A Dirac field can be written as two Weyl fields

[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?

 

[Evalesco]

No, the interpretation is correct. The Dirac field can be written as two Weyl fields stacked on top of each other, and when you use the projection operator $P_L$ on the Dirac field, only the particle field will remain. However, when you write the Dirac field in terms of creation and annihilation operators, some of the antiparticle field will be involved since the projection operator $P_L$ will not necessarily yield zero when applied to the antiparticle field.

 

[Entropia]

I would say that the interpretation of \psi as a particle and \zeta as an antiparticle is not wrong, but it is a simplified view of the Dirac field. The Dirac field can indeed be written as two Weyl fields stacked on top of each other, with \psi representing the particle and \zeta representing the antiparticle. However, the Dirac field also contains both particle and antiparticle components, and they cannot be completely separated in all cases.

When we project the Dirac field with P_L, we are essentially selecting only the left-handed components of the field. This means that only the \psi part of the field will remain, while the \zeta part will be set to zero. However, when we write the Dirac field in terms of creation and annihilation operators, both particle and antiparticle components are involved in the summation. This is because the creation and annihilation operators operate on both particle and antiparticle states.

Therefore, it is not incorrect to interpret \psi as a particle and \zeta as an antiparticle, but it is important to understand that the Dirac field contains both components and they cannot always be completely separated. This is a fundamental aspect of quantum field theory and it is important to consider both particle and antiparticle components in our calculations and interpretations.