Conjecture Fieldbook

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Yang–Mills and mass gap

Canonical problem dossier

Construct a rigorous quantum Yang–Mills theory in four dimensions and prove that it has a positive mass gap.

Status
Open
Field
Mathematical physics
Verification
Reviewed dossier
Source grade
mixed
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

Δ=E1−E0>0\Delta=E_1-E_0>0

Why it matters

Yang–Mills theories describe fundamental interactions. The problem asks for a mathematically rigorous foundation and an explanation of the observed gap above the vacuum.

Context and known approaches

The challenge combines constructive quantum field theory with the non-perturbative behavior of non-abelian gauge fields.

Formal boundary

For any compact simple gauge group G, establish a non-trivial quantum Yang–Mills theory on R⁴ satisfying the stated axioms and a gap Δ > 0.

Known partial results

Pending review. No structured partial-result note has been added to this dossier.

Source trail

  1. Clay Mathematics Institute — Official Millennium problem overview.
  2. Wikipedia — Background and references.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.