the spin-statistics theorem
Why are electrons fermions that refuse to share a state, while photons are bosons that pile up happily in a laser beam? In nonrelativistic quantum mechanics you simply postulate which is which. The spin-statistics theorem does far more: it proves that a particle's spin dictates its statistics, with no freedom left. Half-integer spin means fermion; integer spin means boson. This is one of the deepest results connecting relativity, quantum mechanics, and the structure of matter.
The theorem lives in relativistic quantum field theory. Given a handful of physically sacred assumptions — Lorentz invariance, locality (microcausality: fields at spacelike-separated points must not interfere), positive energy (a stable vacuum), and positive-definite norms (probabilities cannot be negative) — one can prove that fields of half-integer spin MUST be quantized with anticommutators, giving Fermi-Dirac statistics and the exclusion principle, while integer-spin fields MUST be quantized with commutators, giving Bose-Einstein statistics. Choosing the 'wrong' quantization for a given spin produces either states of negative energy or violations of causality. Pauli established the result in 1940.
The consequences organize the material world: because electrons are spin-1/2 fermions they stack into shells, giving the periodic table, chemistry, and the very rigidity of matter; because photons and phonons are integer-spin bosons they permit lasers, superfluids, and Bose-Einstein condensation. An honest caveat: this is genuinely a theorem of RELATIVISTIC field theory — nonrelativistic quantum mechanics cannot derive it and must take the symmetrization postulate as an input. And in two dimensions the theorem's assumptions relax, allowing anyons with fractional statistics.
The electron (spin 1/2) is a fermion, so a white dwarf is held up by electron degeneracy pressure. The photon (spin 1) and the helium-4 atom (integer total spin) are bosons, so light forms coherent laser beams and liquid helium becomes a superfluid — all dictated by spin alone.
Spin alone fixes statistics: half-integer to fermion, integer to boson.
The theorem is not an assumption of nonrelativistic quantum mechanics but a consequence of relativistic field theory plus causality. Its usual proof genuinely requires special relativity; there is no simple nonrelativistic derivation.