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quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, a fermionic field is a quantum field whose quanta are
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p ...
s of bosonic fields. The most prominent example of a fermionic field is the ''Dirac field'', which describes fermions with spin-1/2:
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s,
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s,
quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
, etc. The Dirac field can be described as either a 4-component
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
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 A neutrino ( ; denoted by the Greek letter ) is an elementary particle that interacts via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small ('' -ino'') that i ...
is a Majorana fermion or a
Dirac fermion In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. A vast majority of fermions fall under this category. Description In particle physics, all fermions in the standard model have distinct antipar ...
; observing neutrinoless double-beta decay experimentally would settle this question.


Basic properties

Free (non-interacting) fermionic fields obey canonical anticommutation relations; i.e., involve the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
s = ''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.


Dirac fields

The prominent example of a spin-1/2 fermion field is the Dirac field (named after
Paul Dirac Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...
), and denoted by \psi(x). The equation of motion for a free spin 1/2 particle is the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
, :\left(i\gamma^\mu \partial_\mu - m\right) \psi(x) = 0.\, where \gamma^ are
gamma matrices In mathematical physics, the gamma matrices, \ \left\\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra \ \mathr ...
and m is the mass. The simplest possible solutions \psi(x) to this equation are plane wave solutions, u(p)e^\, and v(p)e^\,. These
plane wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
solutions form a basis for the Fourier components of \psi(x), allowing for the general expansion of the wave function as follows, :\psi_(x) = \int \frac \frac \sum_ \left(a^s_\mathbf u^s_(p) e^ + b^_\mathbf v^s_(p) e^\right).\, Here ''u'' and ''v'' are spinors labelled by their spin ''s'' and spinor indices \alpha \in \. 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. In
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
, \psi(x) is promoted to an operator, so the coefficients of its Fourier modes must be operators too. Hence, a^_ and b^_ are operators. The properties of these operators can be discerned from the properties of the field. \psi(x) and \psi(y)^ obey the anticommutation relations: :\left\ = \delta^(\mathbf - \mathbf)\delta_. We impose an anticommutator relation (as opposed to a
commutation relation In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
as we do for the bosonic field) in order to make the operators compatible with Fermi–Dirac statistics. By putting in the expansions for \psi(x) and \psi(y), the anticommutation relations for the coefficients can be computed. :\left\ = \left\ = (2\pi)^ \delta^3 (\mathbf - \mathbf) \delta^,\, In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that a^_ creates a fermion of momentum p and spin s, and b^_ creates an antifermion of momentum q and spin ''r''. The general field \psi(x) is now seen to be a weighted (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions. Its conjugate field, \overline \ \stackrel\ \psi^ \gamma^, 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 \overline\psi\,. This makes the reason for the choice of \overline = \psi^ \gamma^clear. This is because the general Lorentz transform on \psi is not unitary so the quantity \psi^\psi would not be invariant under such transforms, so the inclusion of \gamma^\, is to correct for this. The other possible non-zero Lorentz invariant quantity, up to an overall conjugation, constructible from the fermionic fields is \overline\gamma^\partial_\psi. 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. :\mathcal_D = \overline\left(i\gamma^\mu \partial_\mu - m\right)\psi\, Such an expression has its indices suppressed. When reintroduced the full expression is :\mathcal_D = \overline_a\left(i\gamma^\mu_ \partial_\mu - m\mathbb_\right)\psi_b\, The
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
(
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
) density can also be constructed by first defining the momentum canonically conjugate to \psi(x), called \Pi(x): :\Pi \ \overset\ \frac = i\psi^\dagger\,. With that definition of \Pi, the Hamiltonian density is: : \mathcal_D = \overline\left i\vec \cdot \vec + m\right\psi\,, where \vec is the standard
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
of the space-like coordinates, and \vec is a vector of the space-like \gamma matrices. It is surprising that the Hamiltonian density doesn't depend on the time derivative of \psi, directly, but the expression is correct. Given the expression for \psi(x) we can construct the Feynman propagator for the fermion field: : D_F(x - y) = \left\langle 0\left, T(\psi(x) \overline(y))\ 0 \right\rangle we define the time-ordered product for fermions with a minus sign due to their anticommuting nature : T\left psi(x) \overline(y)\right\ \overset\ \theta\left(x^0 - y^0\right) \psi(x) \overline(y) - \theta\left(y^0 - x^0\right) \overline\psi(y) \psi(x). Plugging our plane wave expansion for the fermion field into the above equation yields: : D_F(x - y) = \int \frac \frace^ where we have employed the Feynman slash notation. This result makes sense since the factor :\frac is just the inverse of the operator acting on \psi(x) 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 In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...
for the Dirac field, and preserved causality. More complicated field theories involving interactions (such as Yukawa theory, or
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
) can be analyzed too, by various perturbative and non-perturbative methods. Dirac fields are an important ingredient of the
Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
.


See also

*
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
* Spin–statistics theorem *
Spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
* Composite Field * Auxiliary Field


References

* * Peskin, M and Schroeder, D. (1995). ''An Introduction to Quantum Field Theory'', Westview Press. (See pages 35–63.) * Srednicki, Mark (2007).
Quantum Field Theory
'', Cambridge University Press, {{ISBN, 978-0-521-86449-7. * Weinberg, Steven (1995). ''The Quantum Theory of Fields'', (3 volumes) Cambridge University Press. Quantum field theory Spinors