Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question on Fierz identity

  1. Mar 27, 2014 #1
    Hi everyone, I have a doubt on Fierz identities. If we define the following quantities: [itex] S=1,\; V=\gamma_\mu,\; T=\sigma_{\mu\nu},\; A=\gamma_\mu\gamma_5,\;P=\gamma_5[/itex], then we have the identity:

    (\Gamma_i)_{\alpha\beta}(\Gamma_i)_{\gamma\xi}=\sum_j F_{ij}(\Gamma_j)_{\alpha\xi}(\Gamma_j)_{\gamma\beta},
    where [itex]\Gamma_i[/itex] are the matrices define before. Moreover:
    2 & 2 & 1 & -2 & -2 \\
    8&-4&0&-4&-8 \\
    24&0&-4&0&24 \\
    -8&-4&0&-4&8 \\
    Therefore, if we take the VV+AA combination it turns out that [itex]VV+AA=-VV-AA[/itex] with exchanged indices.

    However I usually read the Fierz transformation to be:
    (\psi_1\Gamma P_L\psi_2)(\psi_3\Gamma P_L\psi_4)=(\psi_1\Gamma P_L\psi_4)(\psi_3\Gamma P_L\psi_2).

    Without any minus sign. Does anyone knows why?
  2. jcsd
  3. Mar 27, 2014 #2


    User Avatar
    Science Advisor

    Because the ψ's anticommute? I think it matters whether you just give the relation between matrices, as Wikipedia does, or include the ψ's. Both of these references give the table for Fij including the ψ's, with the opposite sign.
  4. Mar 27, 2014 #3
    I think you are right. Once we write the identity for the matrices then we need to switch the two field and this should give an extra minus sign.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Question Fierz identity Date
I X-ray dose question Wednesday at 3:54 PM
I X-ray tube to gamma tube question Tuesday at 3:51 AM
A Question about a cross section from PDG Mar 8, 2018
A Question about the matrix of a pseudoscalar meson Jan 20, 2018
A Fierz identity Feb 24, 2017