# I Self-dual solutions to Maxwell's equations, Euclidean space

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1. Jul 30, 2017

### nomadreid

I am attempting to understand a question posed to me by an acquaintance, who asked me if I could refer him to literature freely available on the Internet on "self-dual solutions to Maxwell's equations on Euclidean space, or pseudo-Euclidean space, not Minkowski space (where there are none)" and he labeled them "instantons". I'm lost, so my questions:
(a) aren't instantons solutions to equations of motion? Can you consider Maxwell's equations as equations of motion?
(b) In what sense are solutions to such equations self-dual, or more simply, what concept of duality is meant here?
(c) does anyone have any appropriate literature (freely available on the Internet) to recommend to answer my acquaintance's question?
Any indications on any or all of these questions would be greatly appreciated.

2. Aug 5, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Aug 5, 2017

### nomadreid

I made two attempts at providing the literature, but missed the mark both times. Here they are, and the reason they didn't work:
I first sent an article about Maxwell's equations on a space with a Minkowski metric, but the researcher wrote back that he needed solutions on a space with a Euclidean metric. I then sent the attached, specifically section 4.4, The reply:

"Unfortunately, the BPST instanton, built in Section 4.4, is not what we need. We need to find in the Euclidean space a self-dual 2-form F = * F that satisfies Maxwell's equations dF = 0. Another group of Maxwell equations is automatically satisfied because the 2-form F is self-dual. Maxwell equations in this case are reduced to 4 first-order partial differential equations with 3 unknown functions. We have found some solutions, and we would like to know what solutions had already been found before us."

#### Attached Files:

• ###### Lect4.pdf
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