In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Fermions have a half-odd-integer spin (spin 1/2, spin 3/2, etc.) and obey the Pauli exclusion principle. These particles include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.
Some fermions are elementary particles (such as electrons), and some are composite particles (such as protons). For example, according to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons. In contrast, particles with half-integer spin are fermions.
In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin-statistics relation is, in fact, a spin statistics-quantum number relation.
As a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at a given time. Suppose multiple fermions have the same spatial probability distribution. Then, at least one property of each fermion, such as its spin, must be different. Fermions are usually associated with matter, whereas bosons are generally force carrier particles. However, in the current state of particle physics, the distinction between the two concepts is unclear. Weakly interacting fermions can also display bosonic behavior under extreme conditions. For example, at low temperatures, fermions show superfluidity for uncharged particles and superconductivity for charged particles.
Composite fermions, such as protons and neutrons, are the key building blocks of everyday matter.
English theoretical physicist Paul Dirac coined the name fermion from the surname of Italian physicist Enrico Fermi.
For setting the fermionic state ## |n_1 n_2\rangle## where ##n_i## is the number of particles in the orbital ##i## we can use the following both representations for the state ##|1 1\rangle##: $$ \hat {\mathbf c_1} ^{\dagger} \hat {\mathbf c_2} ^{\dagger} |vac\rangle $$ or $$ \hat {\mathbf c_2}...