Dirac Spinor

In quantum field theory, the Dirac spinor is the bispinor in the plane-wave solution

of the free Dirac equation,

where (in the units )

is a relativistic spin-1/2 field,

is the Dirac spinor related to a plane-wave with wave-vector ,

is the four-wave-vector of the plane wave, where is arbitrary,

are the four-coordinates in a given inertial frame of reference.

The Dirac spinor for the positive-frequency solution can be written as

$\omega_\vec{p} = \begin{bmatrix} \phi \\ \frac{\vec{\sigma}\vec{p}}{E_{\vec{p}} + m} \phi \end{bmatrix} \;,$

where

is an arbitrary two-spinor,

are the Pauli matrices,

is the positive square root