Yang-Mills theory and four-dimensional quantum fields

5 problems

• Area law for Wilson loops in 4D lattice gauge theory at weak coupling

  Open

  Quark confinement shows up mathematically as a loop observable that decays with the area enclosed by the loop. This is proven only when the coupling is strong, far from the regime that describes nature.

• Continuum limit of four-dimensional lattice Yang-Mills theory

  Open

  Yang-Mills theory can be defined on a grid of points, and the task is to show that it has a sensible limit as the grid spacing goes to zero. The limit should be interacting, not a free theory.

• Existence and mass gap of four-dimensional quantum Yang-Mills theory

  Open

  Prove that the quantum theory of gluons exists as a well-defined mathematical object and that its lightest particle has a strictly positive mass. This is the Clay Millennium problem on Yang-Mills theory.

• Construct any interacting quantum field theory in four spacetime dimensions

  Open

  Build, with full proof, one quantum field theory in 3+1 dimensions in which particles actually scatter off each other. Every rigorous example so far is either free (no interaction) or lives in fewer dimensions.

• Construction of three-dimensional Euclidean Yang-Mills theory

  Open

  In three dimensions the short-distance problems of Yang-Mills theory are milder, yet even this case has not been built rigorously. It is the natural stepping stone to four dimensions.