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In the quantum thory of a Dirac field we require that parity is implemented by an operator $P$ such that:

$P^{-1}=P^\dagger$

$P^{-1}=P$

$P a_{r}(\vec k) P^{-1}=a_{r}(-\vec{k})$

$P b_{r}(\vec k) P^{-1}=-b_{r}(-\vec{k})$

where $a_{r}(\vec k)$ and $b_{r}(\vec k)$ are distruction operator of particles and antiparticles respectively, thus we get the nice result:

$P \Psi (t, \vec x) P^{-1} = \gamma^{0} \Psi (t, -\vec x) $

But shouldn't we prove that the 4 conditions above are consistent? If so, I haven't found in textbooks such a proof or an explicit expression for the $P$ operator satisfying them. Any idea or reference? Thanks

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