Fermionic field
Encyclopedia
In quantum field theory
, a fermionic field is a quantum field whose quanta are fermion
s; that is, they obey Fermi-Dirac statistics
. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relation
s of bosonic field
s.
The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin
-1/2: electron
s, proton
s, 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 particles, such as neutrino
s, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor.
rather than the commutators 
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.
), and denoted by
. The equation of motion for a free field is the Dirac equation
,
where
are gamma matrices and m 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 Dirac field as follows,
The
and 
labels are spinor indices and the 
indices represent spin labels and so for the electron, a spin 1/2 particle, s = +1/2 or s=−1/2. The energy factor is the result of having a Lorentz invariant integration measure. Since 
can be thought of as an operator, the coefficients of its Fourier modes must be operators too. Hence, 
and 
are operators. The properties of these operators can be discerned from the properties of the field. 
and 
obey the anticommutation relations
By putting in the expansions for
and 
, the anticommutation relations for the coefficients can be computed.
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 
and spin s, and 
creates an antifermion of momentum 
and spin r. The general field 
is now seen to be a weighed (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions. Its conjugate field, 
, is the opposite, a weighted summation over all possible spins and momenta for annihilating fermions and antifermions.
With the field modes understood and the conjugate field defined, it is possible to construct Lorentz invariant quantities for fermionic fields. The simplest is the quantity
. This makes the reason for the choice of 
clear. This is because the general Lorentz transform on 
is not unitary
so the quantity
would not be invariant under such transforms, so the inclusion of 
is to correct for this. 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.
Such an expression has its indices suppressed. When reintroduced the full expression is
Given the expression for
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
in the Dirac equation. 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. As we know from elementary quantum mechanics two simultaneously commuting observables can be measured simultaneously. 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
.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, a fermionic field is a quantum field whose quanta are fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s; that is, they obey Fermi-Dirac statistics
Fermi-Dirac statistics
Fermi–Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relation
Canonical commutation relation
In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:[x,p_x] = i\hbar...
s of bosonic field
Bosonic field
In quantum field theory, a bosonic field is a quantum field whose quanta are bosons; that is, they obey Bose-Einstein statistics. Bosonic fields obey canonical commutation relations, as distinct from the canonical anticommutation relations obeyed by fermionic fields.Examples include scalar fields,...
s.
The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
-1/2: electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s, proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s, quarks, etc. The Dirac field can be described as either a 4-component spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
or as a pair of 2-component Weyl spinors. Spin-1/2 Majorana particles, such as neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
s, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor.
Basic properties
Free (non-interacting) fermionic fields obey canonical anticommutation relations, i.e., involve the anticommutators rather than the commutators 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
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
: two fermionic particles cannot occupy the same state at the same time.
Dirac fields
The prominent example of a spin-1/2 fermion field is the Dirac field (named after Paul DiracPaul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
), and denoted by
. The equation of motion for a free field is the Dirac equationDirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
,
where
are gamma matrices and m is the mass. The simplest possible solutions to this equation are plane wave solutions, and . These plane wavePlane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
solutions form a basis for the Fourier components of
, allowing for the general expansion of the Dirac field as follows,The
and labels are spinor indices and the indices represent spin labels and so for the electron, a spin 1/2 particle, s = +1/2 or s=−1/2. The energy factor is the result of having a Lorentz invariant integration measure. Since can be thought of as an operator, the coefficients of its Fourier modes must be operators too. Hence, and are operators. The properties of these operators can be discerned from the properties of the field. and obey the anticommutation relationsBy putting in the expansions for
and , the anticommutation relations for the coefficients can be computed.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 and spin s, and creates an antifermion of momentum and spin r. The general field is now seen to be a weighed (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions. Its conjugate field, , is the opposite, a weighted summation over all possible spins and momenta for annihilating fermions and antifermions.With the field modes understood and the conjugate field defined, it is possible to construct Lorentz invariant quantities for fermionic fields. The simplest is the quantity
. This makes the reason for the choice of clear. This is because the general Lorentz transform on is not unitaryUnitary
Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...
so the quantity
would not be invariant under such transforms, so the inclusion of is to correct for this. The other possible non-zero Lorentz invariantLorentz covariance
In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space"...
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
Euler-Lagrange equation
In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
of the system recover the Dirac equation.
Such an expression has its indices suppressed. When reintroduced the full expression is
Given the expression for
we can construct the Feynman propagatorPropagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
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
in the Dirac equation. 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. As we know from elementary quantum mechanics two simultaneously commuting observables can be measured simultaneously. We have therefore correctly implemented Lorentz invariance for the Dirac field, and preserved causalityCausality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
.
More complicated field theories involving interactions (such as Yukawa theory, or quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
) can be analyzed too, by various perturbative and non-perturbative methods.
Dirac fields are an important ingredient of the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
.
See also
- Dirac equationDirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
- Einstein-Maxwell-Dirac equationsEinstein-Maxwell-Dirac equationsEinstein-Maxwell-Dirac equations are related to quantum field theory. The current Big Bang Model is a quantum field theory in a curved spacetime. Unfortunately, no such theory is mathematically well-defined; in spite of this, theoreticians claim to extract information from this hypothetical theory...
- Spin-statistics theoremSpin-statistics theoremIn quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...
- SpinorSpinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...