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I am reading Aspects of Symmetry by Sidney Coleman. More specifically I am trying to learn about instantons, and I would like some clarifications.

In chapter 7, section 4. he derives confinement in a 2 dimensional abelian Higgs model using the dilute instanton gas approximation, and than argues that such a mechanism can not possibly work in 4D. He derives confinement in the sense of the Wilson loop criterion.

If I understand correctly, the important difference, is that in 2D, instantons deep inside the Wilson loop have a non negligable contribution to the functional integral, while in 4D this is not the case, there, only instantons that overlap with the loop contribute. What I do not understand is what specific properties of the 4D instanton solutions make this last statement true. What is the difference between 2D and 4D?

In the 2D model, the Wilson loop will be simply determined (at least for large loops) by the winding number that we would get if we only considered instantons inside the loop. Such a simple thing can not be said in 4D where to get the equivalent of the winding number we would need to consider a 3D surface integral.

But than again, all this says to me is that I can not carry out the computation along the same lines in 4D than what was done in 2D. That does not necessarily mean, as I quote "Whatever confines quarks, it is not instantons."

Thanks in advance for any clarifications on the matter.

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  • $\begingroup$ Think about it like this (very handwavy, hence no answer) - the reason you get the confining "area law" in 2D is that the whole area inside the Wilson loop contributes. In 4D, only the overlaps contribute, so you get no area law, hence no confinement. $\endgroup$ – ACuriousMind Feb 19 '15 at 15:42

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