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Dirac spinor question

  1. Jul 24, 2011 #1
    In Qed they replace the current vector [tex]J^{\alpha}[/tex] by [tex]ie\overline{\Psi}\gamma^{\alpha}\Psi[/tex]. I don't understand how this is done. I understand that
    [tex]J^{A\dot{A}}=J^{\alpha}{\sigma^{A\dot{A}}_\alpha}[/tex] but if [tex]J^{A\dot{A}}[/tex] is a rank two matrix then [tex]J^{A\dot{A}}=\psi^{A}\psi^{*\dot{A}}+\phi^{A}\phi^{*\dot{A}}[/tex]. So shouldn't [tex]J^{\alpha}[/tex] be written as something like
    [tex]ie(\bar{\Psi}\gamma^{\alpha}\Psi +\bar{\Phi}\gamma^{\alpha}\Phi)[/tex]?
     
  2. jcsd
  3. Jul 24, 2011 #2
    The two 2-component Weyl spinors (and their complex conjugates) need to be combined into a single Dirac spinor (and its Dirac conjugate). And the Dirac matrices are constructed as an off-block-diagonal combination of the Pauli matrices - the http://en.wikipedia.org/wiki/Gamma_matrices#Weyl_basis".

    This is explained in almost any textbook on supersymmetry (where the 2-component formalism is very common) or, e.g., in http://physics.stackexchange.com/questions/6157/list-of-freely-available-physics-books/6167#6167"
     
    Last edited by a moderator: Apr 26, 2017
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