*This article is the first part of a series about quantum field theory published by Quanta Magazine. Other stories in the series can be found* *here**.*

Over the past century, quantum field theory has proved to be the single most sweeping and successful physical theory ever invented. It is an umbrella term that encompasses many specific quantum field theories—the way “shape” covers specific examples like the square and the circle. The most prominent of these theories is known as the Standard Model, and it is this framework of physics that has been so successful.

“It can explain at a fundamental level literally every single experiment that we’ve ever done,” said David Tong, a physicist at the University of Cambridge.

But quantum field theory, or QFT, is indisputably incomplete. Neither physicists nor mathematicians know exactly what makes a quantum field theory a quantum field theory. They have glimpses of the full picture, but they can’t yet make it out.

“There are various indications that there could be a better way of thinking about QFT,” said Nathan Seiberg, a physicist at the Institute for Advanced Study. “It feels like it’s an animal you can touch from many places, but you don’t quite see the whole animal.”

Mathematics, which requires internal consistency and attention to every last detail, is the language that might make QFT whole. If mathematics can learn how to describe QFT with the same rigor with which it characterizes well-established mathematical objects, a more complete picture of the physical world will likely come along for the ride.

“If you really understood quantum field theory in a proper mathematical way, this would give us answers to many open physics problems, perhaps even including the quantization of gravity,” said Robbert Dijkgraaf, director of the Institute for Advanced Study (and a regular columnist for *Quanta*).

Nor is this a one-way street. For millennia, the physical world has been mathematics’ greatest muse. The ancient Greeks invented trigonometry to study the motion of the stars. Mathematics turned it into a discipline with definitions and rules that students now learn without any reference to the topic’s celestial origins. Almost 2,000 years later, Isaac Newton wanted to understand Kepler’s laws of planetary motion and attempted to find a rigorous way of thinking about infinitesimal change. This impulse (along with revelations from Gottfried Leibniz) birthed the field of calculus, which mathematics appropriated and improved—and today could hardly exist without.

Now mathematicians want to do the same for QFT, taking the ideas, objects and techniques that physicists have developed to study fundamental particles and incorporating them into the main body of mathematics. This means defining the basic traits of QFT so that future mathematicians won’t have to think about the physical context in which the theory first arose.

The rewards are likely to be great: Mathematics grows when it finds new objects to explore and new structures that capture some of the most important relationships—between numbers, equations and shapes. QFT offers both.

“Physics itself, as a structure, is extremely deep and often a better way to think about mathematical things we’re already interested in. It’s just a better way to organize them,” said David Ben-Zvi, a mathematician at the University of Texas, Austin.

For 40 years at least, QFT has tempted mathematicians with ideas to pursue. In recent years, they’ve finally begun to understand some of the basic objects in QFT itself—abstracting them from the world of particle physics and turning them into mathematical objects in their own right.