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Dirac Lagrangian not invariant under rotations?

  1. Jun 24, 2006 #1
    First, I need to be able to do equations in my post but it has been a long time since I posted here. Someone please point me to a resource that gives the how-to.

    If you make a infinitesimal rotation of the free-field Lagrangian for the Dirac equation, you get an extra term because the Dirac gamma matrices and the rotation generator do not commute. I'll show you when I can. So what do we make of this?

    There is more to the question, and I know it has something to do with the Pauli-Lubanski pseudovector operator. Anything you can tell me about the Pauli-Lubanski pseudovector would also be appreciated.

  2. jcsd
  3. Jun 24, 2006 #2
    The solution is to use the full rotation, rather than the infinitesimal one, which isn't a full rotation.

    [tex]\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi [/tex]
    [tex]\psi\rightarrow\Lambda_{\frac{1}{2}}\psi[/tex] (Ignoring the transformation of the coordinate dependence of the Dirac spinor)
    [tex]\mathcal{L}\rightarrow\bar{\psi}\Lambda_{\frac{1}{2}}^{-1}\left(i\gamma^{\mu '}\Lambda^{\mu}_{\mu '}\partial_{\mu}-m\right)\Lambda_{\frac{1}{2}}\psi[/tex]

    Which is exactly the same as the original Lagrangian density because

    [tex]\Lambda_{\frac{1}{2}}^{-1}\gamma^{\mu '}\Lambda_{\frac{1}{2}}=\Lambda^{\mu '}_{\nu}\gamma^{\nu}[/tex]
    Last edited: Jun 24, 2006
  4. Jun 24, 2006 #3
    Thanks, Perturbation. I'll get back as soon as I can figure out doing equations with LaTeX.
    Last edited: Jun 25, 2006
  5. Jun 25, 2006 #4
    I'll go ahead and be verbose. At least I get to have fun with the LaTeX. Maybe someone here can explain where I am going wrong.

    The Lagrangian I am using is


    so that it is symmetric with respect to psi and psi-bar. Just seemed to sit well with me.

    In the particular representation in which the gamma matrices take the form


    a rotation of the four-spinor around the z-axis is given by


    For infinitesimal rotations this is






    The conjugate is


    Here we go.




    and [tex][\tau_{z},\gamma^{\mu}]\neq0[/tex]
    Last edited: Jun 25, 2006
  6. Jun 25, 2006 #5
    Also, I ignored "the transformation of the coordinate dependence of the Dirac spinor" above just as Perturbation did in his post. The result of the coordinate part of the transformation results in a term in the transformed Lagrangian which is the expected scalar transformation. The remaining [tex]\delta\mathcal{L}[/tex] in my calculation above should vanish if [tex]\mathcal{L}[/tex] is a scalar under rotations.

    Looking at Perturbation's post, apparently I have neglected the effect of the rotation on the gamma matrices. I'll have to think on that for a while.

    Last edited: Jun 25, 2006
  7. Jun 25, 2006 #6
    The gamma matrices don't change, you just defined what they are. But what happens with [tex]\partial_\mu[/tex] under a Lorentz transformation?
  8. Jun 26, 2006 #7
    Thanks. I have to spend some more time thinking about rotations. The question of this thread came out trying to understand spin as discussed in my next thread: What is spin?
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