1. Limited time only! Sign up for a free 30min personal 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!

Covariant Bilinears: Fierz Expansion of Dirac gamma matrices products

  1. Apr 4, 2012 #1
    1. The problem statement, all variables and given/known data

    So my question is related somehow to the Fierz Identities.

    I'm taking a course on QFT. My teacher explained in class that instead of using the traces method one could use another, more intuitive, method. He said that we could use the fact that if we garante that we have the same number os indexes at each side of the expression and only use the base matrices (scalar, vector, pseudoscalar, tensor and axial) one would get the same results as using the traces method.

    He then gave an example and advised for us to try with some other example.

    I then tried to write [itex]\gamma_5\gamma^{\alpha}\gamma^{\mu}[/itex] using that method.
    2. Relevant equations

    [itex]\sigma^{\alpha\mu}=\frac{i}{2}[\gamma^{\alpha},\gamma^{\mu}][/itex]
    [itex]\eta^{\alpha \mu}[/itex] is the minkowski metric and [itex]I_{4}[/itex] is the identity matrix in 4-spacetime.

    3. The attempt at a solution

    The attempt of a solution goes as:
    [itex]\gamma_{5}\gamma^{\alpha}\gamma^{\mu}=
    a*\eta^{\alpha\mu}I_{4}+b*\eta^{\alpha\mu}\gamma_{5}
    +c*\sigma^{\alpha\mu}[/itex]

    Is this correct?

    If I contract [itex]\gamma_5\gamma^\alpha\gamma^\mu[/itex] with [itex]\eta_{\alpha \mu}[/itex] I get a=0 and b=1. But if the above expression is correct, how can I get [itex]c[/itex]?

    Please, somebody help me.
     
  2. jcsd
  3. Apr 4, 2012 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    From the definition of [itex]\sigma^{\alpha\mu}[/itex] and the anticommutation relations, you can see that there must be a term [itex]\gamma_5\sigma^{\alpha\mu}[/itex] in the expansion of [itex]\gamma_5\gamma^\alpha\gamma^\mu[/itex]. I guess we'd call this a pseudotensor term.
     
  4. Apr 5, 2012 #3
    Thanks for the prompt reply,

    I thought so, but I have read that such term would not be independent of the other 16 matrices.

    [itex]\sigma^{\alpha \mu}\gamma_{5}=\frac{i}{2}\epsilon^{\alpha \mu \nu \beta}\sigma_{\nu \beta}[/itex]

    ,where [itex]\epsilon^{\alpha \mu \nu \beta}[/itex] is the Levi-Civita symbol

    So the above expression for [itex]\gamma_{5}\gamma^{\alpha} \gamma^{\mu}[/itex] is incomplete and I should add a extra term
    [itex]d* \epsilon^{\alpha \mu \nu \beta}\sigma_{\nu \beta}[/itex] ?
     
    Last edited: Apr 5, 2012
  5. Apr 5, 2012 #4
    I got it now.

    One as to add such term and then get the correct answer. For other readers with a similar doubt I got:

    [itex]\gamma_{5}\gamma^{\alpha}\gamma^{\mu}=\eta^{\alpha\mu}I_{4}+\frac{1}{2}\epsilon^{\alpha\mu\nu\beta}σ_{\nu\beta}[/itex]

    Which I believe, it's the correct answer.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Covariant Bilinears: Fierz Expansion of Dirac gamma matrices products
  1. Dirac gamma matrices (Replies: 2)

Loading...