• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,037 questions , 2,191 unanswered
5,345 answers , 22,706 comments
1,470 users with positive rep
816 active unimported users
More ...

  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

Your answer

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):
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:
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).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights