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Chirality of Dirac adjoint

  1. Jul 11, 2015 #1
    I have a question about chirality.
    When a spinor [itex]\psi[/itex] have plus chirality, namely
    [tex]
    \gamma_5\psi=+\psi,
    [/tex]
    how can I write this condition for the Dirac adjoint [itex]\bar{\psi}=\psi^\dagger i\gamma^0[/itex]?

    Let me choose the signature as [itex]\eta_{\mu\nu}=\mathrm{diag}(-,+,+,+)[/itex] and define [itex]\gamma_5\equiv i\gamma^0\gamma^1\gamma^2\gamma^3[/itex]. Taking the Dirac adjoint of the left hand side of the above equation,
    [tex]
    (\gamma_5\psi)^\dagger i\gamma^0=\psi^\dagger\gamma^\dagger_5i\gamma^0\\
    =\psi^\dagger(-\gamma^0\gamma^0)\gamma_5i\gamma^0\\
    =\bar{\psi}\gamma^0\gamma^0\\
    =-\bar{\psi}\gamma_5,
    [/tex]
    so the chirality condition seems to be rewritten as
    [tex]
    \bar{\psi}\gamma_5=-\bar{\psi}.
    [/tex]

    However, for example, considering a quantity [itex]\bar{\psi}\gamma_5\psi[/itex], this result is inconsistent. Where is wrong in the above calculation?
     
  2. jcsd
  3. Jul 11, 2015 #2

    fzero

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    Science Advisor
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    Gold Member

    It's not inconsistent, since ##\bar{\psi}\gamma_5 \psi =0## if ##\psi## is chiral. Essentially you have already proved it, but you can also verify it in terms of the chiral components.
     
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