Editing C-symmetry (section)

====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>