The choice of course was an inspired one. Feynman’s innate talents allowed him to excel in physics. But he had another talent that mattered even more perhaps, and I don’t know if it was innate or not. This was intuition.
Physical intuition is a fascinating, ephemeral kind of skill. How does one know which avenue of approach will be most fruitful to solve a physics problem? No doubt some aspects of intuition are acquired. This is why physics majors are required to solve so many problems. In this way, they begin to learn which approaches work and which don’t, and increase their toolkit of techniques along the way. But surely some aspect of physical intuition cannot be taught, one that resonates at a certain place and time. Einstein had such intuition, and it served him well for over twenty years, from his epochal work on special relativity to his crowning achievement, general relativity. But his intuition began to fail him as he slowly drifted away from the mainstream of interest in quantum mechanics in the twentieth century.
Feynman’s intuition was unique in a different way. Whereas Einstein developed completely new theories about nature, Feynman explored existing ideas from a completely new and usually more fruitful perspective. The only way he could really understand physical ideas was to derive them using his own language. But because his language was usually also self-taught, the end results sometimes differed radically from what “conventional” wisdom produced. As we shall see, Feynman created his own wisdom.
But Feynman’s intuition was also earned the hard way, based on relentless labor. His systematic approach and the thoroughness with which he examined problems were already evident in high school. He recorded his progress in notebooks, with tables of sines and cosines he had calculated himself, and later on in his comprehensive calculus notebook, titled “The Calculus for the Practical Man,” with extensive tables of integrals, which again he had worked out himself. In later life he would amaze people by proposing a new way to solve a problem, or by grasping immediately the heart of a complex issue. More often than not this was because at some time, in the thousands of pages of notes he kept as he worked to understand nature, Feynman had thought about that very problem and explored not just one, but a host of different ways of solving it. It was this willingness to investigate a problem from every vantage point, and to carefully organize his thinking until he had exhausted all possibilities—a product of his deep intellect and his indefatigable ability to concentrate—that set him apart.
Perhaps willingness is the wrong word here. Necessity would be a better choice. Feynman needed to fully understand every problem he encountered by starting from scratch, solving it in his own way and often in several different ways. Later on, he would try to imbue this same ethic to his students, one of whom later said, “Feynman stressed creativity—which to him meant working things out from the beginning. He urged each of us to create his or her own universe of ideas, so that our products, even if only answers to assigned classwork problems, would have their own original character—just as his own work carried the unique stamp of his personality.”
Not only was Feynman’s ability to concentrate for long periods evident when he was young, but so was his ability to control and organize his thoughts. I remember having a chemistry set when I was a kid and I also remember often randomly throwing things together to see what would happen. But Feynman, as he later emphasized, “never played chaotically with scientific things.” Rather he always carried out his scientific “play” in a controlled manner, always attentive to what was going on. Again, much later, after his death, it became clear from the extensive notes he took that he carefully recorded each of his explorations. He even considered at one point organizing his domestic life with his future wife along scientific lines, before a friend convinced him that he was being hopelessly unrealistic. Ultimately, his naivete in this regard disappeared, and much later he advised a student, “You cannot develop a personality with physics alone. The rest of life must be worked in.” In any case, Feynman loved to play and joke, but when it came to science, starting early on and continuing for the rest of his life, Feynman could be deadly serious.
He may have waited until the end of his first year of university to declare himself a physics major, but the stars aligned when he was still in high school. In retrospect, what might have been the defining moment occurred when his high school teacher, Mr. Bader, introduced him to one of the most subtle and wonderful hidden mysteries of the observable world, a fact that had built on a discovery made three hundred years before he was born by a brilliant and reclusive lawyer-turned-mathematician, Pierre de Fermat.
Like Feynman, Fermat would achieve public recognition late in life for something that was unrelated to his most substantial accomplishments. In 1637, Fermat scrawled a brief note in the margin of his copy of Arithmetica, the masterpiece by the famous Greek mathematician Diophantus, indicating that he had discovered a simple proof of a remarkable fact. The equation xn + yn = zn has no integer solutions if n >2 (for n = 2, this is familiar as the Pythagorean theorem relating lengths of the sides of a right triangle). It is doubtful that Fermat really possessed such a proof, which 350 years later required almost all of the developments of twentieth-century mathematics and several hundred pages to complete. Nevertheless, if Fermat is remembered at all today among the general public, it is not for his many key contributions to geometry, calculus, and number theory, but rather for this speculation in the margin that will forever be known as Fermat’s last theorem.
