Getting Math and Physics on the Same Page

December 19, 2021 • by Marc Airhart

Mathematicians are working to bring quantum field theory (QFT) into mainstream mathematics.

Three colored lines spiral around each other from the top to bottom of the image

In this illustration, three exotic particles called “anyons” circle around each other in a process called braiding. Mathematicians use a method called generalized symmetry to study such interactions.

When physicists want to explain how subatomic particles—such as electrons, photons, quarks and neutrinos—behave and interact, they use a framework called quantum field theory (QFT). QFT might be the most successful physical theory ever invented. It was used to predict the existence of the Higgs boson, antimatter and neutrinos. And it has predicted the results of particle physics experiments accurately to the highest number of decimal places ever recorded.

But because QFT was developed mostly by physicists, outside of mainstream mathematics, it isn't constructed in the language that mathematicians typically use. It's also incomplete. And it isn't even clear what makes a quantum field theory a quantum field theory.

"It's like the elephant in that classic children's story," said Dan Freed, a mathematician at UT Austin who has worked for much of his career at the interface between geometry and theoretical physics. "Quantum field theory has many different facets, depending on where you're standing, but we don't yet know how to describe the whole thing."

A new international collaboration involving 16 principal investigators, including Freed, aims to apply new mathematical tools to symmetry in QFT and to further develop the underlying mathematics. The PIs are roughly half mathematicians and half physicists. Funding provided by the Simons Foundation will support graduate students and a postdoctoral researcher in Freed's group.

If successful, their collaboration could bring physicists closer to solving some of their most vexing problems, like how to merge gravity and quantum mechanics into what Einstein, and every theoretical physicist since, considered the holy grail: quantum gravity. It might even lay the groundwork for completing the Final Theory, which would unify all the known forces in the universe. For mathematicians, QFT is a stimulating new playground of ideas from the physical world that could accelerate research in the more abstract realms they often inhabit.

"It's a dialogue that impacts both fields," said Freed.

Those dream goals may still be a long way off, but at least one area could see results sooner: experimental physicists are currently searching for undiscovered phases of matter predicted by a new class of mathematical theories called invertible QFTs, developed and classified recently by Freed and colleagues.

Learn more about the Simons Collaboration on Global Categorical Symmetries:


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