1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Fierz Rearrengement Identity

  1. Apr 29, 2015 #1
    1. The problem statement, all variables and given/known data

    I have to prove the Fierz rearrengement identity for Weyl Fermions. Eq 2.20 in Martin's supersymmetry primer:

    [itex]\chi_\alpha(\xi\eta)=-\xi_\alpha(\eta\chi)-\eta_\alpha(\chi\xi)[/itex]

    2. Relevant equations
    We have that the antisimetric tensor raises and lowers indices.

    3. The attempt at a solution
    [itex]\chi_\alpha(\xi\eta)=-\xi_\alpha(\eta\chi)-\eta_\alpha(\chi\xi)=-\xi_\alpha(\eta\chi)-\eta_\alpha(\xi\chi)[/itex]
    [itex]\chi_\alpha(\xi\eta)=-\xi_\alpha(\eta^\beta\chi_\beta)-\eta_\alpha(\xi^\beta\chi_\beta)[/itex]
    [itex]\chi_\alpha(\xi\eta)=-\chi_\beta(\xi_\alpha\eta^\beta+\eta_\alpha\xi^\beta)[/itex]

    Now I need to prove that [itex](\xi_\alpha\eta^\beta+\eta_\alpha\xi^\beta)=-\delta^\beta_\alpha(\xi\eta)[/itex]. Can anyone help me with this? Thanks a lot!
     
  2. jcsd
  3. May 1, 2015 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Generally the way to prove these formulae is to manipulate indices so that you can write an expression as a sum of terms involving just epsilons and deltas multiplying the same expression of fermions. For example:

    $$\begin{split} \xi_\alpha\eta^\beta+\eta_\alpha\xi^\beta +\delta^\beta_\alpha(\xi\eta) & = \epsilon_{\alpha\gamma} \xi^\gamma \eta^\beta + \epsilon_{\alpha\gamma} \eta^\gamma \xi^\beta + \delta^\beta_\alpha \epsilon_{\gamma\delta} \xi^\gamma\eta^\delta \\
    &= (\epsilon_{\alpha\gamma} \delta^\beta_\delta - \epsilon_{\alpha\delta} \delta^\beta_\gamma + \delta^\beta_\alpha \epsilon_{\gamma\delta} ) \xi^\gamma\eta^\delta.\end{split}$$

    We can then manipulate the coefficent to see that it vanishes due to the Jacobi identity

    $$ \epsilon_{\alpha\beta} \epsilon_{\gamma\delta} + \epsilon_{\alpha\gamma} \epsilon_{\delta\beta} + \epsilon_{\alpha\delta} \epsilon_{\beta\gamma} =0.$$

    We could have directly applied this method to the original Fierz formula.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Fierz Rearrengement Identity
  1. Commutator Identity (Replies: 2)

  2. Jacobi Identity (Replies: 3)

  3. QM: Identity (Replies: 1)

Loading...