Twenty-five years after making this dubious claim, Fermat did present a complete proof of something else, however: a remarkable and almost otherworldy principle that established an approach to physical phenomena that Feynman would use later to change the way we think about physics in the modern world. The issue to which Fermat turned his attention in 1662 involved a phenomenon the Dutch scientist Willebrord Snell had described forty years earlier. Snell discovered a mathematical regularity in the way light is bent, or refracted, when it crosses between two different media, such as air and water. Today we call this Snell’s law, and it is often presented in high school physics classes as yet one additional tedious fact to be memorized, even though it played a profoundly important role in the history of science.
Snell’s law pertains to the angles that a light ray makes when transmitted across the surface between two media. The exact form of the law is unimportant here; what is important is both its general character and its physical origin. In simple terms, the law states that when light goes from a less dense to a more dense medium, the trajectory of the light ray is bent closer to the perpendicular to the surface between the media (see figure).
Snell’s law
Now, why does the light bend? Well, if light were made up of a stream of particles, as Newton and others thought, one could understand this relationship if the particles speed up as they move from one medium to the other. They would literally be dragged forward, moving more effectively in a direction perpendicular to the surface they had just crossed. However, this explanation seemed fishy even at the time. After all, in a more dense medium any such particles would presumably encounter a greater resistance to their motion, just as cars on a road end up moving more slowly in heavy traffic.
There was another possibility, however, as the Dutch scientist Christiaan Huygens demonstrated in 1690. If light were a wave and not made of particles, then just as a sound wave bends inward when it slows down, the same would occur for light if it too slowed down in the denser medium. As anyone familiar with the history of physics knows, light does indeed slow down in denser media, so that Snell’s law provides important evidence that light behaves, in this instance, like a wave.
Amost thirty years before Huygens’s work, Fermat too reasoned that light should travel more slowly in dense media than in less dense media. Instead of thinking in terms of whether light was a wave or particle, however, Fermat the mathematician showed that in this case one could explain the trajectory of light in terms of a general
mathematical principle, which we now call Fermat’s principle of least time. As he demonstrated, light would follow precisely the same bending trajectory determined by Snell if “light travels between two given points along the path of shortest time.”
Heuristically this can be understood as follows. If light travels more quickly in the less dense medium, then to get from A to B (see figure) in the shortest time, it would make sense to travel a longer distance in this medium, and a shorter distance in the second medium in which it travels more slowly. Now, it cannot travel for too long in the first medium, otherwise the extra distance it travels would more than overcome the gain obtained by traveling at a faster speed. One path is just right, however, and this path turns out to involve a bending trajectory that exactly reproduces the trajectory Snell observed.
Snell’s law
Fermat’s principle of least time is a mathematically elegant way of determining the path light takes without recourse to any mechanistic description in terms of waves or particles. The only problem is that when one thinks about the physical basis of this result, it seems to suggest intentionality, so that, like a commuter in Monday-morning rush-hour listening to the traffic report, light somehow considers all possible paths before embarking on its voyage, and ultimately chooses the one that will get it to its destination fastest.
But the fascinating thing is that we don’t need to ascribe any intentionality to light’s wanderings. Fermat’s principle is a wonderful example of an even more remarkable property of physics, a property that is central to the amazing and a priori unexpected fact that nature is comprehensible via mathematics. If there is any one property that was a guiding light for Richard Feynman’s approach to physics, and essential to almost all of his discoveries, it was this one, which he thought was so important that he referred to it at least two different times during his Nobel Prize address. First, he wrote,
It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but, with a little mathematical fiddling you can show the relationship. . . . it was something I learned from experience. There is always another way to say the same thing that doesn’t look at all like the way you said it before. . . . I think it is somehow a representation of the simplicity of nature. I don’t know what it means, that nature chooses these curious forms, but maybe that is a way of defining simplicity. Perhaps a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing.
And later (and more important for what was to come), he added,
Theories of the known, which are described by different physical ideas, may be equivalent in all their predictions and are hence scientifically indistinguishable. However, they are not psychologically identical when trying to move from that base into the unknown. For different views suggest different kinds of modifications which might be made and hence are not equivalent in the hypotheses one generates from them in one’s attempt to understand what is not yet understood.
Fermat’s principle of least time clearly represents a striking example of this strange redundancy of physical law that so fascinated Feynman, and also of the differing “psychological utilities” of the different prescriptions. Thinking about the bending of light in terms of electric and magnetic forces at the interface between media reveals something about the properties of the media. Thinking about it in terms of the speed of light itself reveals something about light’s intrinsic wavelike character. And thinking about it in terms of Fermat’s principle may reveal nothing about specific forces or about the wave nature of light, but it illuminates something deep about the nature of motion. Happily, and importantly, all of these alternate descriptions result in identical predictions.
