Fundamentals Read online

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  THERE ARE VERY FEW LAWS

  The way that the fundamental* physical laws work is quite different from how human laws work. There are many human laws, and they differ from place to place and change over time. Human laws presuppose that there are different options for behavior, and propose reactions to them. Human laws do not support long chains of reasoning that lead to unambiguous conclusions, and experts often differ about their meaning.

  Fundamental physical laws differ from human laws on each of those counts. There are very few of them, and they are the same everywhere and always. Physical laws simply describe what will happen. Physical laws are expressed as mathematical equations among precisely defined quantities, leaving no room for ambiguity or disagreement among competent experts. Drawing out their consequences is merely a matter of calculation. You can program a computer to do it.

  A child’s conception of how the world works, which most people carry into adulthood by default, stands much closer to the model of human law than to the ideal of physical law. We have the direct experience of weighing options and making choices. Our mental choices seem to make a difference in the physical world. Specifically, they seem to control how our bodies move. We form expectations for how people and things will behave based on rules of thumb, and only rarely through chains of logic and calculation. Nobody walks, rides a bicycle, or catches a fly ball by working up from Newton’s laws of motion, let alone the quantum theory of matter.

  To reach fundamental understanding, we need to rethink those experiences and childlike methods. Only then can we graduate from human law to physical law.

  THE TRIUMPH OF LOCALITY AND THE GLORY OF FIELDS

  Newton’s Principia, published in 1687, established a powerful framework for understanding the physical world that dominated science well into the nineteenth century. Within this framework, laws express how bodies exert forces upon one another. The model of a successful law was Newton’s law of gravity. According to that law, bodies attract one another with a force that is proportional to the product of their masses and decreases as the square of the distance between them.

  When people began to grapple with other kinds of forces— electric and magnetic forces, to be specific—they tried to use the same basic framework. Early results were encouraging. Coulomb’s law for electric forces, for example, echoes Newton’s law for gravitational forces, with electric charge taking the place of mass.

  But it didn’t work as neatly for magnetism. Magnetic forces turned out to depend on velocity, as well as on position, in a complicated way. Then, when people studied situations where both electricity and magnetism operated at the same time, the complications multiplied.

  Michael Faraday (1791–1867), a self-educated experimental genius of humble origins, could not follow the intricate mathematics of these complicated force-laws. He thought for himself, instead, in imagery. He visualized that electrically and magnetically active bodies extend influence through space, as a sort of aura or atmosphere, even where no other bodies are around to feel that influence. Today, we call these activations of space electric and magnetic fields. Faraday used more vivid language; he called them “lines of force.” As James Clerk Maxwell (1831–1879), the spectacularly gifted theorist who became Faraday’s disciple and evangelist, put it, “Faraday, in his mind’s eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium.”

  Guided by his unorthodox ideas, Faraday soon discovered a remarkable effect that was difficult even to state without referring to his fields. This is his law of induction, according to which magnetic fields that change in time produce circulating electric fields. With that discovery, he revealed that fields have a life of their own.

  An everyday experience with water provides a familiar model to illustrate how a space-filling medium can generate forces between distant bodies, through local action, as Faraday envisaged. If a moving boat, or jet ski, creates a disturbance in a lake, the influence of that disturbance spreads gradually through the lake, as moving water at one location pushes water nearby—and only water nearby. And so eventually, even if they’re far from the source, swimmers in the lake will feel a force when the wave arrives. I’ve had that annoying experience many times. It would be worse if it came without warning, but usually I see the wave coming. Locality is a blessing—it means you can’t be taken completely by surprise.

  Faraday’s more complete vision of locality inspired a revolution in physics. Since electromagnetic fields, which fill space, have a life of their own, they must be included among the world’s ingredients. The Newtonian framework, based on particles in space—harkening back to Democritus’s “atoms and the void”—wouldn’t cut it anymore. Thus, our description of the world was profoundly enriched. As Maxwell wrote:

  The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of Space, or produce the slightest flaw in its infinite continuity.

  If Maxwell’s rapturous prose seems excessive, let’s consider how he got there. When Maxwell decided, early in his career, to take up the study of electricity and magnetism, he was galvanized by Faraday’s conceptions and discoveries. He resolved to build upon Faraday’s intuitive field concept, rather than retreat to the much better-developed and more popular Newtonian framework. Maxwell put forward that

  whenever energy is transmitted from one body to another in time, there must be a medium or substance in which the energy exists after it leaves one body and before it reaches the other. . . . And if we admit this medium as an hypothesis, I think it ought to occupy a prominent place in our investigations, and that we ought to construct a mental representation of all the details of its action.

