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

+ 8 like - 0 dislike
290 views

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

edited Jul 18, 2017
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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.