# Particle physics gives mathematics a potentially powerful new tool

Although abstract by nature, mathematics has concrete origins: the greatest advances have been inspired by the natural world. Recently, a new result in linear algebra was discovered by three physicists trying to understand the behavior of neutrinos.

Neutrinos are subatomic particles that interact only weakly with matter, so they easily pass through a wall, the Earth or a star. The American poet John Updike described them beautifully in his poem Cosmic Gall: Neutrinos, they are very small. / They have no charge and no mass/ And don’t interact at all./ The earth is just a stupid ball/ For them, through which they only pass, / Like garbage collectors in a room full of drafts/ Or photons through a sheet of glass.

Neutrinos are produced in large numbers in the sun, and billions pass harmlessly through our bodies every second. Almost nothing can stop them. Dune (The Deep Underground Neutrino Experiment) aims to unlock the mysteries of neutrinos. This international experiment will use particle accelerators to send an intense beam of high-energy neutrinos from Fermilab in Illinois 800 miles across the earth to massive detectors a mile underground in South Dakota. The experiment could lead to vital applications in medicine and could change our understanding of the universe.

#### three kinds

Neutrinos come in three “flavors” and can suddenly change from one form to another as they travel at near the speed of light. Recently, three physicists, Peter Denton, Stephen Parke and Xining Zhang, studying these sudden changes, came across a new algebraic identity. The mathematical description of how neutrinos interact with matter involves square arrays of numbers called matrices. Each matrix has a set of characteristic numbers called eigenvalues; and with each eigenvalue goes a direction in space called an eigenvector.

Eigenvalues and eigenvectors appear in a wide variety of contexts. Operations that stretch, shrink, shear, reflect, or rotate objects in a linear fashion are described by matrices. Eigenvectors give the directions that remain unchanged under transformation, and eigenvalues determine the change in length or angle of rotation.

#### powerful tool

Physicists noticed that the eigenvectors of their matrices could be expressed as combinations of the eigenvalues. Since the latter are much easier to compute than the former, the new identity provides a powerful tool. It applies to symmetric matrices, called Hermitian, named after the French mathematician Charles Hermite. The formula gives the eigenvectors in terms of the eigenvalues of the matrix and the so-called minor sub-matrices.

The basic mathematics had been studied for centuries, and physicists were convinced that the result must already be known. But they couldn’t find it anywhere in the literature, so they contacted a world-renowned mathematician, Terence Tao from the University of California, Los Angeles. Tao responded within hours that he had never seen the result and included three independent proofs of the formula in his email.

The result has already benefited the study of neutrinos. Tao is convinced that it will be important elsewhere too. He is quoted in the online magazine Quanta as saying, “It’s so pretty that I’m sure it will come in handy in the near future.”

Peter Lynch is Emeritus Professor in the School of Mathematics and Statistics at University College Dublin. He blogs at thatsmaths.com

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