The first day everyone had to fill out enrollment cards: green for seniors and brown for graduate students. Feynman was proudly aware of the sophomore-pink card in his own pocket. Furthermore he was wearing an ROTC uniform; officer’s training was compulsory for first- and second-year students. But just as he was feeling most conspicuous, another uniformed, pink-card-carrying sophomore sat down beside him. It was T. A. Welton. Welton had instantly recognized the mathematics whiz from the previous spring’s open house.
Feynman looked at the books Welton was stacking on his desk. He saw Tullio Levi-Civita’s Absolute Differential Calculus, a book he had tried to get from the library. Welton, meanwhile, looked at Feynman’s desk and realized why he had not been able to find A. P. Wills’s Vector and Tensor Analysis. Nervous boasting ensued. The Saratoga Springs sophomore claimed to know all about general relativity. The Far Rockaway sophomore announced that he had already learned quantum mechanics from a book by someone called Dirac. They traded several hours’ worth of sketchy knowledge about Einstein’s work on gravitation. Both boys realized that, as Welton put it, “cooperation in the struggle against a crew of aggressive-looking seniors and graduate students might be mutually beneficial.”
Nor were they alone in recognizing that Introduction to Theoretical Physics now harbored a pair of exceptional young students. Stratton, handling the teaching chores for the first semester, would sometimes lose the thread of a string of equations at the blackboard, the color of his face shifting perceptibly toward red. He would then pass the chalk, saying, “Mr. Feynman, how did you handle this problem,” and Feynman would stride to the blackboard.
The Best Path
A law of nature expressed in a strange form came up again and again that term: the principle of least action. It arose in a simple sort of problem. A lifeguard, some feet up the beach, sees a drowning swimmer diagonally ahead, some distance offshore and some distance to one side. The lifeguard can run at a certain speed and swim at a certain lesser speed. How does one find the fastest path to the swimmer?
The path of least time. The lifeguard travels faster on land than in water; the best path is a compromise. Light-which also travels faster through air than through water-seems somehow to choose precisely this path on its way from an underwater fish to the eye of an observer.
A straight line, the shortest path, is not the fastest. The lifeguard will spend too much time in the water. If instead he angles far up the beach and dives in directly opposite the swimmer—the path of least water—he still wastes time. The best compromise is the path of least time, angling up the beach and then turning for a sharper angle through the water. Any calculus student can find the best path. A lifeguard has to trust his instincts. The mathematician Pierre de Fermat guessed in 1661 that the bending of a ray of light as it passes from air into water or glass—the refraction that makes possible lenses and mirages—occurs because light behaves like a lifeguard with perfect instincts. It follows the path of least time. (Fermat, reasoning backward, surmised that light must travel more slowly in denser media. Later Newton and his followers thought they had proved the opposite: that light, like sound, travels faster through water than through air. Fermat, with his faith in a principle of simplicity, was right.)
Theology, philosophy, and physics had not yet become so distinct from one another, and scientists found it natural to ask what sort of universe God would make. Even in the quantum era the question had not fully disappeared from the scientific consciousness. Einstein did not hesitate to invoke His name. Yet when Einstein doubted that God played dice with the world, or when he uttered phrases like the one later inscribed in the stone of Fine Hall at Princeton, “The Lord God is subtle, but malicious he is not,” the great man was playing a delicate game with language. He had found a formulation easily understood and imitated by physicists, religious or not. He could express convictions about how the universe ought to be designed without giving offense either to the most literal believers in God or to his most disbelieving professional colleagues, who were happy to read God as a poetic shorthand for whatever laws or principles rule this flux of matter and energy we happen to inhabit. Einstein’s piety was sincere but neutral, acceptable even to the vehemently antireligious Dirac, of whom Wolfgang Pauli once complained, “Our friend Dirac, too, has a religion, and its guiding principle is ‘There is no God and Dirac is His prophet.’”
Scientists of the seventeenth and eighteenth centuries also had to play a double game, and the stakes were higher. Denying God was still a capital offense, and not just in theory: offenders could be hanged or burned. Scientists made an assault against faith merely by insisting that knowledge—some knowledge—must wait on observation and experiment. It was not so obvious that one category of philosopher should investigate the motion of falling bodies and another the provenance of miracles. On the contrary, Newton and his contemporaries happily constructed scientific proofs of God’s existence or employed God as a premise in a chain of reasoning. Elementary particles must be indivisible, Newton wrote in his Opticks, “so very hard as never to wear or break in pieces; no ordinary power being able to divide what God himself made one in the first creation.” Elementary particles cannot be indivisible, René Descartes wrote in his Principles of Philosophy:
There cannot be any atoms or parts of matter which are indivisible of their own nature (as certain philosophers have imagined)… . For though God had rendered the particle so small that it was beyond the power of any creature to divide it, He could not deprive Himself of the power of division, because it was absolutely impossible that He should lessen His own omnipotence… .
