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Dual of Maxwell tensor as gauge curvature

  1. Apr 26, 2008 #1
  2. jcsd
  3. Apr 26, 2008 #2

    jtbell

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    Staff: Mentor

    Hi, did you know that we have inline LaTeX capabilities here? Enter your code between [ tex ] and [ \tex ] tags for standalone equations, or [ itex ] and [ /itex ] for inline equations. In either case, remove the extra spaces between the brackets. I had to add them here so the tags would show up as plain text instead of being treated as actual tags.

    You can also try clicking on the sigma icon in the message-composition window, although I suspect you'll find it easier to write the code directly, because you apparently did it for that PDF file.

    This will make it more convenient for readers than downloading a PDF file. Also, they can quote your equations if necessary which makes it easier to reply.
     
  4. Apr 26, 2008 #3
    Like so:

    And so on.
     
    Last edited: Apr 26, 2008
  5. Apr 26, 2008 #4
    I know, but I use the built-in Latex for simple things, not for long complicated posts.
    Laura
     
  6. Apr 26, 2008 #5
    Part 2

    Well there's the whole question anyway, parts 1 & 2. I don't know how hard pdf formatting of latex is?
     
    Last edited: Apr 26, 2008
  7. Apr 27, 2008 #6
    So the book's statement is a statement about electromagnetic duality. That the source free or empty space maxwell's equations remain unchanged when you exchange [tex]E[/tex] and [tex]B[/tex], or equivalently, [tex]F[/tex] and [tex]*F[/tex]. Note that [tex]*F[/tex] is only a 2-form in 4 dimensions, so the statement is only true in 4. In this case, [tex]*F[/tex] is the curvature of a connection corresponding to the electromagnetic dual (electric monopoles become magnetic monopoles, etc.)

    There's some interest in this for Yang-Mills theories and in other contexts. Anyway I think the point is that it has to be source free. Or, approximately source free (say, in some non-simply connected region). See for example the Dirac monopole.
     
  8. Apr 27, 2008 #7

    jtbell

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    Staff: Mentor

    Can't you simply paste the LaTeX code that you have to prepare anyway, in between the tags? Or does that not work for some reason?
     
  9. Apr 27, 2008 #8
    Yes but the book didn't say anything about the field being source-free. Is my guess right, that in the application where the dual connection would be used, I guess on wavefunctions, that you wouldn't be including the local sources in the Maxwell equations, because you are looking at things at too small a scale to consider charge density, current etc.? He might be leaving this unsaid - but it seems like an obligatory aspect of considering the dual of the Maxwell tensor as a curvature. I spent a lot of time thinking about whether there'd be some way to get around that restriction, some other way of defining the covariant derivative that would give the dual as the curvature. But nothing really came up.
    Laura
     
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