Editing Dirac spinor (section)

==Charge conjugation==
[[Charge conjugation]] transforms the positive-energy spinor into the negative-energy spinor. Charge conjugation is a mapping (an [[involution (mathematics)|involution]]) <math>\psi\mapsto\psi_c</math> having the explicit form
<math display="block">\psi_c = \eta C \left(\overline\psi\right)^\textsf{T}</math>
where <math>(\cdot)^\textsf{T}</math> denotes the transpose, <math>C</math> is a 4×4 matrix, and <math>\eta</math> is an arbitrary phase factor, <math>\eta^*\eta = 1.</math> The article on [[charge conjugation]] derives the above form, and demonstrates why the word "charge" is the appropriate word to use: it can be interpreted as the [[electrical charge]]. In the Dirac representation for the [[gamma matrices]], the matrix <math>C</math> can be written as
<math display="block">C = i\gamma^2\gamma^0 =
 \begin{pmatrix} 
   0 & -i\sigma_2 \\
   -i\sigma_2 & 0
 \end{pmatrix}
</math>
Thus, a positive-energy solution (dropping the spin superscript to avoid notational overload)
<math display="block">\psi^{(+)} = u\left(\vec{p}\right) e^{-ip\cdot x}
= \textstyle \sqrt{\frac{E + m}{2m}} 
  \begin{bmatrix} 
    \phi\\ 
    \frac{\vec{\sigma} \cdot \vec{p}}{E + m} \phi
  \end{bmatrix} 
   e^{-ip\cdot x}
</math>
is carried to its charge conjugate
<math display="block">\psi^{(+)}_c
= \textstyle \sqrt{\frac{E + m}{2m}} 
  \begin{bmatrix} 
    i\sigma_2 \frac{\vec{\sigma}^* \cdot \vec{p}}{E + m} \phi^*\\
    -i\sigma_2 \phi^*
  \end{bmatrix}
   e^{ip\cdot x}
</math>
Note the stray complex conjugates. These can be consolidated with the identity
<math display="block">\sigma_2 \left(\vec\sigma^* \cdot \vec k\right) \sigma_2 = - \vec\sigma\cdot\vec k</math>
to obtain
<math display="block">\psi^{(+)}_c
= \textstyle \sqrt{\frac{E + m}{2m}} 
  \begin{bmatrix} 
   \frac{\vec{\sigma} \cdot \vec{p}}{E + m} \chi \\
    \chi 
  \end{bmatrix} 
   e^{ip\cdot x}
</math>
with the 2-spinor being
<math display="block">\chi = -i\sigma_2 \phi^*</math>
As this has precisely the form of the negative energy solution, it becomes clear that charge conjugation exchanges the particle and anti-particle solutions. Note that not only is the energy reversed, but the momentum is reversed as well. Spin-up is transmuted to spin-down. It can be shown that the parity is also flipped. Charge conjugation is very much a pairing of Dirac spinor to its "exact opposite".