19.7 Bose-Einstein and Fermi-Dirac statistics
In classical mechanics, particles aredistinguishable, namely each particle may be tagged and this
tagging survives throughout time evolution since particleidentities are conserved.
In quantum mechanics,particles of the same species are indistinguishable, and cannot be tagged
individually. They can only be characterized by the quantumnumbers of the state of the system.
Therefore, the operation of interchange of any two particles of the same species must be a symmetry
of the quantum system. The square of the interchange operation is the identity.^17 As a result, the
quantum states must have definite symmetry properties underthe interchange of two particles.
All particles in Nature are either bosons or fermions;- BOSONS : the quantum state is symmetric under the interchange of any pair of particles
and obeyBose-Einstein statistics. Bosons haveinteger spin. For example, photons,W±,Z^0 ,
gluons and gravitons are bosonic elementary particles, while the Hydrogen atom, theHe 4
and deuterium nuclei are composite bosons. - FERMIONS : the quantum state is anti-symmetric under the interchange of any pair of
particles and obeyFermi-Dirac statistics. Fermions haveinteger plus half spin. For example,
all quarks and leptons are fermionic elementary particles,while the proton, neutron andHe 3
nucleus are composite fermions.
Remarkably, the quantization of free scalars and free photons carried out in the preceeding
subsections has Bose-Einstein statistics built in. The bosonic creation and annihilation operators
are denoted bya†σ(~k) andaσ(~k) for each speciesσ. The canonical commutation relations inform
us that all creation operators commute with one another [a†σ(~k),a†σ′(~k′)] = 0. As a result, two
states differening only by the interchange of two particle of the same species are identical quantum
mechanically,
a†σ 1 (~k 1 )···a†σi(~ki)···a†σj(~kj)···a†σn(~kn)|∅〉 (19.68)
=a†σ 1 (~k 1 )···a†σj(~kj)···a†σi(~ki)···a†σn(~kn)|∅〉 (19.69)The famousCPT Theoremstates that upon the quantization of a Poincar ́e and CPT invariant
Lagrangian, integer spin fields will always produce particle states that obey Bose-Einstein statistics,
while integer plus half fields always produce states that obey Fermi-Dirac statistics.
- Fermi-Dirac Statistics
It remains to establish how integer plus half spin fields are to be quantized. This will be the
subject of the subsection on the Dirac equation. Here, we shall take a very simple approach whose
point of departure is the fact that fermions obey Fermi-Dirac statistics.
(^17) This statement holds in 4 space-time dimensions but it does not hold generally. In 3 space-time dimen-
sions, the topology associated with the interchange of two particles allows forbraidingand the square of this
operation is not 1. The corresponding particles areanyons.