ABSTRACT
Let us consider a gas of N non-interacting identical quantum particles, with discrete singleparticle states 1, 2, · · · , i, · · · , possessing the eigenenergies ε1 ≤ ε2 ≤ · · · εi ≤ · · · . The quantum state of the whole gas is specified by the set of occupation numbers n1, n2, · · · , ni, · · · , where ni is the number of particles in the single-particle state i. According to quantum mechanics two mutually exclusive classes of (elementary) particles exist. From a quantum statistical point of view, the classes are distinguished on the basis of the possible values of the occupation numbers. In the first class there is as such no restriction on the occupation numbers ni, i.e.,
ni = 0, 1, 2, · · · , ∀i. (31.1) The kind of particles which belong to this class are named bosons, and the quantum statistics they obey is known as Bose-Einstein (BE) statistics. In the second class the occupation numbers are restricted to the values
ni = 0, 1, ∀i, (31.2) so that at most one particle can be in any state. The particles belonging to the second class are named fermions, and the quantum statistics they obey is known as Fermi-Dirac (FD) statistics. The restriction in (31.2) states the Pauli exclusion principle: Two identical fermions cannot be in the same single-particle state. The general formulation of the exclusion principle was first given by Pauli in 1925 [182, 183, 184]. There is a most remarkable connection between the spin (intrinsic angular momentum) of a quantum particle and the statistics: Bosons possess integral spin, and fermions half-integral spin. For the elementary particles the spin is (in units of ~) 0 (Higgs), 1/2 (electron, muon, tauon, neutrinos, quarks; and their antiparticles), or 1 (photons, vector bosons [W+, W−, Z0], gluons). For the unobserved graviton the spin is predicted to be 2.
