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Fierz transformations

  1. Oct 28, 2008 #1
    If I have a product like

    [tex]
    \bar\ psi\gamma^\mu\psi\bar\psi\gamma_\mu\psi
    [tex]

    how can i rearrange with Fierz transformations?
     
  2. jcsd
  3. Oct 29, 2008 #2

    samalkhaiat

    User Avatar
    Science Advisor

    I assume, you meant to write

    [tex]
    V(4,2;3,1) = \bar{\psi}(4) \gamma_{a} \psi(2) \bar{\psi}(3) \gamma^{a} \psi(1)
    [/tex]

    The Fietz transformation is

    [tex]V(4,2;3,1) = \left( - S + \frac{1}{2} V - \frac{1}{2} A + P \right) (4,1;3,2)[/tex]

    where

    [tex]S(4,1;3,2) = \bar{\psi}(4) \psi(1) \bar{\psi}(3) \psi(2)[/tex]

    [tex]
    A(4,1;3,2) = \bar{\psi}(4) \gamma_{5} \gamma_{a} \psi(1) \bar{\psi}(3) \gamma^{a}\gamma_{5} \psi(2)
    [/tex]

    [tex]
    P(4,1;3,2) = \bar{\psi}(4) \gamma_{5} \psi(1) \bar{\psi}(3) \gamma_{5} \psi(2)
    [/tex]

    and similar one for V(4,1;3,2). See page 160-162 in Itzykson & Zuber; QFT.

    regards

    sam
     
    Last edited: Oct 29, 2008
  4. Oct 30, 2008 #3
    Thanks a lot, Sam. This is what I was looking for.

    Jon
     
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