Editing C-symmetry (section)

====Weyl spinors====
For the case of massless Dirac spinor fields, chirality is equal to helicity for the positive energy solutions (and minus the helicity for negative energy solutions).<ref name=Itzykson-Zuber-1980/>{{rp|at=§&nbsp;2-4-3, page&nbsp;87&nbsp;ff}} One obtains this by writing the massless Dirac equation as
:<math>i\partial\!\!\!\big /\psi = 0 </math>

Multiplying by <math>\gamma^5\gamma^0 = -i\gamma^1\gamma^2\gamma^3</math> one obtains
:<math>{\epsilon_{ij}}^m\sigma^{ij}\partial_m \psi = \gamma_5 \partial_t \psi</math>

where <math>\sigma^{\mu\nu} = i\left[\gamma^\mu, \gamma^\nu\right]/2</math> is the [[angular momentum operator]] and <math>\epsilon_{ijk}</math> is the [[totally antisymmetric tensor]]. This can be brought to a slightly more recognizable form by defining the 3D spin operator <math>\Sigma^m\equiv {\epsilon_{ij}}^m\sigma^{ij},</math> taking a plane-wave state <math>\psi(x) = e^{-ik\cdot x}\psi(k)</math>, applying the on-shell constraint that <math>k \cdot k = 0</math> and normalizing the momentum to be a 3D unit vector: <math>{\hat k}_i = k_i/k_0</math> to write

:<math>\left(\Sigma \cdot \hat k\right) \psi = \gamma_5 \psi~.</math> 

Examining the above, one concludes that angular momentum eigenstates ([[helicity (particle physics)|helicity]] eigenstates) correspond to eigenstates of the [[chirality (physics)|chiral operator]]. This allows the massless Dirac field to be cleanly split into a pair of [[Weyl spinor]]s <math>\psi_\text{L}</math> and <math>\psi_\text{R},</math> each individually satisfying the [[Weyl equation]], but with opposite energy:
:<math>\left(-p_0 + \sigma\cdot\vec p\right)\psi_\text{R} = 0</math>

and
:<math>\left(p_0 + \sigma\cdot\vec p\right)\psi_\text{L} = 0</math>

Note the freedom one has to equate negative helicity with negative energy, and thus the anti-particle with the particle of opposite helicity. To be clear, the <math>\sigma</math> here are the [[Pauli matrices]], and <math>p_\mu = i\partial_\mu</math> is the momentum operator.