Identical particles require symmetric or antisymmetric many-body states under exchange. Bosons occupy symmetric states and can macroscopically condense; fermions occupy antisymmetric states and obey the Pauli exclusion principle.

In second quantization, the state is built in Fock space and creation/annihilation operators add or remove particles. For bosons, $[\hat a,\hat a^\dagger]=1$; for fermions, $\{\hat c,\hat c^\dagger\}=1$.

Bose-Einstein and Fermi-Dirac statistics emerge from the exchange symmetry of the many-body wavefunction combined with indistinguishability. In relativistic settings, the spin-statistics theorem provides the deep connection between spin and statistics.