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A Dirac field can be written as two Weyl fields

  1. Oct 22, 2009 #1
    A Dirac field can be written as two Weyl fields stacked on top of each other: [tex] \Psi= \left( \begin{array}{cc} \psi \\ \zeta^{\dagger} \end{array}\right) [/tex], where the particle field is [tex] \psi[/tex] and the antiparticle field is [tex]\zeta[/tex].

    So a term like [tex]P_L\Psi=.5(1-\gamma^5)\Psi=\left( \begin{array}{cc} \psi \\ 0 \end{array}\right) [/tex] should only involve the particle and not the antiparticle?

    However, writing [tex]\Psi=\Sigma_s \int d^3p \mbox{ } [b_s(p)u_s(p)e^{ipx}+d_{s}^{\dagger}(p)v_s(p)e^{-ipx}] [/tex], some of the antiparticle gets involved when projecting it with [tex]P_L [/tex] since [tex]P_L v_s(p) [/tex] is not necessarily zero?

    Is the interpretation that [tex] \psi [/tex] is a particle, and [tex] \zeta [/tex] is an antiparticle, wrong then?
     
  2. jcsd
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