Theorists discover the ‘Rosetta Stone’ for neutrino physics

UChicago, Brookhaven, Fermilab scientists find new math identity while studying particle physics

Usually the way things work is that mathematicians make math discoveries, and physicists borrow and adapt those ideas to explain the universe. But three physicists at the University of Chicago and two national laboratories have discovered a fundamental identity in linear algebra—based on studying particle physics.

Three theoretical physicists—Peter Denton, a scientist at Brookhaven National Laboratory and a scholar at Fermilab’s Neutrino Physics Center; Stephen Parke, theoretical physicist at the UChicago-affiliated Fermi National Accelerator Laboratory; and Xining Zhang, a University of Chicago graduate student working with Parke—were investigating the properties of neutrinos, a type of particle that interacts so rarely with matter that trillions of them zip through your body every day without you noticing.

As they travel, neutrinos flip back and forth between different types, or “flavors.” Scientists describe these oscillations using a type of mathematical expression called eigenvectors and eigenvalues.

Eigenvectors and eigenvalues are two important ways of reducing the properties of a matrix to their most basic components and have applications in many math, physics and real-world contexts. The eigenvectors identify the directions in which a transformation occurs, and the eigenvalues specify the amount of stretching or compressing that occurs.

Denton, Parke and Zhang’s formula, however, relates eigenvectors and eigenvalues in a direct way that hadn't been previously recognized.

While the eigenvalues are somewhat unavoidably tricky, this new result shows that the eigenvectors can be written down in a simple, compact and easy-to-remember form once the eigenvalues are calculated. For this reason, the trio called the eigenvalues “the Rosetta Stone” for neutrino oscillations in their original publication: “Once you have them, you know everything you want to know.”

Experts fully expected this formula to exist somewhere in the literature for centuries, but the team couldn’t find any evidence for it online or in textbooks. The trio of theorists were eventually directed to a similar result by UCLA mathematics professor Terence Tao, who has a Fields Medal and Breakthrough Prize to his name. When they presented Tao with their result, he cheerfully declared that it was, in fact, the discovery of a new identity, and he provided several mathematical proofs, which were recently published online. (Tao also discussed the new identity in his math blog.)

“I couldn’t believe it at first, because linear algebra has been studied for centuries,” Zhang said. “But Prof. Tao’s proofs show that our method, which was created to calculate neutrino oscillations quickly and with better precision, can also be expressed as a very elegant math formula.”

Outside particle physics, eigenvalues and eigenvectors are used in analyzing vibrating systems and facial recognition programs. Linear algebra is used everywhere in quantitative sciences, engineering and economics, much like the simple calculator is used for everyday life.

—Adapted from an article that first appeared on the Fermilab website.