First of all, the Standard Model doesn't treat bosonic fields as classical. They're quantum mechanical i.e. non-classical, they're just not anticommuting or Grassmann-odd. Second, a consistent theory just requires the relationship between spin and statistics, see e.g. the

http://en.wikipedia.org/wiki/Spin-statistics_theorem

Combining integer spin with Fermi statistics leads to ghost or energy or the norm that isn't positively definite, and vice versa (half-integer spin with Bose statistics). It was proved by Pauli.

However, your very question didn't actually talk about the integer vs half-integer spin at all. It was talking about the relationship between fermions and anticommuting fields. This is almost a tautology. A fermion is a particle whose wave function for many particles is antisymmetric, $\psi(x_1,x_2)=-\psi(x_2,x_1)$ etc., so the fields that create these particles must be anticommuting, $a^\dagger(x_1) a^\dagger(x_2)=-a^\dagger(x_2)a^\dagger(x_1)$. The multiparticle state in QFT is written as
$$ |\text{2 fermions}\rangle = \int d^3 x_1 d^3 x_2\, \psi(x_1,x_2) a^\dagger(x_1)a^\dagger(x_2)|0\rangle $$
Because the wave function $\psi$ is antisymmetric, only the antisymmetric combination $a^\dagger(x_1)a^\dagger(x_2) - a^\dagger(x_2)a^\dagger(x_1)$ contributes to the state, and in fact, only this combination is nonzero. The sum – the anticommutator – vanishes. That's why the antisymmetry of $\psi$ is "automatic": if there were a non-antisymmetric part of $\psi$, it would vanish in the integral above because the product of the creation operators is antisymmetric.

The same for bosons and "commuting", without the minus sign.

The answer to your "why" question is that your statement is really a tautology, pretty much a definition of bosons and fermions, up to the possibly confusing comments about "one antisymmetry" implying the "other antisymmetry" above. Of course, you could also ask why one uses commuting or anti-commuting fields to describe particles at all. Well, Nature just works in this way. Quantum fields naturally reduce to multi-body quantum mechanics with the automatic symmetry or antisymmetry – and they may give rise to automatically Lorentz-invariant theories, too (something that would be hard in the "non-relativistically styled" multiparticle quantum mechanics).

This post imported from StackExchange Physics at 2014-04-11 15:50 (UCT), posted by SE-user Luboš Motl