OP-0050OpenMathematical physics$1,000,000 prize

Yang–Mills existence and the mass gap

Construct quantum Yang–Mills theory on R^4 rigorously, and prove it has a positive mass gap.

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Problem statement

For any compact simple gauge group $G$ , prove that a non-trivial quantum Yang–Mills theory exists on $\mathbb{R}^4$ and satisfies the Wightman axioms (or an equivalent set), and that it has a mass gap $\Delta > 0$ : the spectrum of the Hamiltonian has an isolated lowest eigenvalue at $0$ and no spectrum in $(0,\Delta)$ .

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Background

Yang–Mills theory is the mathematical backbone of the Standard Model, and lattice simulations agree with experiment to high precision. Nevertheless no construction of the continuum theory satisfying the axioms of quantum field theory exists in four dimensions. This is a Clay Millennium Prize problem, with the official formulation by Jaffe and Witten.

Why it matters

The mass gap explains why the nuclear force is short-ranged despite the gauge bosons being massless in the classical theory, and it is the rigorous content of confinement. A construction would also validate the constructive-field-theory programme in the physically relevant dimension.

Known partial results

Constructions exist in two and three spacetime dimensions (Glimm–Jaffe and successors).
Lattice gauge theory gives a well-defined regularised theory; controlling the continuum limit is the open step.
Numerical evidence for the mass gap is overwhelming, with lattice glueball masses computed to a few percent.

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