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It seems to me the 'naive' approach to proving the existence of Yang-Mills in a rigorous context (via Osterwalder-Schrader $\to$ Wightman axioms), would be:

  1. Study gauge invariant lattice QCD correlation functions such as $$\langle \mathrm{Tr}(W(\gamma_1)) \cdots \mathrm{Tr}(W(\gamma_n)) \rangle_{a,V} := \int dU e^{-S_\mathrm{Wilson}[U]} (\mathrm{Tr}(W(\gamma_1)) \cdots \mathrm{Tr}(W(\gamma_n))$$ $\gamma_i$ are physical paths in 4d, and $W(\gamma_i)$ are the corresponding wilson loops. The average is done on a lattice with lattice spacing $a$, and physical volume $V$. This observable is fully formal, finite, and something that I can go simulate on my computer right now. We hope that the 'Schwinger Functions': $$\langle \mathrm{Tr}(W(\gamma_1)) \cdots \mathrm{Tr}(W(\gamma_n)) \rangle := \lim_{a \to 0, V \to \infty} \langle \mathrm{Tr}(W(\gamma_1)) \cdots \mathrm{Tr}(W(\gamma_n)) \rangle_{a,V}$$ also exist (note that $\gamma_i$ are physical-sized loops, in other words they grow to contain more lattice links in the limit $a \to 0$)

  2. Prove that the Schwinger Functions satisfy the Osterwalder Schrader axioms (and for the cherry on top, if the connected correlation functions have exponential decay then you have a mass gap)

What is the difficulty in this approach? For example, is it known whether the Schwinger Functions (step 1) even exist? It's certainly clear from numerical simulations that they should. If they are known to exist, step 2 doesn't seem too bad, given results like this.

(On physics.stackexchange, I've mostly seen discussion about other approaches, via algebras of observables and things like that. Sorry if this is a duplicate)

QCD_IS_GOOD
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    Pretty much nothing is known, not even that the Schwinger functions exist. There also is no analog of Osterwalder-Schrader reconstruction for Wilson loops afaik. – Prof. Legolasov Feb 21 '23 at 19:20
  • I see, thanks! Yes I was a bit hand-wavey when choosing the wilson-loops to be my operators of interest - really I could talk $A_\mu$ fields living on each link, would I then need to worry about gauge fixing? On the lattice, gauge fixing leads to gribov copies, which is an uncertainty that I've never really understood how to deal with (seems kind of annoying) – QCD_IS_GOOD Feb 21 '23 at 19:47
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    As I’m sure you’re well aware this is a major unsolved problem so all I can give here is an opinion, albeit a common one to my knowledge. BRST is broken non-perturbatively (Gribov copies are themselves heuristic classical arguments, but it appears that in this case their existence does in fact break gauge fixing schemes beyond perturbation theory). So a solution here has to either be algebraic and manifestly gauge invariant (operator algebras etc), or indeed it could come from Wick-rotated lattice Wilson loops, but it’s very much unclear how to do either – Prof. Legolasov Feb 21 '23 at 20:15
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    Even with the Euclidean Wilson loops on the lattice, to take the continuum limit you need to renormalise, and that’s highly nontrivial and afaik hasn’t been done even in 3d, let alone 4d. – Prof. Legolasov Feb 21 '23 at 20:16
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    Actually another important point I forgot to mention yesterday is extracting the classical limit. Theories based on loops are much harder to build perturbative expansions for, so even if every step above is somehow done, you’ll still have to demonstrate that what you got in the end is Yang-Mills theory and agrees with the well known celebrated perturbative expansion of the latter. LQG is an example where this program has been rigorously constructed for nothing less than General Relativity, yet they struggle to reproduce the classical limit and there are indications it’s anomalous – Prof. Legolasov Feb 22 '23 at 21:10
  • https://arxiv.org/pdf/1903.09829.pdf This paper seems a step to what I was imagining possible; (Note however that it's not accepted in a journal yet it seems) – QCD_IS_GOOD May 20 '23 at 21:46
  • I should also mention that the naive limit I wrote in this question is no good - Wilson loops have both power divergences (in the length), and log divergences (at the cusps), you do have to renormalize them somehow – QCD_IS_GOOD Sep 13 '23 at 15:47

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