Fermionic field

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 canonical commutation relations. The most common example is the Dirac field which can describe spin-1/2 particles: electrons, protons, quarks, etc. The Dirac field is a 4-component spinor. It can also be described by two 2-component Weyl spinors. Spin-1/2 particles that have no antiparticles (possibly the neutrinos) can be described by a single 2-component Weyl spinor (or by a 4-component Majorana spinor, whose components are not independent).

Details
Fermions are particles whose quantum mechanical wavefunction is totally antisymmetric under quantum number interchange. This gives the Pauli exclusion principle, in that any two fermionic particles cannot occupy the same state at the same time. Because of this antisymmetry, fermionic fields have equations which typically involve the anticommutator brackets $$\{a,b\} = ab+ba$$ rather than the commutator brackets of bosonic or standard quantum mechanics.

Here we will consider the most common example of a spin-1/2 fermion field, called Dirac field (named after Paul Dirac), and denoted by $$\psi(x)$$. The equation of motion for a free field is the Dirac equation,


 * $$(i\gamma^{\mu} \partial_{\mu} - m) \psi(x) = 0.\,$$

where $$\gamma^{\mu}\,$$ are gamma matrices. The simplest possible solutions to this equation are plane wave solutions, $$\psi_{1}(x) = u(p)e^{-ip.x}\,$$ and $$\psi_{2}(x) = v(p)e^{ip.x}\,$$. These plane wave solutions form a basis for the Fourier components of $$\psi(x)$$, allowing for the general expansion of the Dirac field as follows,

$$\psi(x) = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{p}}}\sum_{s} \left( a^{s}_{\textbf{p}}u^{s}(p)e^{-ip \cdot x}+b^{s \dagger}_{\textbf{p}}v^{s}(p)e^{ip \cdot x}\right).\,$$

The $$a\,$$ and $$b\,$$ labels are spinor indices and the $$s\,$$ 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 $$\psi(x)\,$$ can be thought of as an operator, the coefficents of its Fourier modes must be operators too. Hence, $$a^{s}_{\textbf{p}}$$ and $$b^{s \dagger}_{\textbf{p}}$$ are operators. The properties of these operators can be discerned from the properties of the field. $$\psi(x)\,$$ and $$\psi(y)^{\dagger}$$ obey the anticommutation relations


 * $$\{\psi_a(\textbf{x}),\psi_b^{\dagger}(\textbf{y})\} = \delta^{(3)}(\textbf{x}-\textbf{y})\delta_{ab},$$

By putting in the expansions for $$\psi(x)\,$$ and $$\psi(y)\,$$, the anticommutation relations for the coefficents can be computed.


 * $$\{a^{r}_{\textbf{p}},a^{s \dagger}_{\textbf{q}}\} = \{b^{r}_{\textbf{p}},b^{s \dagger}_{\textbf{q}}\}=(2 \pi)^{3} \delta^{3} (\textbf{p}-\textbf{q}) \delta^{rs},\,$$

In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that $$a^{s \dagger}_{\textbf{p}}$$ creates a fermion of momentum $$\textbf{p}\,$$ and spin s, and $$b^{r \dagger}_{\textbf{q}}$$ creates an antifermion of momentum $$\textbf{q}\,$$ and spin r. The general field $$\psi(x)\,$$ 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, $$\bar{\psi} \ \stackrel{\mathrm{def}}{=}\ \psi^{\dagger} \gamma^{0}$$, 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 $$\overline{\psi}\psi\,$$. This makes the reason for the choice of $$\bar{\psi} = \psi^{\dagger} \gamma^{0}$$clear. This is because the general Lorentz transform on $$\psi\,$$ is not unitary so the quantity $$\psi^{\dagger}\psi$$ would not be invariant under such transforms, so the inclusion of $$\gamma^{0}\,$$ is to correct for this. The other possible non-zero Lorentz invariant quantity, up to an overall conjugation, constructable from the fermionic fields is $$\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi$$.

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.


 * $$\mathcal{L}_{D} = \bar{\psi}(i\gamma^{\mu} \partial_{\mu} - m)\psi\,$$

Such an expression has its indices suppressed. When reintroduced the full expression is


 * $$\mathcal{L}_{D} = \bar{\psi}_{a}(i\gamma^{\mu}_{ab} \partial_{\mu} - m\mathbb{I}_{ab})\psi_{b}\,$$

Given the expression for $$\psi(x)$$ we can construct the Feynman propagator for the fermion field:


 * $$ D_{F}(x-y) = \langle 0| T(\psi(x) \bar{\psi}(y))| 0 \rangle $$

we define the time-ordered product for fermions with a minus sign due to their anticommuting nature


 * $$ T(\psi(x) \bar{\psi}(y)) \ \stackrel{\mathrm{def}}{=}\ \theta(x^{0}-y^{0}) \psi(x) \bar{\psi}(y)  - \theta(y^{0}-x^{0})\bar\psi(y) \psi(x) .$$

Plugging our plane wave expansion for the fermion field into the above equation yields:


 * $$ D_{F}(x-y) = \int \frac{d^{4}p}{(2\pi)^{4}} \frac{i(p\!\!\!/ + m)}{p^{2}-m^{2}+i \epsilon}e^{-ip \cdot (x-y)}$$

where we have employed the Feynman slash notation. This result makes sense since the factor


 * $$\frac{i(p\!\!\!/ + m)}{p^{2}-m^{2}}$$

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 for the Dirac field, and preserved causality.

The basics of free fermion field theory are sketched here. 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.