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

+ 8 like - 0 dislike

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?

This post imported from StackExchange Physics at 2017-07-18 06:45 (UTC), posted by SE-user Thales

asked Jun 13, 2017 in Theoretical Physics by Thales (40 points) [ revision history ]
edited Jul 18, 2017 by Dilaton
I think a good starting point could be F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory , (World Scientific, Singapore, 1993).

This post imported from StackExchange Physics at 2017-07-18 06:45 (UTC), posted by SE-user Jon
Possibly related on MathOverflow.

This post imported from StackExchange Physics at 2017-07-18 06:45 (UTC), posted by SE-user Keith McClary
Jaffe and Witten's statement of the problem is probably the most precise you can get with our current state of knowledge.

This post imported from StackExchange Physics at 2017-07-18 06:45 (UTC), posted by SE-user SCFT

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