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Homework Help: Dirac gamma matrices

  1. Jan 5, 2012 #1
    Hi!
    I can define
    [itex]\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3[/itex]
    I know that the four gamma matrices [itex]\gamma^i\:\:,\;i=0...3[/itex] are invariant under a Lorentz transformation. So I can say that also [itex]\gamma ^5[/itex] is invariant, because it is a product of invariant matrices.
    But this equality holds:
    [tex]\gamma ^5=\frac{i}{4!}\epsilon_{\mu\nu\rho\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}[/tex]
    and this expression is not invariant!!
    So, is [itex]\gamma^5[/itex] invariant or isn't it?
     
  2. jcsd
  3. Jan 5, 2012 #2

    fzero

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    [itex]\gamma^\mu[/itex] transforms like a 4-vector.
     
  4. Jan 6, 2012 #3
    But [itex]\gamma^5[/itex] transforms like a pseudo-scalar because of [itex]\epsilon_{\mu\nu\rho\sigma}[/itex]
     
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