# A Dirac field can be written as two Weyl fields

1. Oct 22, 2009

### RedX

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

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

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

Is the interpretation that $$\psi$$ is a particle, and $$\zeta$$ is an antiparticle, wrong then?