“This is completely new and very much simpler than anything that has been done before,” said Andrew Hodges, a mathematical physicist at Oxford University who has been following the work.
The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.
“The degree of efficiency is mind-boggling,” said Jacob Bourjaily, a theoretical physicist at Harvard University and one of the researchers who developed the new idea. “You can easily do, on paper, computations that were infeasible even with a computer before.” (...)
The amplituhedron looks like an intricate, multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated, “scattering amplitudes,” which represent the likelihood that a certain set of particles will turn into certain other particles upon colliding. These numbers are what particle physicists calculate and test to high precision at particle accelerators like the Large Hadron Collider in Switzerland.
The 60-year-old method for calculating scattering amplitudes — a major innovation at the time — was pioneered by the Nobel Prize-winning physicist Richard Feynman. He sketched line drawings of all the ways a scattering process could occur and then summed the likelihoods of the different drawings. The simplest Feynman diagrams look like trees: The particles involved in a collision come together like roots, and the particles that result shoot out like branches. More complicated diagrams have loops, where colliding particles turn into unobservable “virtual particles” that interact with each other before branching out as real final products. There are diagrams with one loop, two loops, three loops and so on — increasingly baroque iterations of the scattering process that contribute progressively less to its total amplitude. Virtual particles are never observed in nature, but they were considered mathematically necessary for unitarity — the requirement that probabilities sum to one.
“The number of Feynman diagrams is so explosively large that even computations of really simple processes weren’t done until the age of computers,” Bourjaily said. A seemingly simple event, such as two subatomic particles called gluons colliding to produce four less energetic gluons (which happens billions of times a second during collisions at the Large Hadron Collider), involves 220 diagrams, which collectively contribute thousands of terms to the calculation of the scattering amplitude.
In 1986, it became apparent that Feynman’s apparatus was a Rube Goldberg machine.
To prepare for the construction of the Superconducting Super Collider in Texas (a project that was later canceled), theorists wanted to calculate the scattering amplitudes of known particle interactions to establish a background against which interesting or exotic signals would stand out. But even 2-gluon to 4-gluon processes were so complex, a group of physicists had written two years earlier, “that they may not be evaluated in the foreseeable future.”
Stephen Parke and Tommy Taylor, theorists at Fermi National Accelerator Laboratory in Illinois, took that statement as a challenge. Using a few mathematical tricks, they managed to simplify the 2-gluon to 4-gluon amplitude calculation from several billion terms to a 9-page-long formula, which a 1980s supercomputer could handle. Then, based on a pattern they observed in the scattering amplitudes of other gluon interactions, Parke and Taylor guessed a simple one-term expression for the amplitude. It was, the computer verified, equivalent to the 9-page formula. In other words, the traditional machinery of quantum field theory, involving hundreds of Feynman diagrams worth thousands of mathematical terms, was obfuscating something much simpler. As Bourjaily put it: “Why are you summing up millions of things when the answer is just one function?”
“We knew at the time that we had an important result,” Parke said. “We knew it instantly. But what to do with it?”
by Natalie Wolchover, Quanta | Read more:
Image: Andy Gilmore
