Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields.
The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin-1/2: electrons, protons, quarks, etc.
Spin-1/2 Majorana fermions, such as the hypothetical neutralino, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor.
It is not known whether the neutrino is a Majorana fermion or a Dirac fermion; observing neutrinoless double-beta decay experimentally would settle this question.
Free (non-interacting) fermionic fields obey canonical anticommutation relations; i.e., involve the anticommutators {a, b} = ab + ba, rather than the commutators [a, b] = ab − ba of bosonic or standard quantum mechanics.
Those relations also hold for interacting fermionic fields in the interaction picture, where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states.
It is these anticommutation relations that imply Fermi–Dirac statistics for the field quanta.
They also result in the Pauli exclusion principle: two fermionic particles cannot occupy the same state at the same time.
to this equation are plane wave solutions,
These plane wave solutions form a basis for the Fourier components of
, allowing for the general expansion of the wave function as follows, Here u and v are spinors labelled by their spin s and spinor indices
The energy factor is the result of having a Lorentz invariant integration measure.
obey the anticommutation relations: We impose an anticommutator relation (as opposed to a commutation relation as we do for the bosonic field) in order to make the operators compatible with Fermi–Dirac statistics.
In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that
creates a fermion of momentum p and spin s, and
creates an antifermion of momentum q and spin r. The general field
is now seen to be a weighted (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions.
, is the opposite, a weighted summation over all possible spins and momenta for annihilating fermions and antifermions.
The other possible non-zero Lorentz invariant quantity, up to an overall conjugation, constructible from the fermionic fields is
Since linear combinations of these quantities are also Lorentz invariant, this leads naturally to the Lagrangian density for the Dirac field by the requirement that the Euler–Lagrange equation of the system recover the Dirac equation.
When reintroduced the full expression is The Hamiltonian (energy) density can also be constructed by first defining the momentum canonically conjugate to
It is surprising that the Hamiltonian density doesn't depend on the time derivative of
we can construct the Feynman propagator for the fermion field: we define the time-ordered product for fermions with a minus sign due to their anticommuting nature Plugging our plane wave expansion for the fermion field into the above equation yields: where we have employed the Feynman slash notation.
This result makes sense since the factor is just the inverse of the operator acting on
Note that the Feynman propagator for the Klein–Gordon field has this same property.
Since all reasonable observables (such as energy, charge, particle number, etc.)
are built out of an even number of fermion fields, the commutation relation vanishes between any two observables at spacetime points outside the light cone.
We have therefore correctly implemented Lorentz invariance for the Dirac field, and preserved causality.
More complicated field theories involving interactions (such as Yukawa theory, or quantum electrodynamics) can be analyzed too, by various perturbative and non-perturbative methods.
Dirac fields are an important ingredient of the Standard Model.