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Homework Help: Help: Gordon decompositon of the current

  1. May 14, 2012 #1

    A spinless electron can interact with [itex]A^\mu[/itex] only via its charge; the coupling is proportional to [itex](p_{f} + p_{i})^{\mu}[/itex]. An electron with spin, on the other hand, can also interact with the magnetic field via its magnetic moment. This coupling involves the factor [itex]i\sigma^{\mu\nu}(p_{f} - p_{i})[/itex]The relation between the Dirac current and the Klein-Gordon current can be studied as
    follows: Define the antisymmetric sigma tensor as:

    [tex]i\sigma^{\mu\nu} = \frac{i}{2} (\gamma^{\mu}\gamma^{\nu} - \gamma^{\nu}\gamma^{\mu})[/tex]

    And the Gordon decomposition of the Dirac current can be made:

    [tex]\bar u_{f}\gamma^{\mu}u_{i} = \frac{1}{2m} \bar u_{f} [(p_{f} + p_{i})^{\mu} + i\sigma^{\mu\nu} (p_{f} - p_{i})_{\nu} ] u_{i}[/tex]

    To identify the magnetic moment interaction [itex] (-{\mu} . B)[/itex] it suffices to show that:

    [tex] \int[-\frac{e}{2m} \bar \psi_{f} i\sigma_{\mu\nu} (p_{f} - p_{i})^{\nu}\psi_{i}] A^{\mu} d^{3}x = \int\psi^{f*}_{A}(\frac{e}{2m}{\sigma} . B) \psi^{i}_{A} d^{3}x[/tex]

    Can someone please tell me how I can desmostrate this ecuation?

    P.S.: Exercise 6.2 of "Quarks and Leptons. An Introductory Course in Modern Particle Physics - F.Halzem,A.Martin"
  2. jcsd
  3. May 16, 2012 #2


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    elmerx25, All three of your posts have been asking for solutions to homework problems. You need to take a look at the Forum Rules. In particular where it says, "If you are seeking help with a homework/coursework/textbook related issue please post your problem in the appropriate forum in our Homework & Coursework Questions area."
  4. May 12, 2014 #3
    How can I send this post to the homework/coursework/textbook area?
  5. May 12, 2014 #4


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    Staff: Mentor

    I moved the thread.

    What did you find out so far?
  6. May 12, 2014 #5

    I have had a sabbatical year without physics. Now I try to continue with my study.
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