Knocking on Heaven's Door (6 page)

One of the most important aspects of an effective theory is that while it describes what we can see, it also categorizes what is missing—be it small scale or large. With any effective theory, we can determine how big an effect the unknown (or known) underlying dynamics could possibly have on any particular measurement. Even in advance of new discoveries at different scales, we can mathematically determine the maximal size of the influence any new structure can have on the effective theory at the scale at which we are working. As we will explore further in Chapter 12, it is only when the underlying physics is discovered that anyone fully understands the effective theory’s true limitations.

One familiar example of an effective theory might be thermodynamics, which tells us how refrigerators or engines work and was developed long before atomic or quantum theory. The thermodynamic state of a system is well characterized by its pressure, temperature, and volume. Though we know that fundamentally the system consists of a gas of atoms and molecules—with much more detailed structure than the preceding three quantities can possibly describe—for many purposes we can concentrate on these three quantities to characterize the system’s readily observable behavior.

Temperature, pressure, and volume are real quantities that can be measured. The theory behind their relationships is fully developed and can be used to make successful predictions. The effective theory of a gas makes no mention of the underlying molecular structure. (See Figure 4.) The behavior of those underlying elements determines temperature and pressure, but scientists happily used these quantities to do calculations even before atoms or molecules were discovered.

[
FIGURE 4
]
Pressure and temperature can be understood at a more fundamental level in terms of the physical properties of individual molecules.

Once the fundamental theory is understood, we can relate temperature and pressure to properties of the underlying atoms and also understand when the thermodynamic description should break down. But we can still use thermodynamics for a wide variety of predictions. In fact, many phenomena are only understood from a thermodynamic point of view, since without huge computing power and memory, well beyond what exists, we can’t track the paths of all the individual atoms. The effective theory is the only way at this point to understand some important physical phenomena that are pertinent to solid and liquid
condensed matter.

This example teaches us another critical aspect of effective theories. We sometimes treat “fundamental” as a relative term. From the perspective of thermodynamics, the atomic and molecular description is fundamental. But from a particle physics description that details the quarks and electrons inside the atoms, the atom is
composite
—made up of smaller elements. Its use from a particle physics perspective is as an effective theory.

This description of the clean developmental progression in science from the well understood to regimes at the frontier of knowledge applies best to fields such as physics and cosmology, where we have a clear understanding of the functional units and their relationships. Effective theories won’t necessarily work for newer fields such as systems biology, where the relationships between activities at the molecular and more macroscopic levels, as well as the relevant feedback mechanisms, are yet to be fully understood.

Nonetheless, the effective theory idea applies in a broad range of scientific contexts. The mathematical equations that govern the evolution of species won’t change in response to new physics results, as I discussed with the mathematical biologist Martin Nowak in response to a question he had asked. He and his colleagues can characterize the parameters independently of any more fundamental description. They might ultimately relate to more basic quantities—physical or otherwise—but that doesn’t change the equations that mathematical biologists use to evolve the behavior of populations over time.

For particle physicists, effective theories are essential. We isolate simple systems at different scales and relate them to each other. In fact, the very invisibility of underlying structure that allows us to focus on observable scales and ignore more fundamental effects keep underlying interactions so well hidden that only with tremendous resources and effort can we ferret them out. The tininess of effects of more fundamental theories on observable scales is the reason that physics today is so challenging. We need to directly explore smaller scales or make increasingly precise measurements if we are to perceive the effects of the more fundamental nature of matter and its interactions. Only with advanced technology can we access very tiny or extremely vast length scales. That is why we need to conduct elaborate experiments—such as those at the Large Hadron Collider—to make advances today.

PHOTONS AND LIGHT

The story of theories of light nicely exemplifies the ways in which effective theories are used as science evolves, with some ideas being discarded while others are retained as approximations appropriate to their specified domains. From the time of the ancient Greeks, people studied light with geometrical optics. It is one of the topics any aspiring physics graduate student is tested on when taking the physics GRE (the exam that is a prerequisite for graduate school). This theory assumes that light travels in rays or lines and tells you how those rays behave as they travel through different media, as well as how instruments use and detect them.

The strange thing is that virtually no one—at least no one at Harvard where I now teach and was once a student—actually studies classical and geometrical optics. Maybe geometrical optics is taught a little bit in high school, but it is certainly no big part of the curriculum.

Geometrical optics is an old-fashioned subject. It hit its heyday several centuries ago with Newton’s famous
Opticks
, continuing into the 1800s when William Rowan Hamilton made what is perhaps the first real mathematical prediction of a new phenomenon.

The classical theory of optics still applies to areas such as photography, medicine, engineering, and astronomy, and is used to develop new mirrors, telescopes, and microscopes. Classical optical scientists and engineers work out different examples of various physical phenomena. However, they are simply applying optics—not discovering new laws.

