#renormalization theory

 Back in 1930, Robert Oppenheimer (who would later lead the project to build the atomic bomb) realized something profoundly disturbing. When one tried to describe the quantum theory of an electron interacting with a photon, one found that the quantum corrections actually diverged, yielding useless, infinite results. Quantum corrections were supposed to be small—that had been the guiding principle for decades. So there was an essential flaw in simply combining the Dirac equation of electrons and Maxwell’s theory of photons. This haunted physicists for nearly two decades. Many physicists worked on this problem, but little progress was made.  Finally, in 1949, three young physicists working independently, Richard Feynman and Julian Schwinger in the United States, and Shin’Ichiro Tomonaga in Japan, cracked this long-standing problem.  They were spectacularly successful, able to compute things like the magnetic property of the electron with enormous accuracy. But the way they did it was controversial and still causes physicists some unease and consternation even today.  They started with the Dirac equation and Maxwell’s equation, where the mass and charge of the electron are given certain initial values (called the “bare mass and bare charge”). Then they calculated the quantum corrections to the bare mass and charge. These quantum corrections were infinite. This was the problem found earlier by Oppenheimer.  But the magic occurs here. If we assume that the original bare mass and charge were actually infinite to start with, and then calculate the infinite quantum corrections, we find that these two infinite numbers can cancel each other out, leaving a finite result! In other words, infinity minus infinity equals zero!  This was a crazy idea, but it worked. The strength of the magnetic field of the electron could be calculated using QED to an astonishing accuracy—that is, one part in one hundred billion.  “The numerical agreement between theory and experiment here is perhaps the most impressive in all science,” Steven Weinberg noted. It is like calculating the distance from Los Angeles to New York to within the diameter of a hair. Schwinger was so proud of this that he had the symbol for this result carved on his gravestone.  This method is called renormalization theory. The procedure, however, is arduous, complex, and mind-numbing. Literally thousands of terms have to be computed exactly, and they all have to cancel precisely. The tiniest error in this thick book of equations can throw off the entire calculation. (It is no exaggeration to say that some physicists spend their entire lives calculating quantum corrections to the next decimal place using renormalization theory.)  Because the process of renormalization is so difficult, even Dirac, who helped to create QED in the first place, did not like it. Dirac felt that it seemed totally artificial, like brushing things under the rug. He once said, “This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out to be small—not neglecting it just because it is infinitely great and you do not want it!”  Renormalization theory, which could combine Einstein’s special relativity with Maxwell’s electromagnetism, is indeed supremely ugly. One has to master an encyclopedia of mathematical tricks in order to cancel thousands of terms. But you cannot argue with results.