Yang-Mills Existence and Mass Gap
Asks for a rigorous quantum Yang-Mills theory with a positive mass gap.
The Yang-Mills existence and mass gap problem requires proving that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on four-dimensional Euclidean space satisfying the Wightman axioms (or equivalently the Osterwalder-Schrader axioms), and that the mass spectrum has a strictly positive lower bound (mass gap Delta > 0). Yang-Mills gauge theory is the mathematical foundation of the Standard Model of particle physics, and the mass gap explains why the strong nuclear force has finite range despite gluons being classically massless. The existence of a mass gap is confirmed by experiment and lattice simulations, but no rigorous mathematical proof exists. The problem was formulated for the Clay Institute by Arthur Jaffe (Harvard) and Edward Witten (IAS Princeton).
Formula
The lowest energy excitation above the vacuum has strictly positive mass — confinement in pure Yang–Mills theory.
The classical action for a gauge field with curvature F; quantizing this action and proving a mass gap is the Millennium problem.
The problem requires constructing a quantum Yang–Mills theory on R⁴ satisfying rigorous axioms — not just perturbative expansion.
Summary
The Yang-Mills existence and mass gap problem requires proving that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on four-dimensional Euclidean space satisfying the Wightman axioms (or equivalently the Osterwalder-Schrader axioms), and that the mass spectrum has a strictly positive lower bound (mass gap Delta > 0). Yang-Mills gauge theory is the mathematical foundation of the Standard Model of particle physics, and the mass gap explains why the strong nuclear force has finite range despite gluons being classically massless. The existence of a mass gap is confirmed by experiment and lattice simulations, but no rigorous mathematical proof exists. The problem was formulated for the Clay Institute by Arthur Jaffe (Harvard) and Edward Witten (IAS Princeton).
