Genius_ The Life and Science of Richard Feynman - James Gleick [44]
Yet the Klein-Gordon equation still produced mismatches with experiments when calculations were carried out carefully. It also had what seemed to Dirac a painful logical flaw. It implied that the probability of certain events must be negative, less than zero. Negative probabilities, Dirac said, “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 called spin, as well 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 texts they embarked on a program of self-study. Their collaboration 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 filled a notebook, mailing it back and forth, and in a period of months they recapitulated nearly the full 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’ll probably get an equation that I can’t solve anyway,” Welton added ruefully. (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 well? “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