Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear.

Source : Clay Mathematics Institute

The 'Standard Model' (SM) of particle physics is completely reliant on Yang-Mills quantum field theory, which was devised in 1953 as a way of explaining the behaviour of fundamental atomic particles.

It has been spectacularly successful - not only confirmed by computer simulations, but also by countless experiments.

A full mathematical model of the theory, however, has not yet been found.*

In addition, most of the particles have mass, and yet the 'waves' of the 'Yang-Mills field' are assumed to travel at the speed of light. Relativity theory, however, insists that any particle with mass cannot travel at light speed.

The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the ‘mass gap’: the quantum particles have positive masses, even though the classical waves travel at the speed of light.

[ Source as above ]

Further reading : A detailed technical description of the Yang Mills theory, by Professor David Tong, Trinity College, Cambridge university.

* Note : In year 2000, The Clay Mathematics Institute offered a $1 million prize for a full mathematical description. It remains unclaimed.