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The top answer in this post Why are there no instantons in the gauge group $U(1)$? stated that, the reason why there is no U(1) instanton in R^4 is the second cohomology group of R^4 is trivial. I can understand the logic of this answer(and I think the answer is great), but I also remember that there actually are non-trivial solutions to the free maxwell equation, that is just plane-waves.

So my question is, can the plane-waves solution be counted as "instantons"? (Very probably not, since I believe the answer above is right). And if not, why? It seems they really are non-trivial solutions to classcial equation of motions. Mathematically, why they can't be an element in the second cohomology group?

Thanks for any help. :)

Gu Xia
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In two-dimensional the topological defect that is analogous to an instanton is the vortex. A plane wave cannot produce a vortex by itself, but one can find solutions for the free Maxwell equations that contain vortices.

A vortex appear when there is a complex null in a two-dimensional complex field. A way to think about it is to separate the complex function into an amplitude and a phase $$ \psi(x,y) = A(x,y) \exp(i\theta(x,y)) . $$ The phase function $\theta(x,y)$ is given by a mapping from the two-dimensional plane to the unit circle. (Remember a phase of 0 is equal to a phase of $2\pi$.) The unit circle has a nontrivial second homotopy group. What this mean is that when a consider any closed contour on the two-dimensional plane and a consider how the phase function maps this contour onto the unit circle I can find cases where the mapping of the contour is warped around the circle so that when a shrink the contour to a point on the two-dimensional plane the mapping on the circle must jump from one side to the other to become a point on the circle. This jump is not allowed from a topological point of view. Therefore the must be a topological defect (a vortex) enclosed by the initial contour. When I consider this complex field in three-dimensional space, the vortices become lines, and in 4D they become sheets.

Now let's generalize this picture to four dimensional to consider instantons. In this case we need to change the circle to a hypersphere embedded in 4D. Here we are looking for fields with a nontrivial fourth homotopy group. Note that when I have a circle on such a hypersphere, I can shrink it to a point without having to jump from one side to another. Therefore, a phase function does not contain the ability to form instantons. So the solutions of the free Maxwell equations do not contain instantons.

To get an instanton I must give the field more internal degrees of freedom, like what we would have with the gauge fields of non-abelian gauge interactions. Such fields can be represented by an amplitude times a more complicated structure (the generalization of a phase) that is parameterized by a function given as a mapping from $\mathbb{R}^4$ to a hypersphere. Provided that this hypersphere has a nontrivial fourth homotopy group, this field can produce instantons.

flippiefanus
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