Yang-Mills theory

Yang-Mills theory is a mathematical model from physics. It describes how forces between the smallest known building blocks of matter come about. The basic idea: forces are not transmitted directly, but via so-called fields, i.e. quantities that have a value at every point in space. Everyone knows an example of such a field from everyday life: the magnetic field of a magnet, which also acts at a distance. In 1954, the two physicists Chen Ning Yang and Robert Mills set up equations that describe such fields for forces inside atomic nuclei. This work has grown into an entire toolbox with which physicists today treat three of the four known fundamental forces.

The foundation of the Standard Model

Physics knows four fundamental forces. Gravity and electromagnetism are familiar from everyday life. In addition, there are two forces that only act over tiny distances: the strong nuclear force holds atomic nuclei together, and the weak nuclear force is behind certain forms of radioactivity. Both are described using Yang-Mills-type equations.

This makes the theory the backbone of the Standard Model of particle physics. This model is the most accurate description of matter that science currently has. Its predictions agree with measurements in some cases to more than ten decimal places. Without the framework of Yang and Mills, this precision would not exist.

The theory is also of interest to mathematics, however. To this day, it has not been rigorously proven that the equations even have a mathematically clean solution. This question, the so-called Yang-Mills mass gap problem, is one of the seven Millennium Problems. A million US dollars is offered for its solution. So physics already works here, while the underlying mathematics remains incomplete.

Symmetry, gauge fields, and force carriers

The core of the idea is a symmetry. Symmetry here means: you may change something about the description of a particle without the measurable physics changing. Similar to rotating a die, which looks the same afterward as before. Yang and Mills required that this freedom hold independently at every point in space.

This requirement has a consequence. For the equations to remain valid, additional fields must be introduced, the gauge fields. These are precisely the fields that then transmit the force. Their associated particles are called gauge bosons, such as the gluons of the strong nuclear force. The force is thus not built in by hand, but arises from the symmetry requirement.

One difference from electromagnetism is important. Light particles practically do not influence one another. Gluons, on the other hand, act on themselves because they themselves carry the charge for which they are responsible. As a result, the equations are no longer easily solvable, but become highly intertwined. This self-interaction is the reason why the mathematics remains so stubbornly difficult.

From particle accelerator to supercomputer

The theory becomes practically visible at facilities like the Large Hadron Collider near Geneva. There, protons are made to collide with very high energy. Researchers use Yang-Mills equations to calculate in advance what arises from such collisions. The detection of the Higgs particle in 2012 was a confirmation of this framework.

Because the equations cannot be solved with pen and paper, computing power comes into play. In lattice gauge theory, space and time are broken down into a fine grid of points, and the fields are calculated numerically on it. Such simulations run on the world’s largest supercomputers and are among the most expensive computational tasks in basic research. In recent years, machine learning has also been used experimentally to speed up these calculations.

In the news, the term is usually encountered in two contexts. Either it concerns advances in particle physics, such as new measurements at CERN. Or it concerns the Millennium Problem, when someone publishes an attempted proof. A common misconception is that the theory is mere speculation. Physically, it is extremely well tested; only its rigorous mathematical foundation remains open.