Editing Dirac spinor (section)

==Orthogonality==
The Dirac spinors provide a complete and orthogonal set of solutions to the [[Dirac equation]].<ref>{{cite book |first=James D. |last=Bjorken |first2=Sidney D. |last2=Drell |year=1964 |title=Relativistic Quantum Mechanics |publisher=McGraw-Hill }} ''See Chapter 3.''</ref><ref name="iz">{{cite book |first=Claude |last=Itzykson |first2=Jean-Bernard |last2=Zuber |year=1980 |title=Quantum Field Theory |publisher=McGraw-Hill |isbn=0-07-032071-3 }} ''See Chapter 2.''</ref> This is most easily demonstrated by writing the spinors in the rest frame, where this becomes obvious, and then boosting to an arbitrary Lorentz coordinate frame.  In the rest frame, where the three-momentum vanishes: <math>\vec p = \vec 0,</math> one may define four spinors
<math display="block">u^{(1)}\left(\vec{0}\right) = 
  \begin{bmatrix}
    1 \\
    0 \\
    0 \\
    0
  \end{bmatrix} 
\qquad 
  u^{(2)}\left(\vec{0}\right) = 
  \begin{bmatrix}
    0 \\
    1 \\
    0 \\
    0 
  \end{bmatrix}
\qquad 
  v^{(1)}\left(\vec{0}\right) = 
  \begin{bmatrix}
    0 \\
    0 \\
    1 \\
    0 
  \end{bmatrix}
\qquad 
  v^{(2)}\left(\vec{0}\right) =
  \begin{bmatrix}
    0 \\
    0 \\
    0 \\
    1 
  \end{bmatrix}
</math>

Introducing the [[Feynman slash notation#With four-momentum|Feynman slash notation]]
<math display="block">{p\!\!\!/} = \gamma^\mu p_\mu</math>

the boosted spinors can be written as 
<math display="block">u^{(s)}\left(\vec{p}\right) = 
\frac{{p\!\!\!/} + m}{\sqrt{2m(E+m)}} u^{(s)}\left(\vec{0}\right) 
= \textstyle \sqrt{\frac{E+m}{2m}}
  \begin{bmatrix}
    \phi^{(s)}\\
    \frac {\vec\sigma \cdot \vec p} {E+m} \phi^{(s)}
  \end{bmatrix} 
</math>
and
<math display="block">
  v^{(s)}\left(\vec{p}\right) =
  \frac{-{p\!\!\!/} + m}{\sqrt{2m(E+m)}} v^{(s)}\left(\vec{0}\right) 
= \textstyle \sqrt{\frac{E+m}{2m}}
  \begin{bmatrix}
    \frac {\vec\sigma \cdot \vec p} {E+m} \chi^{(s)} \\
    \chi^{(s)}
  \end{bmatrix}
</math>

The conjugate spinors are defined as <math>\overline \psi = \psi^\dagger \gamma^0</math> which may be shown to solve the conjugate Dirac equation
<math display="block">\overline \psi (i{\partial\!\!\!/} + m) = 0</math>

with the derivative understood to be acting towards the left.  The conjugate spinors are then
<math display="block">
  \overline u^{(s)}\left(\vec{p}\right) = 
  \overline u^{(s)}\left(\vec{0}\right) \frac{{p\!\!\!/} + m}{\sqrt{2m(E+m)}}  
</math>
and
<math display="block">
  \overline v^{(s)}\left(\vec{p}\right) =
  \overline v^{(s)}\left(\vec{0}\right) \frac{-{p\!\!\!/} + m}{\sqrt{2m(E+m)}}  
</math>

The normalization chosen here is such that the scalar invariant <math>\overline\psi \psi</math> really is invariant in all Lorentz frames. Specifically, this means 
<math display="block">
  \begin{align}
     \overline u^{(a)} (p) u^{(b)} (p) &=  \delta_{ab} & \overline u^{(a)} (p) v^{(b)} (p) &= 0 \\
     \overline v^{(a)} (p) v^{(b)} (p) &= -\delta_{ab} & \overline v^{(a)} (p) u^{(b)} (p) &= 0
  \end{align}
</math>