Editing C-symmetry (section)

===Charge conjugation, chirality, helicity ===
The interplay between chirality and charge conjugation is a bit subtle, and requires articulation. It is often said that charge conjugation does not alter the [[Chirality (physics)|chirality]] of particles. This is not the case for ''fields'', the difference arising in the "hole theory" interpretation of particles, where an anti-particle is interpreted as the absence of a particle. This is articulated below.

Conventionally, <math>\gamma_5</math> is used as the chirality operator. Under charge conjugation, it transforms as
:<math>C\gamma_5 C^{-1} = \gamma_5^\textsf{T}</math>

and whether or not <math>\gamma_5^\textsf{T}</math> equals <math>\gamma_5</math> depends on the chosen representation for the gamma matrices. In the Dirac and chiral basis, one does have that <math>\gamma_5^\textsf{T} = \gamma_5</math>, while <math>\gamma_5^\textsf{T} = -\gamma_5</math> is obtained in the Majorana basis. A worked example follows.

====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.

====Charge conjugation in the chiral basis====
Taking the [[Gamma matrices#Weyl representation|Weyl representation]] of the gamma matrices, one may write a (now taken to be massive) Dirac spinor as
:<math>\psi = \begin{pmatrix} \psi_\text{L}\\ \psi_\text{R} \end{pmatrix}</math>

The corresponding dual (anti-particle) field is
:<math>\overline{\psi}^\textsf{T}
    = \left( \psi^\dagger \gamma^0 \right)^\textsf{T}
    = \begin{pmatrix} 0 & I \\ I & 0\end{pmatrix} \begin{pmatrix} \psi_\text{L}^* \\ \psi_\text{R}^* \end{pmatrix}
    = \begin{pmatrix} \psi_\text{R}^* \\ \psi_\text{L}^* \end{pmatrix}
</math>

The charge-conjugate spinors are
:<math>\psi^c
    = \begin{pmatrix} \psi_\text{L}^c\\ \psi_\text{R}^c \end{pmatrix} 
    = \eta_c C \overline\psi^\textsf{T}
    = \eta_c \begin{pmatrix} -i\sigma^2 & 0 \\ 0 & i\sigma^2\end{pmatrix} \begin{pmatrix} \psi_\text{R}^* \\ \psi_\text{L}^* \end{pmatrix}
    = \eta_c \begin{pmatrix} -i\sigma^2\psi_\text{R}^* \\ i\sigma^2\psi_\text{L}^* \end{pmatrix}
</math>

where, as before, <math>\eta_c</math> is a phase factor that can be taken to be <math>\eta_c=1.</math>  Note that the left and right states are inter-changed. This can be restored with a parity transformation. Under [[P-symmetry|parity]], the Dirac spinor transforms as
:<math>\psi\left(t, \vec x\right) \mapsto \psi^p\left(t, \vec x\right) = \gamma^0 \psi\left(t, -\vec x\right)</math>

Under combined charge and parity, one then has
:<math>\psi\left(t, \vec x\right) \mapsto \psi^{cp}\left(t, \vec x\right) 
     =  \begin{pmatrix} \psi_\text{L}^{cp} \left(t, \vec x\right)\\ \psi_\text{R}^{cp}\left(t,\vec x\right) \end{pmatrix} 
     = \eta_c \begin{pmatrix} -i\sigma^2\psi_\text{L}^*\left(t, -\vec x\right) \\ i\sigma^2\psi_\text{R}^*\left(t, -\vec x\right) \end{pmatrix}</math>

Conventionally, one takes <math>\eta_c = 1</math> globally. See however, the note below.

====Majorana condition====
The [[Majorana equation|Majorana condition]] imposes a constraint between the field and its charge conjugate, namely that they must be equal: <math>\psi = \psi^c.</math> This is perhaps best stated as the requirement that the Majorana spinor must be an eigenstate of the charge conjugation involution.

Doing so requires some notational care. In many texts discussing charge conjugation, the involution <math>\psi\mapsto\psi^c</math> is not given an explicit symbolic name, when applied to ''single-particle solutions'' of the Dirac equation. This is in contrast to the case when the ''quantized field'' is discussed, where a unitary operator <math>\mathcal{C}</math> is defined (as done in a later section, below). For the present section, let the involution be named as <math>\mathsf{C}:\psi\mapsto\psi^c</math> so that <math>\mathsf{C}\psi = \psi^c.</math> Taking this to be a linear operator, one may consider its eigenstates. The Majorana condition singles out one such: <math>\mathsf{C}\psi = \psi.</math>  There are, however, two such eigenstates: <math>\mathsf{C}\psi^{(\pm)} = \pm \psi^{(\pm)}.</math> Continuing in the Weyl basis, as above, these eigenstates are
:<math>\psi^{(+)} = \begin{pmatrix} \psi_\text{L}\\ i\sigma^2\psi_\text{L}^* \end{pmatrix}</math>

and
:<math>\psi^{(-)} = \begin{pmatrix} i\sigma^2\psi_\text{R}^*\\ \psi_\text{R} \end{pmatrix}</math>

The Majorana spinor is conventionally taken as just the positive eigenstate, namely <math>\psi^{(+)}.</math> The chiral operator <math>\gamma_5</math> exchanges these two, in that
:<math>\gamma_5\mathsf{C} = - \mathsf{C}\gamma_5</math>

This is readily verified by direct substitution. Bear in mind that <math>\mathsf{C}</math> ''does '''not''' have'' a 4×4 matrix representation! More precisely, there is no complex 4×4 matrix that can take a complex number to its complex conjugate; this inversion would require an 8×8 real matrix. The physical interpretation of complex conjugation as charge conjugation becomes clear when considering the complex conjugation of scalar fields, described in a subsequent section below.

The projectors onto the chiral eigenstates can be written as <math>P_\text{L} = \left(1 - \gamma_5\right)/2</math> and <math>P_\text{R} = \left(1 + \gamma_5\right)/2,</math> and so the above translates to
:<math>P_\text{L}\mathsf{C} = \mathsf{C}P_\text{R}~.</math>

This directly demonstrates that charge conjugation, applied to single-particle complex-number-valued solutions of the Dirac equation flips the chirality of the solution. The projectors onto the charge conjugation eigenspaces are <math>P^{(+)} = (1 + \mathsf{C})P_\text{L}</math> and <math>P^{(-)} = (1 - \mathsf{C})P_\text{R}.</math>

====Geometric interpretation====
The phase factor <math>\ \eta_c\ </math> can be given a geometric interpretation. It has been noted that, for massive Dirac spinors, the "arbitrary" phase factor <math>\ \eta_c\ </math> may depend on both the momentum, and the helicity (but not the chirality).<ref>{{harvp|Itzykson|Zuber|1980|loc=§&nbsp;2-4-2 ''Charge Conjugation'', page&nbsp;86, equation&nbsp;2-100}}</ref> This can be interpreted as saying that this phase may vary along the fiber of the [[spinor bundle]], depending on the local choice of a coordinate frame. Put another way, a spinor field is a local [[section (fiber bundle)|section]] of the spinor bundle, and Lorentz boosts and rotations correspond to movements along the fibers of the corresponding [[frame bundle]] (again, just a choice of local coordinate frame). Examined in this way, this extra phase freedom can be interpreted as the phase arising from the electromagnetic field. For the [[Majorana spinor]]s, the phase would be constrained to not vary under boosts and rotations.