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The commutation relation for neutral Klein Gordan field is

$$[\phi(x,t),\pi(x',t)]=i\delta^3(x-x')$$ with all other commutators zero;

The commutation relation for charged Klein Gordan field is $$[\phi(x,t),\pi(x',t)]=[\phi^\dagger(x,t),\pi^\dagger(x',t)]=i\delta^3(x-x')$$ with all other commutation zero;

but the anticommutation relation for the Dirac field is

$$\{\psi_a(x,t),\psi_b^\dagger(x',t)\}=\delta_{ab}\delta^3(x-x').$$

My question is, why is the anticommutation relation for the Dirac field between $\psi_a$ and $\psi^\dagger_b$, not between $\psi_a$ and $\pi_b$ as one would expect looking at the Klein Gordan field?

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