For identical particles, the many-body state must be symmetric for bosons and antisymmetric for fermions under exchange. This is not merely a labeling convention; it changes physical predictions such as interference and ground-state structure.

A fermionic $N$-particle wavefunction is built from a Slater determinant, guaranteeing antisymmetry. Bosonic states are symmetrized sums, and their combinatorics underlie phenomena such as Bose enhancement and condensation.

Introduce creation and annihilation operators $a_k^\dagger$ and $a_k$ satisfying commutation or anticommutation relations depending on statistics. This formalism compactly describes variable particle number and is indispensable in many-body theory.

Exchange symmetry produces effective interactions in atoms, quantum dots, and spin systems. The Heisenberg model and Hubbard model are canonical examples of many-body physics built from indistinguishable particles.

In two dimensions, exchange can realize fractional statistics beyond bosons and fermions. Anyons are central to topological phases and fault-tolerant quantum computation.