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

Click For Summary
SUMMARY

This discussion focuses on self-dual solutions to Maxwell's equations specifically in Euclidean space, distinguishing them from Minkowski space where such solutions do not exist. The term "instantons" is used to describe these solutions, which are indeed considered as solutions to equations of motion. The key concept of duality in this context refers to the self-dual 2-form F that satisfies the condition F = *F, leading to a reduction of Maxwell's equations to four first-order partial differential equations with three unknown functions. The participants seek freely available literature that addresses these specific solutions.

PREREQUISITES
  • Understanding of Maxwell's equations
  • Familiarity with self-dual forms in differential geometry
  • Knowledge of Euclidean and pseudo-Euclidean spaces
  • Basic concepts of instantons in theoretical physics
NEXT STEPS
  • Research self-dual forms in the context of differential geometry
  • Explore literature on instantons and their applications in field theory
  • Study the reduction of Maxwell's equations to first-order partial differential equations
  • Investigate freely available resources on Euclidean metrics and their implications in physics
USEFUL FOR

This discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and researchers interested in advanced topics related to Maxwell's equations and instantons.

nomadreid
Gold Member
Messages
1,762
Reaction score
248
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.
 
Physics news on Phys.org
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."
 

Attachments

Similar threads

  • · Replies 9 ·
Replies
9
Views
9K
  • · Replies 4 ·
Replies
4
Views
1K
  • · Replies 11 ·
Replies
11
Views
5K
  • · Replies 12 ·
Replies
12
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
Replies
8
Views
2K
Replies
17
Views
3K
  • · Replies 7 ·
Replies
7
Views
5K
Replies
20
Views
4K