In 2009, I was honored to be asked to give the Hamilton lecture at the University of Dublin—a lecture several of my most respected colleagues had given before me. It is named after Sir William Rowan Hamilton, the remarkable nineteenth-century Irish mathematician and physicist. I confess that the name Hamilton is so universally present in physics that I foolishly didn’t initially make the connection with an actual person who was in fact Irish. But I was fascinated by the many areas of math and physics that Hamilton had revolutionized, including, among them, geometrical optics.

The celebration of Hamilton Day is really quite something. The day’s activities include a procession down the Royal Canal in Dublin where everyone stops at the Broom Bridge to watch the youngest member of the party write down the same equations on the bridge that Hamilton, in the excitement of discovery, had many years past carved into the bridge’s side. I visited the College Observatory of Dunsink where Hamilton lived and got to see the pulleys and wooden structure of a telescope from two centuries ago. Hamilton arrived there after his graduation from Trinity College in 1827, when he was made the chair of astronomy and Astronomer Royal of Ireland. Locals joke that despite Hamilton’s prodigious mathematical talent, he had no real knowledge of or interest in astronomy, and that despite his many theoretical advances, he might have set back observational astronomy in Ireland fifty years.

Hamilton Day nonetheless pays homage to this great theorist’s many accomplishments. These included advances in optics and dynamics, the invention of the mathematical theory of
quaternions
(a generalization of complex numbers), as well as definitive demonstrations of the predictive power of math and science. The development of quaternions was no small advance. Quaternions are important for vector calculus, which underlies the way we mathematically study all three-dimensional phenomena. They are also now used in computer graphics and hence in the entertainment industry and video games. Anyone with a PlayStation or Xbox can thank Hamilton for some of the fun.

Among his numerous and substantial contributions, Hamilton significantly advanced the field of optics. In 1832, he showed that light falling at a certain angle on a crystal that has two independent axes would be refracted to form a hollow cone of emergent rays. He thereby made predictions about
internal
and
external
conical refraction of light through a crystal. In a tremendous—and perhaps the first—triumph of mathematical science, this prediction was verified by Hamilton’s friend and colleague Humphrey Lloyd. It was a very big deal to see verified a mathematical prediction of a never-before-seen phenomenon and Hamilton was knighted for his achievement.

When I visited Dublin, the locals proudly described this mathematical breakthrough—worked out purely on the basis of geometrical optics. Galileo helped pioneer observational science and experiments, and Francis Bacon was an initial advocate of
inductive science
—where one predicts what will happen based on what came before. But in terms of using math to describe a never-before-seen phenomenon, Hamilton’s prediction of conical refraction was probably the first. For this reason, at the very least, Hamilton’s contribution to the history of science is not to be ignored.

Nonetheless, despite the significance of Hamilton’s discovery, classical geometrical optics is no longer a research subject. All the important phenomena were worked out long ago. Soon after Hamilton’s time, in the 1860s, the Scottish scientist James Clerk Maxwell, among others, developed the electromagnetic description of light. Geometrical optics, though clearly an approximation, is nonetheless a good description for a wave with wavelength small enough for interference effects to be irrelevant, and for the light to be treated as a linear ray. In other words, geometrical optics is an effective theory, valid in a limited regime.

That doesn’t mean we keep every idea that has ever been developed. Sometimes ideas are just proved wrong. Euclid’s initial description of light, resurrected in the Islamic world in the ninth century by Al-Kindi, which claimed that light was emitted by our eyes, was one such example. Although others, such as the Persian mathematician Ibn Sahl, correctly described phenomena such as refraction based on this false premise, Euclid and Al-Kindi’s theory—which predates science and modern scientific methods—was simply incorrect. It wasn’t absorbed into future theories. It was simply abandoned.

Newton didn’t anticipate a different aspect of the theory of light. He had developed a “corpuscular” theory that was inconsistent with the wave theory of light developed by his rival Robert Hooke in 1664 and Christian Huygens in 1690. The debate between them lasted a long time. In the nineteenth century, Thomas Young and Augustin-Jean Fresnel measured light interference, providing a clear verification that light had the properties of a wave.

Later developments in quantum theory demonstrated that Newton was correct in some sense too. Quantum mechanics now tells us that light is indeed composed of individual particles called
photons
that are responsible for communicating the electromagnetic force. But the modern theory of photons is based on light quanta, the individual particles of which light is made, that have a remarkable property. Even an individual particle of light, a photon, acts like a wave. That wave gives the probability of a single photon being found in any region of space. (See Figure 5.)

Geometrical Optics

Light travels in straight lines
.

Wave Optics

Light travels in waves
.