I'm reading the Wikipedia article about Fermionic field and have some troubles to understand the meaning following phrase:
We impose an anticommutator relation (as opposed to a commutation relation as we do for the bosonic field) in order to make the operators compatible with Fermi–Dirac statistics.
I understand that the (anti)commutator relations for operatorys are posed as algebraic conditions in the operator algebra with "composition" as binary relation there. So up to now that's a purely "algebraic data".
But what does it mean that the operators should be compatible with Fermi–Dirac statistics? Compatible in which sense? I not understand in what is the concrete connection between the obtained the operators obtained from second quantization procedure and the Fermi–Dirac statistic, in mathematical terms a distribution. How these concepts fit together?