Could God make atoms so flawed that they could break? Could God make atoms so perfect that they would defy His power to break them? It was only one of the difficulties thrown up by God’s omnipotence, even before relativity placed a precise upper limit on velocity and before quantum mechanics placed a precise upper limit on certainty. The natural philosophers wished to affirm the presence and power of God in every corner of the universe. Yet even more fervently they wished to expose the mechanisms by which planets swerved, bodies fell, and projectiles recoiled in the absence of any divine intervention. No wonder Descartes appended a blanket disclaimer: “At the same time, recalling my insignificance, I affirm nothing, but submit all these opinions to the authority of the Catholic Church, and to the judgment of the more sage; and I wish no one to believe anything I have written, unless he is personally persuaded by the evidence of reason.”
The more competently science performed, the less it needed God. There was no special providence in the fall of a sparrow; just Newton’s second law, f = ma. Forces, masses, and acceleration were the same everywhere. The Newtonian apple fell from its tree as mechanistically and predictably as the moon fell around the Newtonian earth. Why does the moon follow its curved path? Because its path is the sum of all the tiny paths it takes in successive instants of time; and because at each instant its forward motion is deflected, like the apple, toward the earth. God need not choose the path. Or, having chosen once, in creating a universe with such laws, He need not choose again. A God that does not intervene is a God receding into a distant, harmless background.
Yet even as the eighteenth-century philosopher scientists learned to compute the paths of planets and projectiles by Newton’s methods, a French geometer and philosophe, Pierre-Louis Moreau de Maupertuis, discovered a strangely magical new way of seeing such paths. In Maupertuis’s scheme a planet’s path has a logic that cannot be seen from the vantage point of someone merely adding and subtracting the forces at work instant by instant. He and his successors, and especially Joseph Louis Lagrange, showed that the paths of moving objects are always, in a special sense, the most economical. They are the paths that minimize a quantity called action—a quantity based on the object’s velocity, its mass, and the space it traverses. No matter what forces are at work, a planet somehow chooses the cheapest, the simplest, the best of all possible paths. It is as if God—a parsimonious God—were a
fter all leaving his stamp.
None of which mattered to Feynman when he encountered Lagrange’s method in the form of a computational shortcut in Introduction to Theoretical Physics. All he knew was that he did not like it. To his friend Welton and to the rest of the class the Lagrange formulation seemed elegant and useful. It let them disregard many of the forces acting in a problem and cut straight through to an answer. It served especially well in freeing them from the right-angle coordinate geometry of the classical reference frame required by Newton’s equations. Any reference frame would do for the Lagrangian technique. Feynman refused to employ it. He said he would not feel he understood the real physics of a system until he had painstakingly isolated and calculated all the forces. The problems got harder and harder as the class advanced through classical mechanics. Balls rolled down inclines, spun in paraboloids—Feynman would resort to ingenious computational tricks like the ones he learned in his mathematics-team days, instead of the seemingly blind, surefire Lagrangian method.
Feynman had first come on the principle of least action in Far Rockaway, after a bored hour of high-school physics, when his teacher, Abram Bader, took him aside. Bader drew a curve on the blackboard, the roughly parabolic shape a ball would take if someone threw it up to a friend at a second-floor window. If the time for the journey can vary, there are infinitely many such paths, from a high, slow lob to a nearly straight, fast trajectory. But if you know how long the journey took, the ball can have taken only one path. Bader told Feynman to make two familiar calculations of the ball’s energy: its kinetic energy, the energy of its motion, and its potential energy, the energy it possesses by virtue of its presence high in a gravitational field. Like all high-school physics students Feynman was used to adding those energies together. An airplane, accelerating as it dives, or a roller coaster, sliding down the gravity well, trades its potential energy for kinetic energy: as it loses height it gains speed. On the way back up, friction aside, the airplane or roller coaster makes the same conversion in reverse: kinetic energy becomes potential energy again. Either way, the total of kinetic and potential energy never changes. The total energy is conserved.
Bader asked Feynman to consider a less intuitive quantity than the sum of these energies: their difference. Subtracting the potential energy from the kinetic energy was as easy as adding them. It was just a matter of changing signs. But understanding the physical meaning was harder. Far from being conserved, this quantity—the action, Bader said—changed constantly. Bader had Feynman calculate it for the ball’s entire flight to the window. And he pointed out what seemed to Feynman a miracle. At any particular moment the action might rise or fall, but when the ball arrived at its destination, the path it had followed would always be the path for which the total action was least. For any other path Feynman might try drawing on the blackboard—a straight line from the ground to the window, a higher-arcing trajectory, or a trajectory that deviated however slightly from the fated path—he would find a greater average difference between kinetic and potential energy.
