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Rotating Dirac particle current

  1. Nov 1, 2006 #1
    There is a step that bothers me in my book (Ryder) on QFT and I can't seem to figure it out. It concerns the (spatial) rotation of the spatial part of the Dirac four current:

    [tex]\bar{\psi} \gamma \psi [/tex]

    The crucial step here is

    [tex]\frac{1}{4}(\vec{\sigma} \cdot \vec{\theta}) \vec{\sigma}(\vec{\sigma} \cdot \vec{\theta}) = -\vec{\theta} \times \vec{ \sigma} [/tex]

    With [itex]\vec{\theta}[/tex] the (infinitesimal) rotation and [itex]\vec{\sigma}[/tex] the vector consiting of the three Pauli matrices.

    I tried writing it in tensor notation and using the commutation reltaions as in the supplied tip, but I can't figure it out... Can anybody show me this?
  2. jcsd
  3. Nov 1, 2006 #2
    What are you trying to prove, invariance of the four current under rotation? This is not that difficult.
  4. Nov 3, 2006 #3
    Well, thx :wink: Actually, that the spatial part of the Dirac four-current bahves as a vector under rotations. BUt I finally figured it out myself at last!
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