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“are of course quite absurd.”

It remained only for Dirac to invent—or was it “design” or

“discover”?—a new equation for the electron. This was exceedingly beautiful in its formal simplicity and the sense of inevitability it conveyed, after the fact, to sensitive physicists. The equation was a triumph. It correctly predicted (and so, to a physicist, “explained”) the newly discovered quantity cal ed spin, as wel as the hydrogen spectrum. For the rest of his life Dirac’s equation remained his signal achievement. It was 1927. “That is the way in which quantum mechanics was started,” Dirac said.

These were the years of Knabenphysik , boy physics.

When they began, Heisenberg was twenty-three and Dirac twenty-two. (Schrödinger was an elderly thirty-seven, but, as one chronicler noted, his discoveries came “during a late erotic outburst in his life.”) A new Knabenphysik began at MIT in the spring of 1936. Dick Feynman and T. A. Welton were hungry to make their way into quantum theory, but no course existed in this nascent science, so much more obscure even than relativity. With guidance from just a few

obscure even than relativity. With guidance from just a few texts they embarked on a program of self-study. Their col aboration began in one of the upstairs study rooms of the Bay State Road fraternity house and continued past the end of the spring term. Feynman returned home to Far Rockaway, Welton to Saratoga Springs. They fil ed a notebook, mailing it back and forth, and in a period of months they recapitulated nearly the ful sweep of the 1925–27 revolution.

“Dear R. P… .” Welton wrote on July 23. “I notice you write your equation:

This was the relativistic Klein-Gordon equation. Feynman had rediscovered it, by correctly taking into account the tendency of matter to grow more massive at velocities approaching the speed of light—not just quantum mechanics, but relativistic quantum mechanics. Welton was excited. “Why don’t you apply your equation to a problem like the hydrogen atom, and see what results it gives?” Just as Schrödinger had done ten years before, they worked out the calculation and saw that it was wrong, at least when it came to making precise predictions.

“Here’s something, the problem of an electron in the gravitational field of a heavy particle. Of course the electron would contribute something to the field …”

“I wonder if the energy would be quantized? The more I think about the problem, the more interesting it sounds. I’m

going to try it …

“… I’l probably get an equation that I can’t solve anyway,”

Welton added rueful y. (When Feynman got his turn at the notebook he scrawled in the margin, “Right!”) “That’s the trouble with quantum mechanics. It’s easy enough to set up equations for various problems, but it takes a mind twice as good as the differential analyzer to solve them.”

General relativity, barely a decade old, had merged gravity and space into a single object. Gravity was a curvature of space-time. Welton wanted more. Why not tie electromagnetism to space-time geometry as wel ? “Now you see what I mean when I say, I want to make electrical phenomena a result of the metric of a space in the same way that gravitational phenomena are. I wonder if your equation couldn’t be extended to Eddington’s affine geometry…” (In response Feynman scribbled: “I tried it. No luck yet.”)

Feynman also tried to invent an operator calculus, writing rules of differentiation and integration for quantities that did not commute. The rules would have to depend on the order of the quantities, themselves matrix representations of forces in space and time. “Now I think I’m wrong on account of those darn partial integrations,” Feynman wrote. “I oscil ate between right and wrong.”

“Now I know I’m right … In my theory there are a lot more

‘fundamental’ invariants than in the other theory.”

And on they went. “Hot dog! after 3 wks of work … I have at last found a simple proof,” Feynman wrote. “It’s not important to write it, however. The only reason I wanted to

do it was because I couldn’t do it and felt that there were some more relations between the A n & their derivatives that I had not discovered … Maybe I’l get electricity into the metric yet! Good night, I have to go to bed.”

The equations came fast, penciled across the notebook pages. Sometimes Feynman cal ed them “laws.” As he worked to improve his techniques for calculating, he also kept asking himself what was fundamental and what was secondary, which were the essential laws and which were derivative. In the upside-down world of early quantum mechanics, it was far from obvious. Heisenberg and Schrödinger had taken starkly different routes to the same physics. Each in his way had embraced abstraction and renounced visualization. Even Schrödinger’s waves defied every conventional picture. They were not waves of substance or energy but of a kind of probability, rol ing through a mathematical space. This space itself often resembled the space of classical physics, with coordinates specifying an electron’s position, but physicists found it more convenient to use momentum space (denoted by Pα), a coordinate system based on momentum rather than position—or based on the direction of a wavefront rather than any particular point on it. In quantum mechanics the uncertainty principle meant that position and momentum could no longer be specified simultaneously. Feynman in the August after his sophomore year began working with coordinate space (Qα)—less convenient for the wave point of view, but more directly visualizable.

“Pα is no more fundamental than Qα nor vice versa—

why then has Pα played such an important role in theory and why don’t I try Qα instead of Pα in certain generalizations of equations …” Indeed, he proved that the customary approach could be derived directly from the theory as cast in terms of momentum space.

In the background both boys were worrying about their health. Welton had an embarrassing and unexplained tendency to fal asleep in his chair, and during the summer break he was taking naps, mineral baths, and sunlamp treatments—doses of high ultraviolet radiation from a large mercury arc light. Feynman suffered something like nervous exhaustion as he finished his sophomore year. At first he was told he would have to stay in bed al summer. “I’d go nuts if it were me I,” T. A. wrote in their notebook. “Anyhow, I hope you get to school al right in the fal . Remember, we’re going to be taught quantum mechanics by no less an authority than Prof. Morse himself. I’m real y looking forward to that.” (“Me too,” Feynman wrote.)

They were desperately eager to be at the front edge of physics. They both started reading journals like the Physical Review . (Feynman made a mental note that a surprising number of articles seemed to be coming from Princeton.) Their hope was to catch up on the latest discoveries and to jump ahead. Welton would set to work on a development in wave tensor calculus; Feynman would tackle an esoteric application of tensors to electrical engineering, and only after wasting several months did they begin to realize that the journals made poor Baedekers.

Much of the work was out of date by the time the journal article appeared. Much of it was mere translation of a routine result into an alternative jargon. News did sometimes break in the Physical Review , if belatedly, but the sophomores were il equipped to pick it out of the mostly inconsequential background.

Morse had taught the second half of the theoretical physics course that brought Feynman and Welton together, and he had noticed these sophomores, with their penetrating questions about quantum mechanics. In the fal of 1937 they, along with an older student, met with Morse once a week and began to fit their own blind discoveries into the context of physics as physicists understood it. They final y read Dirac’s 1935 bible, The Principles of Quantum Mechanics . Morse put them to work calculating the properties of different atoms, using a method of his own devising. It computed energies by varying the parameters in equations known as hydrogenic radial functions—