It is almost impossible for a physicist to talk about the principle of least action without inadvertently imputing some kind of volition to the projectile. The ball seems to choose its path. It seems to know all the possibilities in advance. The natural philosophers started encountering similar minimum principles throughout science. Lagrange himself offered a program for computing planetary orbits. The behavior of billiard balls crashing against each other seemed to minimize action. So did weights swung on a lever. So, in a different way, did light rays bent by water or glass. Fermat, in plucking his principle of least time from a pristine mathematical landscape, had found the same law of nature.
Where Newton’s methods left scientists with a feeling of comprehension, minimum principles left a sense of mystery. “This is not quite the way one thinks in dynamics,” the physicist David Park has noted. One likes to think that a ball or a planet or a ray of light makes its way instant by instant, not that it follows a preordained path. From the Lagrangian point of view the forces that pull and shape a ball’s arc into a gentle parabola serve a higher law. Maupertuis wrote, “It is not in the little details … that we must look for the supreme Being, but in phenomena whose universality suffers no exception and whose simplicity lays them quite open to our sight.” The universe wills simplicity. Newton’s laws provide the mechanics; the principle of least action ensures grace.
The hard question remained. (In fact, it would remain, disquieting the few physicists who continued to ponder it, until Feynman, having long since overcome his aversion to the principle of least action, found the answer in quantum mechanics.) Park phrased the question simply: How does the ball know which path to choose?
Socializing the Engineer
“Let none say that the engineer is an unsociable creature who delights only in formulae and slide rules.” So pleaded the MIT yearbook. Some administrators and students did worry about the socialization of this famously awkward creature. One medicine prescribed by the masters of student life was Tea, compulsory for all freshmen. (“But after they have conquered their initial fears and learned to balance a cup on a saucer while conversing with the wife of a professor, compulsion is no longer necessary.”) Students also refined their conversational skills at Bull Session Dinners and their other social skills at an endless succession of dances: Dormitory Dinner Dances, the Christmas Dance and the Spring Dance, a Monte Carlo Dance featuring a roulette wheel and a Barn Dance offering sleigh rides, dances to attract students from nearby women’s colleges like Radcliffe and Simmons, dances accompanied by the orchestras of Nye Mayhew and Glenn Miller, the traditional yearly Field Day Dance after the equally traditional Glove Fight, and, in the fraternity houses that provided the most desirable student quarters, formal dances that persuaded even Dick Feynman to put on a tuxedo almost every week.
The fraternities at MIT, as elsewhere, strictly segregated students by religion. Jews had a choice of just two, and Feynman joined the one called Phi Beta Delta, on Bay State Road in Boston, in a neighborhood of town houses just across the Charles River from campus. One did not simply “join” a fraternity, however. One enjoyed a wooing process that began the summer before college at local smokers and continued, in Feynman’s case, with insistent offers of transportation and lodging that bordered on kidnapping. Having chosen a fraternity, one instantly underwent a status reversal, from an object of desire to an object of contempt. New pledges endured systematic humiliation. Their fraternity brothers drove Feynman and the other boys to an isolated spot in the Massachusetts countryside, abandoned them beside a frozen lake, and left them to find their way home. They submitted to wrestling matches in mud and allowed themselves to be tied down overnight on the wooden floor of a deserted house—though Feynman, still secretly afraid that he would be found out as a sissy, made a surprising show of resisting his sophomore captors by grabbing at their legs and trying to knock them over. These rites were tests of character, after all, mixed with schoolboy sadism that colleges only gradually learned to restrain. The hazing left many boys with emotional bonds both to their tormentors and to their fellow victims.
Walking into the parlor floor of the Bay State Road chapter house of Phi Beta Delta, a student could linger in the front room with its big bay windows overlooking the street or head directly for the dining room, where Feynman ate most of his meals for four years. The members wore jackets and ties to dinner. They gathered in the anteroom fifteen minutes before and waited for the bell that announced the meal. White-painted pilasters rose toward the high ceilings. A stairway bent gracefully up four flights. Fraternity members often leaned over the carved railing to shout down to those below, gathered around the wooden radio console in one corner or waiting to use the pay telephone on an alcove wall. The telephone provided an upperclassman with one of his many opportunities to harass freshmen: they were obliged to carry nickels for making change. They also carried individual black notebooks for keeping a record of their failures, among other things, t
o carry nickels. Feynman developed a trick of catching a freshman nickel-less, making a mark in his black book, and then punishing the same freshman all over again a few minutes later. The second and third floors were given over entirely to study rooms, where students worked in twos and threes. Only the top floor was for sleeping, in double-decker bunks crowded together.