- #1

The Tortoise-Man

- 95

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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.

Very roughly the story with second quantization is one start with a field ( as honest "function" in variables ##x_1, x_2,...##, so before running the quantization machinery) ##\psi(x_1,x_2,..., x_n)## and - depending on how the sign of the field changes ( symmetrically or antisym) when we permute the arguments ##x_i## - the field is called bosonic or fermionic, more less "by definition".

And my question is how this fits in the picture that if we have a fermionic ( resp bosonic) field given, then the "only right" bracket relation in the quatization procedure for this field must be the be anti-commutating (resp commutating) one?

In other words, what would run into troubles if we not set the bracket relation exactly that way? What does that exactly mean ( as definition) for an operator to be "compatible" with certain given statistic/ distribution?