Spin-statistics theorem

The spin-statistics theorem dictates that particles with half-integer spin, known as fermions, must have an antisymmetric quantum state and obey Fermi-Dirac statistics. Conversely, particles with integer spin, known as bosons, have symmetric states and follow Bose-Einstein statistics.

Because fermions obey antisymmetric wavefunctions under exchange, as proven by the spin-statistics theorem, they are subject to the Pauli exclusion principle. This fundamental rule prevents two identical fermions, like electrons in an atom, from having the exact same set of quantum numbers simultaneously.

The spin-statistics theorem shows that bosons, which have integer spin, do not obey the Pauli exclusion principle and can occupy the exact same quantum state. This symmetric behavior allows for macroscopic quantum phenomena, such as the creation of Bose-Einstein condensates.

The theorem was mathematically proven by Wolfgang Pauli in 1940 within the framework of relativistic quantum field theory. Pauli's proof relies heavily on the principles of Lorentz invariance and the positivity of energy in quantum mechanics.