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

In summary, the acquaintance is looking for literature on self-dual solutions to Maxwell's equations on Euclidean or pseudo-Euclidean space (not Minkowski space) called "instantons". They are unsure if instantons are solutions to equations of motion and are seeking clarification on the concept of self-duality. They are also asking for any recommended literature on this topic. The first attempt at providing literature was on Minkowski space, and the second attempt, specifically section 4.4, was not suitable as it did not involve self-dual 2-forms on Euclidean space. They are specifically looking for solutions that satisfy Maxwell's equations and reduce them to 4 first-order partial differential equations with 3 unknown functions.
  • #1
nomadreid
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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.
 
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  • #2
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."
 

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1. What are self-dual solutions to Maxwell's equations?

Self-dual solutions to Maxwell's equations refer to solutions that are invariant under the action of duality transformations. In other words, these solutions remain unchanged when electric and magnetic fields are interchanged and the sign of one of them is flipped.

2. How are self-dual solutions related to Euclidean space?

Self-dual solutions to Maxwell's equations are defined in Euclidean space, which is a mathematical space that has a flat geometry and obeys the rules of Euclidean geometry. This space is commonly used in physics and engineering to describe physical phenomena.

3. What significance do self-dual solutions hold in physics?

Self-dual solutions to Maxwell's equations have many important applications in physics, particularly in the study of electromagnetism. They can help us understand the behavior of electromagnetic fields and have been used to describe phenomena such as the propagation of light and the behavior of superconductors.

4. Can self-dual solutions exist in non-Euclidean spaces?

No, self-dual solutions are only defined in Euclidean space. In non-Euclidean spaces, the concept of duality transformations does not hold, so self-dual solutions do not have any meaning.

5. Are self-dual solutions unique?

No, there can be multiple self-dual solutions to Maxwell's equations for a given set of boundary conditions. However, these solutions may have different physical interpretations or properties, so it is important to carefully consider the specific problem at hand when analyzing self-dual solutions.

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