Editing Spin–statistics theorem (section)

===Exchange symmetry or permutation symmetry===

[[Boson]]s are particles whose wavefunction is symmetric under such an exchange or permutation, so if we swap the particles, the wavefunction does not change. [[Fermion]]s are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the [[Pauli exclusion principle]]: two identical fermions cannot occupy the same state. This rule does not hold for bosons.

In quantum field theory, a state or a wavefunction is described by [[field operator]]s operating on some basic state called the [[Vacuum state|''vacuum'']]. In order for the operators to project out the symmetric or antisymmetric component of the creating wavefunction, they must have the appropriate commutation law. The operator

:<math>
\iint \psi(x,y) \phi(x)\phi(y)\,dx\,dy
</math>

(with <math>\phi</math> an operator and <math>\psi(x,y)</math> a numerical function with complex values) creates a two-particle state with wavefunction <math>\psi(x,y)</math>, and depending on the commutation properties of the fields, either only the antisymmetric parts or the symmetric parts matter.

Let us assume that <math>x \ne y</math> and the two operators take place at the same time; more generally, they may have [[spacelike]] separation, as is explained hereafter.

If the fields '''commute''', meaning that the following holds:

:<math>\phi(x)\phi(y)=\phi(y)\phi(x),</math>

then only the symmetric part of <math>\psi</math> contributes, so that <math>\psi(x,y) = \psi(y,x)</math>, and the field will create bosonic particles.

On the other hand, if the fields '''anti-commute''', meaning that <math>\phi</math> has the property that

:<math>\phi(x)\phi(y)=-\phi(y)\phi(x),</math>

then only the antisymmetric part of <math>\psi</math> contributes, so that <math>\psi(x,y) = -\psi(y,x)</math>, and the particles will be fermionic.