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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?

Qmechanic
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user267839
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1 Answers1

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My understanding is that the anticommutator relationship shows that the many-particle wave function is asymmetric relative to particle exchange, which is related to the Pauli exclusion principle and also half-integer spin, which, in turn, defines Fermi-Dirac statistic, where the maximal value can't exceed one, unlike in the case of the Bose-Einstein statistics.

The spin statistic theorem talks about this, you can find it here.

To answer your question, they are compatible in the sense that two particles can't occupy the same quantum state.

freude
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