Why do the anticommutation relations imply Fermi–Dirac statistics (Pauli Exclusion Principle) for the field quanta?

Question

I was reading the following article Fermion FIelds and discovered the following passage not fully explained to me :

It is these anticommutation relations that imply Fermi–Dirac statistics for the field quanta. They also result in the Pauli exclusion principle: two fermionic particles cannot occupy the same state at the same time.

What is the proof that the anticommutation relations of the Fermion Field gives rise to the Pauli Exclusion Principle?

Answer 1

Keeping it simple, let's asume that $\psi(a)$ creates a particle in the state $a$ (i.e., characterized by some collection of quantum numbers that we call $a$),

$$ \psi(a)|0\rangle=|a\rangle .$$

and $\psi(b)$ does the same for $b$. We can create a state with two particles:

$$ \psi(b)\psi(a)|0\rangle = \psi(b)|a\rangle = |a;b\rangle $$

$$ \psi(a)\psi(b)|0\rangle = \psi(a)|b\rangle = |b;a\rangle $$

Since $\psi$ is anti-commutative,

$$ \psi(a)\psi(b) + \psi(b)\psi(a) = 0 .$$

So,

$$ |a;b\rangle = -|b;a\rangle, $$

that is, the state is antisymmetric under particle change, it has fermionic statistic.

In particular, if $b=a$,

$$ \psi(a)\psi(a) + \psi(a)\psi(a) = 2\psi(a)\psi(a) = 0$$

so,

$$ |a;a\rangle=0 .$$

This is Pauli's Exclusion Principle.