Editing Dirac spinor (section)

== Definition==
The '''Dirac spinor''' is the [[bispinor]] <math>u\left(\vec{p}\right)</math> in the [[plane wave|plane-wave]] ansatz
<math display="block">\psi(x) = u\left(\vec{p}\right)\; e^{-i p \cdot x} </math>
of the free [[Dirac equation]] for a [[spinor]] with mass <math>m</math>,
<math display="block">\left(i\hbar\gamma^\mu \partial_\mu - mc\right)\psi(x) = 0</math>
which, in [[natural units]] becomes
<math display="block">\left(i\gamma^\mu \partial_\mu - m\right)\psi(x) = 0</math>
and with [[Feynman slash notation]] may be written
<math display="block">\left(i\partial\!\!\!/ - m\right)\psi(x) = 0</math>

An explanation of terms appearing in the ansatz is given below.
* The Dirac field is <math>\psi(x)</math>, a [[theory of relativity|relativistic]] [[spin-1/2]] [[field (physics)|field]], or concretely a function on [[Minkowski space]] <math>\mathbb{R}^{1,3}</math> valued in <math>\mathbb{C}^4</math>, a four-component complex vector function.
* The '''Dirac spinor''' related to a plane-wave with [[wave-vector]] <math>\vec{p}</math> is <math>u\left(\vec{p}\right)</math>, a <math>\mathbb{C}^4</math> vector which is constant with respect to position in spacetime but dependent on momentum <math>\vec{p}</math>.
* The inner product on Minkowski space for vectors <math>p</math> and <math>x</math> is <math>p \cdot x \equiv p_\mu x^\mu \equiv E_\vec{p} t - \vec{p} \cdot \vec{x}</math>.
* The four-momentum of a plane wave is <math display="inline">p^\mu = \left(\pm\sqrt{m^2 + \vec{p}^2},\, \vec{p}\right) := \left(\pm E_\vec{p}, \vec{p}\right)</math> where <math>\vec{p}</math> is arbitrary,
* In a given [[inertial frame]] of reference, the coordinates are <math>x^\mu</math>. These coordinates parametrize Minkowski space. In this article, when <math>x^\mu</math> appears in an argument, the index is sometimes omitted.

The Dirac spinor for the positive-frequency solution can be written as
<math display="block">
  u\left(\vec{p}\right) = \begin{bmatrix}
    \phi \\ \frac{\vec{\sigma} \cdot \vec{p}}{E_\vec{p} + m} \phi
  \end{bmatrix} \,,
</math>
where
* <math>\phi</math> is an arbitrary two-spinor, concretely a <math>\mathbb{C}^2</math> vector.
* <math>\vec{\sigma}</math> is the [[Pauli matrices#Pauli vectors|Pauli vector]],
* <math>E_\vec{p}</math> is the positive square root <math display="inline">E_\vec{p} = + \sqrt{m^2 + \vec{p}^2}</math>. For this article, the <math>\vec{p}</math> subscript is sometimes omitted and the energy simply written <math>E</math>.

In natural units, when {{math|''m''<sup>2</sup>}} is added to {{math|''p''<sup>2</sup>}} or when {{math|''m''}} is added to <math>{p\!\!\!/}</math>, {{math|''m''}} means {{math|''mc''}} in ordinary units; when {{math|''m''}} is added to {{math|''E''}}, {{math|''m''}} means {{math|''mc''<sup>2</sup>}} in ordinary units. When ''m'' is added to <math>\partial_\mu</math> or to <math>\nabla</math> it means <math display="inline">\frac{mc}{\hbar}</math> (which is called the ''inverse reduced [[Compton wavelength]]'') in ordinary units.