## Introduction

Recently, *Quanta* has explored the collaboration between physics and mathematics on one of the most important ideas in science: quantum field theory. The basic objects of a quantum field theory are quantum fields, which spread across the universe and, through their fluctuations, give rise to the most fundamental phenomena in the physical world. We’ve emphasized the unfinished business in both physics and mathematics — the ways in which physicists still don’t fully understand a theory they wield so effectively, and the grand rewards that await mathematicians if they can provide a full description of what quantum field theory actually is.

This incompleteness, however, does not mean the work has been unsatisfying so far.

For our final entry in this “Math Meets QFT” series, we’re exploring the most prominent quantum field theory of them all: the Standard Model. As the Cambridge physicist David Tong puts it in the accompanying video, it’s “the most successful scientific theory of all time” despite being saddled with a “rubbish name.”

The Standard Model describes physics in the three spatial dimensions and one time dimension of our universe. It captures the interplay between a dozen quantum fields representing fundamental particles and a handful of additional fields representing forces. The Standard Model ties them all together into a single equation that scientists have confirmed countless times, often with astonishing accuracy. In the video, Professor Tong walks us through that equation term by term, introducing us to all the pieces of the theory and how they fit together. The Standard Model is complicated, but it is easier to work with than many other quantum field theories. That’s because sometimes the fields of the Standard Model interact with each other quite feebly, as writer Charlie Wood described in the second piece in our series.

The Standard Model has been a boon for physics, but it’s also had a bit of a hangover effect. It’s been extraordinarily effective at explaining experiments we can do here on Earth, but it can’t account for several major features of the wider universe, including the action of gravity at short distances and the presence of dark matter and dark energy. Physicists would like to move beyond the Standard Model to an even more encompassing physical theory. But, as the physicist Davide Gaiotto put it in the first piece in our series, the glow of the Standard Model is so strong that it’s hard to see beyond it.

And that, maybe, is where math comes in. Mathematicians will have to develop a fresh perspective on quantum field theory if they want to understand it in a self-consistent and rigorous way. There’s reason to hope that this new vantage will resolve many of the biggest open questions in physics.

The process of bringing QFT into math may take some time — maybe even centuries, as the physicist Nathan Seiberg speculated in the third piece in our series — but it’s also already well underway. By now, math and quantum field theory have indisputably met. It remains to be seen what happens as they really get to know each other.