Dirac Field

The Dirac field describes relativistic spin-one-half particles and antiparticles. Its free theory combines a first-order wave equation with fermionic statistics, providing the building block for electrons in QED and quarks in QCD. The profile here concerns one massive Dirac species in $3+1$ dimensions.

Dirac dynamics and degrees of freedom

With $\bar\psi=\psi^\dagger\gamma^0$, the Lagrangian is

$$\mathcal L=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi, \qquad m>0.$$

The four complex components of $\psi$ form a Lorentz spinor; they are not four independent particle species. On shell, there are two particle spin states and two antiparticle spin states. The field dimension is $[\psi]=3/2$, making the mass term dimension four as required.

For a plane-wave solution $u(p)e^{-ip\cdot x}$, define $p\!\!\!/\equiv\gamma^\mu p_\mu$. The equation is $(p\!\!\!/-m)u=0$. Multiplication by $p\!\!\!/+m$ gives

$$(p^2-m^2)u=0,$$

because the antisymmetric part of $\gamma^\mu\gamma^\nu$ drops out against $p_\mu p_\nu$. Each component therefore obeys the Klein–Gordon dispersion relation even though the spin and statistics differ.

Quantization and the propagator

Equal-time anticommutation, rather than commutation, makes the particle and antiparticle energies positive while preserving relativistic locality. A single fermion mode can be occupied zero or one times. The vacuum propagator is

$$\langle0|T\psi(x)\bar\psi(0)|0\rangle =\int\frac{\mathrm d^4p}{(2\pi)^4} \frac{i(p\!\!\!/+m)e^{-ip\cdot x}} {p^2-m^2+i0}.$$

Multiplying its numerator by $p\!\!\!/-m$ returns $p^2-m^2$, so the same scalar mass shell appears with the spinor inverse operator. This elementary check fixes the relative mass sign. See Srednicki 2007, §§39 and 42 for the quantized field and propagator.

Symmetries, probes, and limits

The phase symmetry preserves the vector current $j^\mu=\bar\psi\gamma^\mu\psi$. Coupling that current to electromagnetism leads to QED. A scalar coupling $\phi\bar\psi\psi$ instead gives a Yukawa interaction.

At $m=0$, left- and right-handed components decouple in the free equations. A Majorana field is a different choice of reality condition, possible only when the relevant charge assignments permit it; setting the Dirac mass to zero does not impose that condition. Nor should “negative-frequency solution” be interpreted as a physical negative-energy particle: the quantized expansion assigns it to antiparticle creation.

Develop the calculation

The Dirac Field explains the representation and action. Canonical Quantization of the Free Dirac Field develops the oscillator algebra. Compare Yukawa Theory and Quantum Electrodynamics to see how the same free field enters different interactions.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§36–42. DOI.