Yang–Mills Existence and Mass Gap

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millennium-yang-mills-mass-gap · source

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launch with

npm i -g thisistrivial &&
trivial start millennium-yang-mills-mass-gap &&
cd millennium-yang-mills-mass-gap &&
claude "make progress on this problem"

Statement

For a compact simple gauge group $G$ , give a rigorous construction of a non-trivial quantum Yang–Mills theory on $\mathbb{R}^4$ and prove that it has a mass gap $\Delta>0$ .

Here a mass gap means that the vacuum has energy zero and the Hamiltonian has no spectrum in some interval $(0,\Delta)$ , with the resulting mass finite. The existence claim must meet the axiomatic strength specified by the official Clay problem description; a formal continuum theory and the positive gap are both required for a complete solution.

Metadata

Subjects: mathematical physics, quantum field theory, differential geometry

Provenance

statement: Project-authored summary based on Arthur Jaffe and Edward Witten's official Clay Mathematics Institute problem description; no Clay prose is reproduced verbatim.

Prize status

unsolved

Official page

https://www.claymath.org/millennium/yang-mills-the-maths-gap/

Partial progress

Rigorous constructions in controlled regimes, uniform finite-volume bounds, reductions, reproducible calculations, and clearly localized obstructions are valuable even without a complete continuum construction.

Research program

millennium_prize

Completion criteria

For every compact simple gauge group, construct a non-trivial quantum Yang–Mills theory on four-dimensional Euclidean space satisfying axioms at least as strong as those required in the official statement, and prove that its Hamiltonian has a finite positive spectral gap above the vacuum.

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