The Pauli exclusion principle states that fermions are antisymmetric under particle exchange, and that as a consequence no two fermions may occupy the same quantum state. Mathematically, the exchange operator for a two-body wavefunction is

Normalisation considerations tell us that the eigenvalue, g must be either ±1 (as the operator must conserve probability). The Pauli exclusion principle then states that the eigenvalue is +1 for bosons and −1 for fermions, and that a wavefunction with an eigenvalue of −1 describes particles that cannot occupy the same quantum state. The spin-statistics theorem states that these particles are fermions, with half-integer spin.