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Concept
Fermionic field
Summary
In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. 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. The Dirac field can be described as either a 4-component spinor or as a pair of 2-component Weyl spinors. 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.
The prominent example of a spin-1/2 fermion field is the Dirac field (named after Paul Dirac), and denoted by . The equation of motion for a free spin 1/2 particle is the Dirac equation,
where are gamma matrices and is the mass. The simplest possible solutions to this equation are plane wave solutions, and . These plane wave solutions form a basis for the Fourier components of , allowing for the general expansion of the wave function as follows,
u and v are spinors, labelled by spin, s and spinor indices . For the electron, a spin 1/2 particle, s = +1/2 or s=−1/2.
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