  Upon spelling out this new viewpoint mathematically, Maxwell discovered that in order to get consistent equations he needed to supplement Faraday’s law of induction with another one, in which the roles of electric and magnetic fields are reversed. According to Maxwell’s law of induction, electric fields that change in time produce circulating magnetic fields.

  When he married the two field-based induction laws—Faraday’s and his own—Maxwell discovered that they gave birth to a dramatic new effect. One could have a self-restoring, permanent, traveling disturbance in electric and magnetic fields. Changing electric fields induce changing magnetic fields induce changing electric fields induce changing magnetic fields . . . Those disturbances, he calculated, should travel at the speed of light, which had been measured independently. Maxwell immediately proposed, “The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.”

  He was right.

  The possible electromagnetic disturbances include visible light—all the wavelengths perceptible to our eyes—and much more. Maxwell predicted the existence of stretched and compressed versions of visible light, including new forms of radiation that were totally unknown and unexpected at the time. Today, we call them radio waves, microwaves, infrared and ultraviolet radiation, x-rays, and gamma rays.

  The decisive experimental test of Maxwell’s equations came more than twenty years after they were first proposed. To achieve it, Heinrich Hertz designed and built the first radio transmitters and receivers. Hertz’s goal was to turn beautiful ideas into physical realities. He felt he had succeeded at that. “One cannot escape the feeling,” he wrote, “that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their di
scoverers, that we get more out of them than was originally put into them.”

  The work of Faraday, Maxwell, and Hertz spanned most of the nineteenth century. It established space-filling fields as a new kind of ingredient in the fundamental description of the world.

  Force and Substance: Quantum Fields

  At first, fields were considered an additional ingredient in the recipe for the physical world, supplementing particles. Over the twentieth century, fields took over completely. We now understand particles as manifestations of a deeper, fuller reality. Particles are avatars of fields.

  As we mentioned earlier, Einstein, building on the work of Planck, proposed that light comes in discrete units, particles that Einstein called light-quanta, and which we now call photons. Einstein’s proposal initially got a chilly reception from the physics community, because it seemed difficult to reconcile the idea that light comes in particles with Maxwell’s field-based understanding of light. Maxwell’s theory had scored many triumphs, including Hertz’s epochal discovery, and was reinforced by detailed study of the new forms of radiation it predicted.

  Fields, being continuously extended through space, appear to be very different from particles. It was hard to imagine how light could be both, yet experimental facts demanded it.

  The different aspects of light—field and particle—get reconciled in the concept of a quantum field. Quantum fields, as their name suggests, are still fields (that is, space-filling media). There are quantum versions of both electric and magnetic fields. They continue to satisfy the same equations—Maxwell’s equations—that nineteenth-century physicists proposed for electric and magnetic fields, before anybody knew about quantum mechanics.

  But the quantum versions of the electric and magnetic fields satisfy additional equations. The additional equations usually go by the rather forbidding name “commutation relations,” but I will use the less formal name “quantum conditions.” Whatever you call them, these additional equations express the essence of quantum theory in mathematical form. Werner Heisenberg introduced the general idea of quantum conditions in 1925, when he was twenty-four years old. Paul Dirac introduced the specific quantum conditions that apply to electric and magnetic fields shortly afterward, in 1926. He, too, was twenty-four years old.

  With more equations to satisfy, there are fewer solutions. As we’ve discussed, Maxwell discovered that light is a kind of self-perpetuating, moving excitation among electric and magnetic fields. Not all of his solutions, however, also satisfy the quantum conditions. The allowed solutions must satisfy a specific relationship between their energy and their frequency (that is, the rate at which the fields oscillate). I will state that important relationship both in words and, alternatively, as a simple equation. The relationship is that the energy of the excitation must be equal to a nonzero constant, called Planck’s constant, multiplied by the frequency. As an equation, it reads E = hν, where E is the energy, ν is the frequency, and h is Planck’s constant. This relationship, not coincidentally, is the one that Planck proposed in 1900 and that Einstein seized on in 1905, to predict the existence of photons. It is called the Planck-Einstein formula. It took twenty years to digest their revolutionary proposal, closely based on experimental results, before physicists reached a consistent theoretical interpretation, as presented here. We have both Maxwell’s equations and discrete units of light.

  This grand story of electromagnetic fields and photons leads directly to another key insight. It explains why, and how, Nature produces vast numbers of interchangeable parts.

  If our account of the fundamental ingredients had ended at the level of elementary particles, it would leave a basic question unanswered. For at that level, we must postulate that each kind of elementary particle exists in many identical copies: many identical photons, many identical electrons, and so forth.

