Identical quantum particles cannot be labeled by physically observable identities. Exchanging two identical particles must therefore leave all measurement probabilities unchanged. The many-particle state may change only by a phase under exchange; in ordinary three-dimensional quantum mechanics this leads to two possibilities.

For identical bosons, the state is symmetric: $$\Psi(1,2)=+\Psi(2,1).$$ For identical fermions, it is antisymmetric: $$\Psi(1,2)=-\Psi(2,1).$$

For two particles in one-particle states $a$ and $b$, the properly symmetrized states are $$\Psi_B=\frac{1}{\sqrt2}[\phi_a(1)\phi_b(2)+\phi_b(1)\phi_a(2)],$$ $$\Psi_F=\frac{1}{\sqrt2}[\phi_a(1)\phi_b(2)-\phi_b(1)\phi_a(2)].$$

If two fermions try to occupy the same one-particle state, set $a=b$. The antisymmetric combination becomes $$\Psi_F=0.$$ Thus two identical fermions cannot occupy exactly the same quantum state: this is the Pauli exclusion principle.

Exchange symmetry produces physical effects even without a classical force between particles. It determines electronic shell structure, quantum statistics, degeneracy pressure and many properties of matter. Spin is tied to particle statistics by relativistic quantum field theory: particles with half-integer spin are fermions, while integer-spin particles are bosons.