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The Millennium Museum·Mathematical physics

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Yang–Mills Existence and Mass Gap

Does quantum Yang–Mills theory exist rigorously on four-dimensional space-time, with a positive mass gap, for every compact simple gauge group?

Open. No Lean statement exists yet; one is being drafted.

The exhibit

The problem, three ways

Curious

The strong nuclear force is described by a quantum Yang–Mills theory. Physicists compute with it to remarkable accuracy, yet nobody has built the theory with mathematical rigour, or proved why its particles have mass when the classical waves do not.

Undergraduate

For each compact simple gauge group, construct a quantum Yang–Mills theory on ℝ⁴ satisfying axioms at least as strong as those of Wightman or of Osterwalder and Schrader, and prove it has a mass gap: the energy spectrum has nothing between the vacuum and some Δ > 0.

Specialist

Lattice gauge theory (Wilson) gives a regularisation with strong numerical evidence for confinement and a gap. Constructive field theory has built interacting theories in two and three dimensions, with partial progress in four (Balaban). Asymptotic freedom is why a continuum limit is expected to exist.

What would count

  • Both parts: a theory satisfying the axioms, and a proof of the mass gap, as in Arthur Jaffe and Edward Witten's Clay description.
  • Clay's conditions: publication in a qualifying outlet, two years, general acceptance, then a decision by its board.

Fidelity traps

  • Results on a lattice, in two or three dimensions, or for an abelian group do not count.
  • Numerical evidence for a gap is not a proof.
  • Any formal statement must encode the axioms of quantum field theory, which is itself hard to state faithfully.

In the register

Formal record: unclassified

No Lean statement is held yet.

No AI contribution is recorded against this problem.

The full record →

Who is attacking it

  • Formal Conjectures: Drafting Lean statements for Yang–Mills, Hodge, and Birch and Swinnerton-Dyer. Source
  • OpenAI: Says it began large-scale work on all the open Millennium problems on 1 September 2026. Source

The history

7 events
  1. 1950s
  2. 1954

    posed

    Yang and Mills

    Chen-Ning Yang and Robert Mills publish non-abelian gauge theory.

    Source

  3. 1970s
  4. 1973

    progress

    Asymptotic freedom

    David Gross, Frank Wilczek and David Politzer discover that the interaction weakens at short distances.

    Source

  5. 1974

    progress

    Lattice gauge theory

    Kenneth Wilson formulates gauge theory on a lattice, giving a route to confinement and to numerical computation.

    Source

  6. 1980s
  7. 1980s

    progress

    Constructive work in four dimensions

    Tadeusz Balaban carries out a renormalisation-group analysis of lattice Yang–Mills, the most substantial constructive progress in four dimensions.

    Source

  8. 2000s
  9. 2000-05-24

    prize

    A Millennium Prize Problem

    The Clay Mathematics Institute names the problem one of seven carrying a $1,000,000 prize, with an official description by Arthur Jaffe and Edward Witten.

    Source

  10. 2026
  11. 2026-09

    formal

    A Lean statement in draft

    Formal Conjectures begins drafting Lean statements for Yang–Mills, Hodge, and Birch and Swinnerton-Dyer.

    Source

  12. 2026-09-01

    AI

    OpenAI turns to the Millennium problems

    OpenAI says it began large-scale work on all the open Millennium problems.

    Source

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