Fermionic field

 ( Quantum Field Theory ) 

In quantum field theory, a fermionic field is a quantum field whose Quantum are ; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of .

The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin-1/2: , , 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 , 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.

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Basic properties Free (non-interacting) fermionic fields obey canonical anticommutation relations; i.e., involve the { a, b} = ab + ba, rather than the commutators a, = 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.

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Dirac fields The prominent example of a spin-1/2 fermion field is the Dirac field (named after Paul Dirac), and denoted by $\backslash psi(x)$. The equation of motion for a free spin 1/2 particle is the Dirac equation,

$\backslash left(i\backslash gamma^\backslash mu\; \backslash partial\_\backslash mu\; -\; m\backslash right)\; \backslash psi(x)\; =\; 0.\backslash ,$

where $\backslash gamma^\{\backslash mu\}$ are gamma matrices and $m$ is the mass. The simplest possible solutions $\backslash psi(x)$ to this equation are plane wave solutions, $u(p)e^\{-ip\backslash cdot\; x\}\backslash ,$ and $v(p)e^\{ip\backslash cdot\; x\}\backslash ,$. These plane wave solutions form a basis for the Fourier components of $\backslash psi(x)$, allowing for the general expansion of the wave function as follows,

$\backslash psi\_\{\backslash alpha\}(x)\; =\; \backslash int\; \backslash frac\{d^3\; p\}\{(2\backslash pi)^3\}\; \backslash frac\{1\}\{\backslash sqrt\{2E\_p\}\}\; \backslash sum\_\{s\}\; \backslash left(a^s\_\backslash mathbf\{p\}\; u^s\_\{\backslash alpha\}(p)\; e^\{-ip\; \backslash cdot\; x\}\; +\; b^\{s\backslash dagger\}\_\backslash mathbf\{p\}\; v^s\_\{\backslash alpha\}(p)\; e^\{ip\; \backslash cdot\; x\}\backslash right).\backslash ,$

Here u and v are spinors labelled by their spin s and spinor indices $\backslash alpha\; \backslash in\; \backslash \{0,1,2,3\backslash \}$. 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, $\backslash psi(x)$ is promoted to an operator, so the coefficients of its Fourier modes must be operators too. Hence, $a^\{s\}\_\{\backslash mathbf\{p\}\}$ and $b^\{s\; \backslash dagger\}\_\{\backslash mathbf\{p\}\}$ are operators. The properties of these operators can be discerned from the properties of the field. $\backslash psi(x)$ and $\backslash psi(y)^\{\backslash dagger\}$ obey the anticommutation relations:

$\backslash left\backslash \{\backslash psi\_\{\backslash alpha\}(\backslash mathbf\{x\}),\; \backslash psi\_\{\backslash beta\}^\backslash dagger(\backslash mathbf\{y\})\backslash right\backslash \}\; =\; \backslash delta^\{(3)\}(\backslash mathbf\{x\}\; -\; \backslash mathbf\{y\})\backslash delta\_\{\backslash alpha\backslash beta\}.$

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

$\backslash left\backslash \{a^r\_\backslash mathbf\{p\},\; a^\{s\; \backslash dagger\}\_\backslash mathbf\{q\}\backslash right\backslash \}\; =\; \backslash left\backslash \{b^r\_\backslash mathbf\{p\},\; b^\{s\backslash dagger\}\_\backslash mathbf\{q\}\backslash right\backslash \}\; =\; (2\backslash pi)^\{3\}\; \backslash delta^3\; (\backslash mathbf\{p\}\; -\; \backslash mathbf\{q\})\; \backslash delta^\{rs\},\backslash ,$

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

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.

$\backslash mathcal\{L\}\_D\; =\; \backslash overline\{\backslash psi\}\backslash left(i\backslash gamma^\backslash mu\; \backslash partial\_\backslash mu\; -\; m\backslash right)\backslash psi\backslash ,$

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

$\backslash mathcal\{L\}\_D\; =\; \backslash overline\{\backslash psi\}\_a\backslash left(i\backslash gamma^\backslash mu\_\{ab\}\; \backslash partial\_\backslash mu\; -\; m\backslash mathbb\{I\}\_\{ab\}\backslash right)\backslash psi\_b\backslash ,$

The Hamiltonian (energy) density can also be constructed by first defining the momentum canonically conjugate to $\backslash psi(x)$, called $\backslash Pi(x):$

$\backslash Pi\; \backslash \; \backslash overset\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\_\{D\}\}\{\backslash partial\; (\backslash partial\_0\; \backslash psi)\}\; =\; i\backslash psi^\backslash dagger\backslash ,.$

With that definition of $\backslash Pi$, the Hamiltonian density is:

$\backslash mathcal\{H\}\_D\; =\; \backslash overline\{\backslash psi\}\backslash left-i\backslash vec\{\backslash gamma\}\; \backslash psi\backslash ,,$

where $\backslash vec\{\backslash nabla\}$ is the standard gradient of the space-like coordinates, and $\backslash vec\{\backslash gamma\}$ is a vector of the space-like $\backslash gamma$ matrices. It is surprising that the Hamiltonian density doesn't depend on the time derivative of $\backslash psi$, directly, but the expression is correct.

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

$D\_F(x\; -\; y)\; =\; \backslash left\backslash langle\; 0\backslash left|\; T(\backslash psi(x)\; \backslash overline\{\backslash psi\}(y))\backslash right|\; 0\; \backslash right\backslash rangle$

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

$$

 T\left[\psi(x) \overline{\psi}(y)\right] \ \overset{\text{def}}{=}\
   \theta\left(x^0 - y^0\right) \psi(x) \overline{\psi}(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:

$D\_F(x\; -\; y)\; =\; \backslash int\; \backslash frac\{d^4\; p\}\{(2\backslash pi)^4\}\; \backslash frac\{i(\{p\backslash !\backslash !\backslash !/\}\; +\; m)\}\{p^2\; -\; m^2\; +\; i\backslash epsilon\}e^\{-ip\; \backslash cdot\; (x\; -\; y)\}$

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

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

is just the inverse of the operator acting on $\backslash psi(x)$ 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.

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.

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See also

• Dirac equation
• Spin–statistics theorem
• Spinor
• Composite field
• Auxiliary field

• 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, .
• Weinberg, Steven (1995). The Quantum Theory of Fields, (3 volumes) Cambridge University Press.

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