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

Read more about Dirac Spinor:  Derivation From Dirac Equation, Four-spinor For Particles, Four-spinor For Anti-particles, Completeness Relations, Dirac Spinors and The Dirac Algebra