# What is a precise mathematical statement of the Yang-Mills and mass gap Clay problem?

I am a mathematician writing a statement of each of the Clay Millennium Prize problems in a formal proof assistant.  For the other problems, it seems quite routine to write the conjectures formally, but I am having difficulty stating the problem on Yang Mills and the mass gap.

To me, it seems the Yang-Mills Clay problem is not a mathematical conjecture at all, but an under-specified request to develop a theory in which a certain theorem holds.  As such, it is not capable of precise formulation.  But a physicist I discussed this with believes that a formal mathematical conjecture should be possible.

I understand the classical Yang-Mills equation with gauge group $G$, as well as the Wightman axioms for QFT (roughly at the level of the IAS/QFT program), but I do not understand the requirements of the theory that link YM with Wightman QFT.

The official Clay problem from page 6 of Jaffe and Witten states the requirements (in extremely vague terms) as follows:

"To establish existence of four-dimensional quantum gauge theory with gauge group $G$ one should define a quantum field theory (in the above sense) with local quantum field operators in correspondence with the gauge-invariant local polynomials in the curvature $F$ and its covariant derivatives […]. Correlation functions of the quantum field operators should agree at short distances with the predictions of asymptotic freedom and perturbative renormalization theory, as described in textbooks. Those predictions include among other things the existence of a stress tensor and an operator product expansion, having prescribed local singularities predicted by asymptotic freedom."

A few phrases are somewhat clear to me like "gauge-invariant local polynomials...", but I do not see how to write much of this with mathematical precision. Can anyone help me out?

## 1 Answer

Not quite an answer but basically, I think one needs to prove the analogues of the results in

1. "All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory" by Fröb, Holland and Hollands
2. "All-order existence of and recursion relations for the operator product expansion in Yang-Mills theory" by Fröb and Holland

but in the sense of true Schwartz distributions instead of formal power series in h-bar. One also needs to show at least one 2-point function decays like $e^{-m|x-y|}$ with $m>0$.

• This is not so clear. Due to confinement, it may well be that such estimates are impossible outside of perturbation theory. I discussed what needs to be proved in physicsoverflow.org/21846 – Arnold Neumaier Dec 31 '17 at 11:17
• @ArnoldNeumaier: If you want the \$million you have to prove estimates "outside perturbation theory". Proofs about correlations as elements in$\mathbb{R}[[\hbar]]\$ are far below the standards required by the Clay problem formulation. You are of course free to believe that this is an impossible task. – Abdelmalek Abdesselam Jan 1 '18 at 21:39
• Your comment does not contradict my statement. I was refering to the fact that the vacuum sector of the theory (to whom the Wightman axioms apply) does not contain the Yang-Mills fields themselves. Thus there is no reason to expect that 2-point correlation functions of these fields should satisfy any decay law. – Arnold Neumaier Jan 2 '18 at 10:14
• but then how to see a mass gap if no correlation decays exponentially? – Abdelmalek Abdesselam Jan 2 '18 at 14:41
• It must be correlations of appropriate composite glueball fields. – Arnold Neumaier Jan 2 '18 at 15:17