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Wedge products and Hodge dual

  1. Jul 27, 2013 #1
    Hi,

    I'm trying to get my head around the Hodge dual and how it exactly works. In the book "Gauge Fields, Knots and Gravity" by John Baez and Javier P. Muniain they define:

    \begin{equation}
    \omega \wedge * \mu = \langle \omega , \mu \rangle \mathrm{vol}
    \end{equation}

    for two p-forms. This implies that:

    \begin{equation}
    \omega \wedge * \mu = * \mu \wedge \omega
    \end{equation}

    Therefore, if we consider a vector space with basis dx, dy, dz, and

    \begin{equation}
    \omega = \omega_x \mathrm{d}x
    \end{equation}

    \begin{equation}
    *\mu = \mu_y \mathrm{d} y
    \end{equation}

    Then the definition by Baez and Muniain yields:

    \begin{equation}
    \omega_x \mu_y \mathrm{d}x \wedge \mathrm{d} y = \mu_y \omega_x \mathrm{d} y \wedge \mathrm{d}x
    \end{equation}

    However, if I would try to calculate the above equation using 'anti-commuting' property of the wedge product, then I get:

    \begin{equation}
    \omega \wedge * \mu = \omega_x \mu_y \mathrm{d}x \wedge \mathrm{d} y = - \omega_x \mu_y \mathrm{d}y \wedge \mathrm{d} x \neq * \mu \wedge \omega
    \end{equation}

    So clearly, I am going wrong somewhere, but I can't see where I'm going wrong.
     
  2. jcsd
  3. Jul 27, 2013 #2

    martinbn

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    [itex] \omega[/itex] and [itex] \mu[/itex] have to be p-forms. In your example one is 1-form, the other is a 2-form.
     
  4. Jul 27, 2013 #3
    Thanks for your reply, I will think about it over the next couple of days. I've the feeling that in my example they are both 1-forms so they should satisfy that equation.
     
  5. Jul 27, 2013 #4

    atyy

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    Science Advisor

    If you interchange ##\omega## and ##\mu##, shouldn't the second line be ##\omega \wedge * \mu = \mu \wedge * \omega## ?

    Winitzki gives something like that in the first equation at the top of the right column on p4 of https://sites.google.com/site/winitzki/linalg.
     
    Last edited: Jul 27, 2013
  6. Jul 27, 2013 #5

    micromass

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    You have made ##\ast \eta## a ##1##-form. Thus ##\eta## is then a ##n-1##-form.
    However, you also made ##\omega## a ##1##-form. This is incompatible.
    Both ##\omega## and ##\eta## should be ##1##-forms.
     
  7. Jul 27, 2013 #6
    Ok, thank micromass, that makes sense.
     
  8. Jul 27, 2013 #7
    atty, yeah but in other sources I have also seen

    \begin{equation}
    (*\omega) \wedge \mu = \langle \omega , \mu \rangle \mathrm{vol}
    \end{equation}

    so I think they are equivalent (although it seems that everybody is using different conventions so I might be wrong). However, what micromass said made sense, so I think that is where I went wrong.
     
  9. Jul 27, 2013 #8
    Are you looking for a definite, mathematically rigorous way of thinking of it, or are you looking for intuition? Intuition is often good for wrapping your head around something, but I get the feeling that you aren't looking for that.
     
  10. Jul 28, 2013 #9
    Yeah, it would be nice to have a more rigorous way of thinking about it. If you have any good sources where I can learn this stuff then that would be great (I thought Sean Carroll's book was quite disappointing regarding differential forms, so now I have the aforementioned book from which I try to learn it). Thanks

    Edit: FYI I study physics and am mainly interested in their application in gauge theory and general relativity.
     
  11. Jul 28, 2013 #10

    WannabeNewton

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    I would recommend Nakahara for a physics oriented viewpoint (it is primarily gauge theory) as well as Frankel. I personally don't know of any GR text that treats differential forms in a completely modern (i.e. index free) way. Wald's GR text has a nice appendix on differential forms but it is index based hence rather classical in nature.

    If you want something on the pure math side, Spivak and Kobayashi are the reigning kings.

    Links: https://www.amazon.com/books/dp/0471157333
    https://www.amazon.com/Comprehensive-Introduction-Differential-Geometry-Edition/dp/0914098705
    https://www.amazon.com/Geometry-Topology-Physics-Edition-Graduate/dp/0750306068
    https://www.amazon.com/The-Geometry-Physics-An-Introduction/dp/1107602602
     
    Last edited by a moderator: May 6, 2017
  12. Jul 29, 2013 #11
    Thanks WannabeNewton for your recommendations!
     
  13. Aug 14, 2013 #12
    Another very good book is by Marian Fecko "Differential Geometry and Lie Groups for Physicists". In particular the Hodge dual is discussed there.
     
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