Charge conjugation

Charge conjugation $\hat{C}$ is a transformation that inverts the sign of all quantum numbers of a particle, transforming it into its antiparticle. It is a discrete symmetry. Despite being called "charge" conjugation, it does not just invert charge. It operates as

where $Q$, $B$ and $S$ are quantum numbers associated with a particle. Using ket notation for particle states, we can write it as

using electron/positron and proton/antiproton as examples.

Charge conjugation is a unit operator: $\hat{C}^{2}=\hat{\mathbf{1}}$. Its eigenvalues are therefore $P=\pm1$.

Intrinsic charge conjugation#

Each particle can be assigned an eigenvalue $C=\pm1$ depending on whether it is symmetric under charge conjugation or not. For example, conjugating the charge of the electron gives the positron. Because it changes, the electron and positron get $C=-1$. Similarly, the Photon does change, $\hat{C}\ket{\gamma}=-\ket{\gamma}$ (polarization is flipped) so $C=-1$. Meanwhile, the neutral pion's state does not change, so $\hat{C}\ket{\pi^{0}}=\ket{\pi^{0}}$. Then $\pi^{0}$ gets $C=1$.

All particles that are $C=1$ are their own antiparticles (e.g. $\pi^{0}$). The converse is not true (e.g. $\gamma$).

Intrinsic charge conjugation is conserved by all electromagnetism and the strong force. It is violated by the weak force. This proves useful to disallow some otherwise fine-looking decays, like $\pi^{0}\to\gamma+\gamma+\gamma$, which violates the C conservation of electromagnetism ($C_{\pi^{0}}=1$ at the start, $C_\text{end}=(C_{\gamma})^{3}=(-1)^{3}=-1$ at the end).