  In the history of human manufacturing, the introduction of standardized, interchangeable parts was a great innovation. To achieve it, new kinds of machines and materials had to be invented so that accurate templates could be made and maintained. And even so, the parts, once made, are subject to wear and tear, and eventually cease to be identical.

  Photons, on the other hand, are observed to have the same properties, whenever and wherever they are found. The light of a given color is the same thing—it has the same properties, and interacts with matter in the same way—whatever its source. Likewise, electrons are precisely the same wherever they are found. If the electrons within different atoms of carbon, for example, did not have identical properties, then each carbon atom would have different properties, and chemistry would not work.

  How does Nature do it? By tracing the common origin of all photons to a common, universal electromagnetic field, we come to understand their otherwise baffling sameness. And we are led, by analogy, to introduce a field—call it the electron field—whose excitations are electrons. All electrons have the same properties, because each one is an excitation in the same universal field.

  Fields are necessary to achieve locality, and quantum fields produce particles. Following this chain of logic, we obtain a deeper understanding of why particles exist, and of their amazing interchangeability. There is no need to introduce two different sorts of fundamental ingredients, fields and particles, after all. Fields rule. Quantum fields, that is.

  Going back to the origin of the field concept, in Faraday’s attempts to picture electric and magnetic influences in space, we can recognize another way that quantum fields unify our picture of the world. The same quantum electric and magnetic fields that produce photons also produce, according to Faraday’s visions—and Maxwell’s equations—electric and magnetic forces.

  To sum up:

  From forces we are led to fields, and from (quantum) fields, we are led to particles.

  From particles we are led to (quantum) fields, and from fields, we are led to forces.

  Thus, we come to understand that substance and force are two aspects of a common underlying reality.

  FOUR FORCES

  In this section, I will briefly sketch our best understanding of the nature of the four known forces, using the framework we discussed in the preceding chapter: principles and properties, embodied in a few kinds of particles. One layer deeper, the particles get replaced by fields, as we just discussed.

  The four forces are:

  electromagnetism, or in its full quantum glory, quantum electromagnetism (QED)

  the strong force, or in its full quantum glory, quantum chromodynamics (QCD)

  gravity, as captured in Einstein’s general relativity

  the weak force

  The electromagnetic and strong forces dominate our understanding of terrestrial matter. The electromagnetic force holds atoms together and governs their structure. It also describes how they interact with light. The strong force holds atomic nuclei together and governs their structure.

  Gravity, as it acts between elementary particles, is very feeble. But when many particles are involved, its influence accumulates, and it comes to dominate interactions between large bodies.

  The weak force governs processes of transformation. It causes some otherwise stable particles to decay, as in some forms of radioactivity. Notably, too, it mediates energy-releasing interactions that power stars, including our Sun.

  Before we plunge into more detail, I’d like to explain two choices I’ve made. The first is simply a choice of words. Physicists often speak of the four “interactions,” as opposed to the four “forces.” There is a legitimate argument for that choice. “Force” has a precise technical meaning in Newtonian mechanics, where it denotes a potential cause of motion. But in the phrase “weak force,” for example, the same word must be understood differently, to include interactions that do other things (namely, processes that change one kind of particle into another). Nevertheless, I’ll stick to “weak force,” since it is less stilted* than “weak i
nteraction.”

  The second choice I’ve made gets to the heart of what I hope to accomplish in this book. The crowning glory of our theories of the four forces is that they can be expressed precisely and accurately in a few mathematical equations. This means something concrete, philosophically, which you do not need mathematical training to understand. It means that it is possible to translate the theories, without loss of content, into reasonably short computer programs. You could, of course, then combine the four programs for the separate forces into one master program. The master program—the operating system of the physical world—would still be much shorter than, say, the operating system that runs your computer.

  But the flip side of that extraordinary “data compression” is that its information is encoded in something very different from any natural human language. The raw equations, or their equivalent in a computer program, use symbols and concepts that are quite remote from the everyday experiences that natural language builds on. It takes a lot of calculation and interpretation to get from the raw equations to consequences that are easy to talk about. So here I had to make a choice—a whole series of choices, actually—about how raw to get and which consequences to emphasize. The overarching message remains—that a very few laws suffice to govern the physical world.

  QUANTUM ELECTRODYNAMICS (QED)

  The Electric Atom

  The basic rules for electromagnetic interactions, starting with Coulomb’s law for electric forces and culminating in Maxwell’s equations, were deduced from experiments with human-sized objects. Nevertheless, as people began to explore the subatomic world, they assumed, by default, that the only important forces in atomic physics are electromagnetic forces, and that they could continue to use Maxwell’s equations to describe those forces. It was the radically conservative thing to do.