Fermionic field

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In quantum field theory, a free fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi-Dirac statistics. Fermionic fields (which are free fields) obey canonical anticommutation relations rather than canonical commutation relations. The simplest example is the Dirac field which correctly describes electrons, neutrinos, muons, and tau particles. In general, one must use supergeometry, and one obtains a graded partial algebra whose elements are not observables; instead, the partial algebra has an integral or trace whose values determine greens functions which determine local scattering matrices.Thus, the Fermionic field is a mathematical device for determining a local scattering theory (see D.Edwards).

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Fermions are particles whose quantum mechanical wavefunction is totally antisymmetric under quantum number interchange. Here we will only consider the most elementary example of a spin one-half fermion, often called a Dirac spinor field. We will denote the fermion (or Dirac) field by $ψ(x)$ . This requirement imposes the anticommutation rules

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

where $a^{s \dagger}_{\textbf{p}}$ creates a fermion of momentum $\textbf{p}$ and spin $s$ , and $b^{s \dagger}_{\textbf{p}}$ creates an antifermion. These anticommutation relations require that the plane wave expansion for $ψ(x)$ create an antifermion while annihilating a fermion:

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

The equal-time anticommutation relations for the field operators are

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

where a and b are spinor indices. Given the right Dirac spinors u and v, our expression for $ψ(x)$ will satisfy the Dirac equation

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

Note that the Dirac equation can be derived from the fermion Lagrangian

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

where "psi-bar" is defined as $\bar{\psi} \equiv \psi^{\dagger} \gamma^{0}$ .

Given the expression for $ψ(x)$ 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

$T(\psi(x) \bar{\psi}(y)) \equiv \theta(x^{0}-y^{0}) \psi(x) \bar{\psi}(y) - \theta(y^{0}-x^{0})\bar\psi(y) \psi(x) .$

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 $ψ(x)$ 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 Fermion field, and preserved causality.

The basics of free fermion field theory are sketched here. For more complicated field theories involving interactions (such as Yukawa theory, or quantum electrodynamics), the results are known only perturbatively.

[edit] See also

D. Edwards, The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories, International J. of Theor. Phys., Vol. 20, No. 7 (1981).
Peskin & Schroeder, Introduction to Quantum Field Theory, pgs. 35-63.
Bosonic field
Spin-statistics theorem
Spinor

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