Can MathGPT solve the Yang–Mills existence and mass gap?

No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

UnsolvedMathematical physicsMillennium Prize Problem

Yang–Mills existence and mass gap

Q: Is there a prize?

Prize of US \$1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).

Construct quantum Yang–Mills theory rigorously and prove its lightest particle has positive mass.

Exact statement

F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu],\qquad S=\frac{1}{4g^{2}}\int_{\mathbb R^{4}}\operatorname{tr}\big(F_{\mu\nu}F^{\mu\nu}\big)\,d^4x

\text{Prove: }\exists\ \Delta>0\ \text{with}\ \operatorname{spec}(H)\subset\{0\}\cup[\Delta,\infty)

For any compact simple gauge group G, prove that a quantum Yang–Mills theory on four-dimensional space-time exists and satisfies axioms at least as strong as the Wightman (or Osterwalder–Schrader) axioms, and that it has a mass gap Δ > 0.

Status

Open. Physicists have strong numerical and experimental evidence; there is no rigorous mathematical construction in four dimensions.

Posed: Yang and Mills (1954); Millennium formulation by Arthur Jaffe and Edward Witten (2000).

Prize of US $1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).

In plain words

Yang–Mills theory is the mathematics behind the strong and weak nuclear forces; quantum chromodynamics (QCD) is a Yang–Mills theory with group SU(3). Its classical field equations are $D_\mu F^{\mu\nu}=0$ .

Two things are missing mathematically. First, nobody has constructed the quantum theory in four dimensions with full mathematical rigour. Second, experiments and computer simulations show that the force-carrying particles bind into massive states (glueballs, and confinement of quarks), so the energy spectrum has a gap above the vacuum; proving that gap is the 'mass gap' half of the problem.

Key results so far

1973: Gross, Wilczek and Politzer discover asymptotic freedom (Nobel Prize 2004).
1974: Wilson introduces lattice gauge theory; lattice simulations later give strong numerical evidence for confinement and a mass gap.
1980s: Bałaban's lattice renormalization-group work proves ultraviolet stability estimates for 4D lattice Yang–Mills, a partial step toward a construction.
—: Two-dimensional Yang–Mills theory is understood rigorously; the four-dimensional theory is not.

What our solver can and cannot do

No computer program or AI can solve the Yang–Mills existence and mass gap by being asked. If you paste it into the MathGPT solver, you get this reference information, labelled as such, never an invented proof. The solver can still compute related special cases and examples. Open the solver.

Frequently asked questions

What does the Yang–Mills existence and mass gap say?: For any compact simple gauge group G, prove that a quantum Yang–Mills theory on four-dimensional space-time exists and satisfies axioms at least as strong as the Wightman (or Osterwalder–Schrader) axioms, and that it has a mass gap Δ > 0.
Has the Yang–Mills existence and mass gap been solved?: Open. Physicists have strong numerical and experimental evidence; there is no rigorous mathematical construction in four dimensions.
Is there a prize?: Prize of US $1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).
Can MathGPT solve the Yang–Mills existence and mass gap?: No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

References

A. Jaffe and E. Witten, 'Quantum Yang–Mills Theory', Clay Mathematics Institute (2000).
C. N. Yang and R. Mills, Phys. Rev. 96 (1954).
K. Wilson, 'Confinement of quarks', Phys. Rev. D 10 (1974).

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