Thus we can rest easy. Light does not know it is taking the shortest path. It just acts like it does.
IT WASN’T THE principle of least time, however, but an even subtler idea that changed Feynman’s life that fateful day in high school. As Feynman later described it, “When I was in high school, my physics teacher—whose name was Mr. Bader—called me down one day after physics class and said, ‘You look bored; I want to tell you something interesting.’ Then he told me something that I found absolutely fascinating, and have, since then, always found fascinating . . . the principle of least action.” Least action may sound like an expression that is more appropriate to describing the behavior of a customer service representative at the phone company than a field like physics, which is, after all, centered around describing actions. But the least action principle is very similar to Fermat’s principle of least time.
The principle of least time tells us that light always takes the path of shortest time. But what about baseballs and cannonballs, planets, and boomerangs? They don’t necessarily behave so simply. Is there something other than time that is minimized whenever these objects follow the paths prescribed by the forces acting on them?
Consider any object in motion, say, a falling weight. Such an object is said to possess two different kinds of energy. One is kinetic energy, and it is related to the motion of objects (and derives from the Greek word for movement). The faster an object moves, the larger the kinetic energy. The other part of an object’s energy is much subtler to ascertain, as reflected in its name: potential energy. This kind of energy may be hidden, but it is responsible for the ability of an object to do work later on. For example, a heavy weight falling off the top of a tall building will do more damage (and hence more work) smashing the roof of a car, than will a similar weight dropped from several inches above the car. Clearly the higher the object, the greater its potential to do work, and hence the greater its potential energy.
Now, what the least action principle states is that the difference between the kinetic energy of an object at any instant and its potential energy at the same instant, when calculated at each point along a path and then added up along the path, will be smaller for the actual path the object takes than for any other possible trajectory. An object somehow adjusts its motion so the kinetic energy and the potential energy are as closely matched, on average, as is possible.
If this seems mysterious and unintuitive, that is because it is mysterious and unintuitive. How on earth would anyone ever come up with this combination in the first place, much less apply it to the motion of everyday objects?
For this we thank the Italian mathematician-physicist Joseph Louis Lagrange, who is best known for his work on celestial mechanics. For example, he determined the points in the solar system where the gravitational attraction from the different planets precisely cancels the frame of reference of the orbiting body. They are called Lagrange points. NASA now sends numerous satellites out to these points so that they can remain in stable orbits and study the universe.
Lagrange’s greatest contribution to physics, however, may have involved his reformulation of the laws of motion. Newton’s laws relate the motion of objects to the net forces acting on them. However, Lagrange managed to show that Newton’s laws of motion were precisely reproduced if one used the “action,” which is the sum over a path of the differences between kinetic and potential energy, now appropriately called a Lagrangian, and then determined precisely what sorts of motion would produce those paths that minimized this quantity. The process of minimization, which required the use of calculus (also invented by Newton), gave very different mathematical descriptions of motion from Newton’s laws, but, in the spirit of Feynman, they were mathematically identical, even if “psychologically” very different.
IT WAS THIS strange principle of least action, often called Lagrange’s principle, that Mr. Bader introduced the teenaged Feynman to. Most teens would not have found it fascinating or even comprehensible, but Feynman did, or so he remembered when he was older.
However, if the young Feynman had any inkling at the time that this principle would return to completel
y color his own life story, he certainly didn’t behave that way as he began to learn more about physics once he entered MIT. Quite the contrary. His best friend as an undergraduate at MIT, Ted Welton, with whom he worked through much of undergraduate and even graduate physics, later described Feynman’s “maddening refusal to concede that Lagrange might have something useful to say about physics. The rest of us were appropriately impressed with the compactness, elegance, and utility of Lagrange’s formulation, but Dick stubbornly insisted that real physics lay in identifying all the forces and properly resolving them into components.”
Nature, like life, takes all sorts of strange twists and turns, and most important, it is largely insensitive to one’s likes and dislikes. As much as Feynman tried early on to focus on understanding motion in a way that meshed with his naive intuition, his own trajectory to greatness involved a very different path. There was no unseen hand guiding him. Instead, he forced his intuition to bend to the demands of the problems of the time, rather than vice versa. The challenge required endless hours and days and months of hard work training his mind to wrap around a problem that the greatest minds in twentieth-century physics had, up to that point, not been able to solve.
When he really needed it, Feynman would find himself returning once again to the very principle that had turned him on to physics in the first place.
Quantum Man: Richard Feynman's Life in Science Page 2