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A question about the Exterior Product in Yang-Mills theory

  1. Aug 9, 2010 #1
    Right, so in Yang-Mills theory, the vector potential is modified from:

    [tex]F = dA[/tex]​


    [tex]F = dA + A\wedge A[/tex]​

    However, it is my understanding that the exterior/wedge product is anticommutitive, so that for a given exterior algebra over a vector space, V:

    [tex]\omega \wedge \omega = 0, \forall \omega \epsilon \Lambda(V) [/tex]​

    Why then is the second term in the curvature, F, not non-zero? I assume I'm missing something, could someone fill me in?

    (Sorry, this probably fits better in Topology & Geometry section, but the question technically is a question about multilinear algebra)
  2. jcsd
  3. Aug 10, 2010 #2

    George Jones

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    Argh! I was almost done a long latex post, but I accidentally closed the window.

    [itex]A[/itex] is a Lie algebra-valued one-form. not a "normal" one-form. Roughy, think of [itex]A[/itex] as a matrix of "normal" one-forms, and [itex]A \wedge A[/itex] as matrix multiplication, where exterior products of matrix elements (normal one-forms) are used instead of multiplication of numbers.
  4. Aug 10, 2010 #3
    Thank you George Jones for your reply.

    Is there a particular name for the topic of Lie-algebra valued one-forms? (I assume it's actually a differentiable form, i.e. a section of the cotangent bundle). If there is, I would like to study it a little more.

    I'm assuming there's a little more to it than this. For instance, the Lie algebra here, is it the Lie algebra of the Lie group of diffeomorphisms/isometries on the differentiable manifold/Riemannian manifold (this is the only straight foreword way, that I can imagine, that they'd be matrices)? Or are we assuming the manifold for which our A-field is defined over is also has smooth group structure?
    Last edited: Aug 10, 2010
  5. Aug 10, 2010 #4

    George Jones

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    They are treated in The Geometry of Physics: An Introduction by Theodore Frankel, and in Geometry, Topology, and Physics by M. Nakahara.
    A Yang-Mills theory involves a internal (i.e., not spacetime) symmetry group G, and uses a structure called a principal G bundle. The field strength that you gave in the first post is the curvature (again, internal, not spacetime) of the principal G bundle.
  6. Aug 11, 2010 #5
    I own M. Nakahara's book. Is it contained in the section on the theory of connections on fiber bundles?
  7. Aug 11, 2010 #6

    George Jones

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    Nakahar covers this stuff briefly (maybe too briefly, I prefer Frankels's treament) in the section on curvature of fiber bundles.
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