spinstatistics

The spin-statistics theorem is a fundamental result in quantum mechanics and quantum field theory that links the intrinsic angular momentum (spin) of particles to the type of quantum statistics they obey. In three spatial dimensions, particles with integer spin are bosons and follow Bose-Einstein statistics, while particles with half-integer spin are fermions and follow Fermi-Dirac statistics. This connection explains why photons and other gauge bosons behave collectively, whereas electrons and other matter particles obey the Pauli exclusion principle.

The standard derivation relies on relativistic quantum field theory and key assumptions such as locality (microcausality), Lorentz invariance, and a positive-energy spectrum. The theorem was first presented by Wolfgang Pauli in 1940 and has since been given rigorous treatments by several mathematicians and physicists. It implies that the wavefunction of a system of identical bosons is symmetric under particle exchange, whereas the fermionic wavefunction is antisymmetric.

Consequences of the spin-statistics connection are pervasive: fermions obey the Pauli exclusion principle, shaping atomic structure, chemistry, and the electronic properties of matter; bosons can occupy the same quantum state, enabling phenomena like Bose-Einstein condensation and laser action. In quantum field theory, the theorem also constrains the algebra of creation and annihilation operators and underpins the stability of matter.

Exceptions and extensions exist in reduced dimensions or under nonstandard conditions. In two-dimensional systems, particles called anyons can exhibit fractional or braid statistics, and the spin-statistics theorem does not apply in the same form. In non-relativistic or nonlocal theories, alternative statistics or generalized frameworks—such as parastatistics—can arise. See also Bose-Einstein statistics, Fermi-Dirac statistics, Pauli exclusion principle, anyons, and parastatistics.