
A fermion is a type of elementary particle that follows Fermi-Dirac statistics and obeys the Pauli exclusion principle, such as electrons and quarks.Fermions are divided into leptons (electron, neutrino) and quarks (which make up protons and neutrons). They share a spin-½, which forces them to follow Fermi–Dirac statistics—a mathematical expression of the exclusion principle. When a gas of fermions is cooled, all lower energy levels are filled up to a boundary—the Fermi energy. This Fermi surface explains the electrical conductivity of metals: only electrons near the boundary can move. The pressure of degenerate fermions prevents white dwarfs from collapsing under their own gravity—stars are held up by the exclusion force.
The name “fermion” honors Enrico Fermi, who together with Paul Dirac in 1926 developed the statistics for particles with half-integer spin. A year earlier, Wolfgang Pauli formulated the exclusion principle by analyzing atomic spectra. The Dirac equation (1928) combined quantum mechanics and special relativity for the electron and led to the discovery of the positron by Carl Anderson in 1932. The quark model, proposed by Gell-Mann and Zweig in 1964, organized the zoo of strongly interacting particles.
The Pauli principle is a consequence of the antisymmetry of the wave function of a fermion system: Ψ(…,x_i,…,x_j,…) = −Ψ(…,x_j,…,x_i,…). When coordinates and spins coincide, the wave function vanishes. In a free electron gas at zero temperature, all levels below the Fermi energy E_F are filled, and those above are empty, according to the Fermi–Dirac distribution f(E) = 1/(e^{(E−μ)/k_B T} + 1). The degeneracy pressure of nonrelativistic fermions p ∝ n^{5/3} supports white dwarfs, while in the relativistic regime the Chandrasekhar limit restricts their mass.
Electrons are fermions. Their stubbornness prevents atoms from collapsing and creates all the diversity of chemistry. Without this rule, there would be no solid table, nor your palm.
The human body contains about 7×10²⁷ fermions (electrons and quarks). If the Pauli principle vanished, all these particles would clump into a single point—and you would shrink to the size of a grain of sand.
| \Gamma_{i\to f} | variable | Transition rate from initial state i to final state f — transition rate per second |
| \psi_i | variable | Initial quantum state — wave function |
| \psi_f | variable | Final quantum state — wave function |
| H' | variable | Perturbation Hamiltonian — perturbation hamiltonian joule |
| \rho(E_f) | variable | Density of final states at energy E_f — density of states per joule |
| \hbar | constant | Reduced Planck constant, relating energy to angular frequency — reduced planck constant = 1.054571817e-34 joule second |
| \langle \psi_f | H' | operator | Matrix element of the perturbation between initial and final states — matrix element |
| H | variable | Hamiltonian operator — hamiltonian joule |
| t | variable | hopping amplitude — energy joule |
| γ_{2j} | variable | Majorana operator at site 2j — majorana operator |
| γ_{2j+1} | variable | Majorana operator at site 2j+1 — majorana operator |
| ∑ | operator | sum over lattice sites j — summation |
















