down in size and standard eight-foot ceilings would become two-foot ceilings.

Why the latter three constants, the charge and masses of the electron and proton, have their particular values is not understood. Why is the mass of the electron 9.1093897 X 10-31 kg and not something else? This is an active area of investigation. One line of reasoning provides at least a rationale: if their observed values were much different, the human species would not be here to observe them. This reasoning purports to show that certain physical and biological processes are required for life, so these constants must have the particular values they have to support such forms of life. In other words, a standard two-foot ceiling could not have little Bohrs and Sommerfelds working under them.

The fine-structure constant is another fundamental constant, which first appeared in Sommerfeld's work on the hydrogen atom. Its value is a = 0.00729735308. More often, this constant is written as follows:

137.0359895'

The fine-structure constant is more than just a number. This constant can be expressed in terms of other constants; namely,

2e0 hc

The fine-structure constant derives its name from its origin. It first appeared in Sommerfeld's work to explain the fine details of the hydrogen spectrum. Recall that Bohr's model of the hydrogen atom provided a mechanism for the origin of spectral lines; namely, quantum jumps from one energy state to another. Bohr's model successfully accounted for the principal features of the hydrogen spectrum. On closer examination, however, the Ha spectral line in the hydrogen spectrum was, in fact, a set of distinct lines. Sommerfeld accounted for this fine structure by complementing Bohr's circular orbits with elliptical orbits. By treating the electron in these elliptical orbits relativistically, Sommerfeld accounted for the observed fine structure. Since Sommerfeld expressed the energy states of the hydrogen atom in terms of the constant a, it came to be called the fine-structure constant.

This constant explains far more than the appearance of the hydrogen atom's spectrum, however. The fine-structure constant is recognized as one of the most important constants in physics. We know, for example, that the fine-structure constant is a measure of the strength of the interaction between photons and electrons. Thus, this constant will appear in all situations that reveal quantum and relativistic properties of electrically charged particles. If electrons and light did not interact, the fine-structure constant would be zero.

The fine-structure constant is endowed with special significance because it is dimensionless.1 In this regard, a is like the dimensionless constant n, the ratio of a circle's circumference to its diameter: 2nr/2r = 3.14159. . . . The fine-structure constant, a, is unlike the speed of light, which has units of m/s, or the charge of the electron, which is measured in coulombs. The fine-structure constant, a, is independent of units: all intelligent beings, everywhere in the universe, share the same numerical value for the fine-structure constant. Citizens on a planet in another galaxy would express the speed of light as a different numerical value because their units would differ from ours, but the fine-structure constant would be identical—1/137.0359895.

Another characteristic adding to the mystique of the fine-structure constant is its ubiquitous nature. It emerges from a number of distinct physical situations, each of which permits a rather precise evaluation of the value of this dimensionless constant. For example:

(a) from measurements of h/mn (h is Planck's constant and m„ is the mass of the neutron) comes the value a = 1/137.03601082;

(b) from measurements of the alternating current Josephson effect at a superconducting junction comes the value a = 1/137.0359770;

(c) from measurements of e2/h (e is the charge of the electron) in the quantum Hall effect comes the value a = 1/137.0360037;

(d) from measurements of the magnetic moment of the electron and positron comes the value a = 1/137.03599993;

(e) from measurements of the energy states of the muonium "atom" consisting of an electron in orbit around a muon2 where the muon serves as the nucleus comes the value a = 1/137.0359940;

(f) from measurements of the helium spectrum comes the value a = 1/137.035853.3

The details of the experiments that give rise to the results above are not important for our considerations. What is important is that each of the measurements (which contain experimental uncertainties that are not shown) arise from the different physical systems and, as is apparent, the values of the fine-structure constant emerging from these experiments are essentially in agreement.

The quest for ever greater precision and accuracy of experi mental measurements is a staple not only of physics, but of any other science that has conceptual models or theoretical systems. The test of all science is experiment. Conceptual models can fascinate the minds of scientists and stir their emotions. They may regard a theoretical idea as so beautiful and so provocative that they cling to it with the eager anticipation that it will bring fresh insights to the secrets of nature. But, if the model or idea does not provide the opportunity for experiments to test its validity, even the most stubborn scientist will eventually abandon the idea regardless of its inherent charm. And here is where the fine-structure constant comes to the fore.

Quantum electrodynamics (QED) is one of the most successful, unifying theories of physics.In fact, the theory of QED underlies all the experiments I have just listed. Furthermore, with QED and the fine-structure constant, physicists can predict the values of many physical parameters to a high level of precision. For these reasons, QED is highly regarded by physicists. Nonetheless, QED, like all theories of physics, is always vulnerable. Since the theory of QED underlies all the various experiments shown above, the measured values of the fine-structure constant from these different experiments should be the same. If these experiments revealed different values of a, even slightly different values, questions as to the validity of QED would automatically follow. That's the way physics and other quantitative sciences work.

The fine-structure constant is badly named. Although it did originate from a study of hydrogen fine structure, the constant's significance far transcends its origin. We have no theory that allows us either to predict or to calculate the fine-structure constant. Edward M. Purcell wrote, "With respect to a we are in the rather humiliating position of people who have to wrap a string around a cylinder to determine n."4 Attempts to find meaning in the fine-structure constant have stimulated flights of fancy and mystical musings. In turn, these abortive attempts were the target of what has been called "arguably the best physics joke ever to slip by an editor of a first-rate physics journal."5 The joke was perpetrated by three young postdoctoral fellows at the Cavendish Laboratory. One of the comedians was Hans Bethe who, eleven years later, was to head the theoretical division at Los Alamos Laboratory. Bethe and his two cohorts wrote a paper purporting to show that the fine-structure constant was exactly 1/137. The paper was nonsense and the editor of Naturwissenschaften published it unawares. He was understandably furious and demanded an apology. On March 1, 1931, an apology appeared. "The Note," they wrote, "was not meant to be taken seriously. It was intended to characterize a certain class of papers in theoretical physics of recent years which are purely speculative and based on spurious numerical agreements."6

The meaning of the fine-structure constant will not come through "spurious numerical agreements." Perhaps if we receive an extraterrestrial message that we decipher as 137.0359895, we shall wonder whether the understanding of physicists somewhere else in the universe is deeper than our own. If such a provocative signal is received and if we haven't learned how to derive and interpret this constant from basic principles, we might convince the intelligent beings out there to tell us what's up.

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