Perturbative vs. non-perturbative approaches to a well-defined Yang-Mills theory in 4 dimensions

Question

Another question regarding the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf). Does the problem require that the "construction" of a four dimensional quantum Yang-Mills be non-perturbative? I get the feeling that this problem is to make notions such as the renormalization group rigorous, and thus is perturbative, but isn't lattice gauge theory already mathematically well-defined? If so, why can this not be used as an approach to this problem? Essentially, which is the preferable approach as specified by the problem: perturbative or non-perturbative?

This post imported from StackExchange Physics at 2014-06-27 11:25 (UCT), posted by SE-user user47299

Comments

It is believed that the mass gap is a purely non-pertubative effect hence a pertubative approach does not work.

This post imported from StackExchange Physics at 2014-06-27 11:25 (UCT), posted by SE-user Tobias Diez

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@TobiasDiez Why is this?

This post imported from StackExchange Physics at 2014-06-27 11:25 (UCT), posted by SE-user user47299

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The reason is, that the mass gap is proportional to $exp(- g^2)$, where $g$ is the coupling constant. So in perturbation theory you send $g \rightarrow 0$ and hence the mass gap also vanishes. See for example, page 29 in media.scgp.stonybrook.edu/presentations/20120117_3_Witten.pdf

This post imported from StackExchange Physics at 2014-06-27 11:25 (UCT), posted by SE-user Tobias Diez

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@TobiasDiez Perfect, thanks.

This post imported from StackExchange Physics at 2014-06-27 11:25 (UCT), posted by SE-user user47299

Answer 1

An important correction to the answer of Tobias Diez. the correct expression is exp(-1/g²), as Witten points out in his talk.

Comments

That is why it is extremely important to start from a qualitatively better initial approximation where some part of permanent interaction is taken into account exactly ;-). Then the perturbative corrections will be smaller and will not affect the qualitative part of solutions.

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@Vladimir: don't know why your comment is downvoted. It is useful to start from an approximation where the mass-gap is explicit at long-distances. The renormalization issues are at short-distances and can be separated out and dealt with essentially perturbatively.

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@Ron Miamon: Thanks, Ron, for your support. Indeed, mathematically, if one has a series $f(x) = f(0) + f'(0)\cdot x + ...$ and manages to sum up a part of it into a another function $f1(x)$ like this $f(x) = f1(x) + a\cdot x + b\cdot x^2+...$, then convergence of the new series may be different, in particular, improved. It means starting the series expansion for $f(x)$ from another initial approximation $f1(x)$, which takes into account the expansion parameter $x$ exactly. If one manages to choose it conceptually from the very beginning, then one gets a better series only giving small quantitative corrections. I myself had in the past such examples. In physics it is well known, for example, in the BCS superconductivity description (Cooper pairs as the initial approximation). In my opinion, those downvoters are just not that experienced.

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@VladimirKalitvianski @RonMaimon Oops, sorry for the downvote, it was accidental. I have removed it now.