# Charge Conjugation

### Charge Conjugation

In physics, C-symmetry means the symmetry of physical laws under a charge-conjugation transformation. Electromagnetism, gravity and the strong interaction all obey C-symmetry, but weak interactions violate C-symmetry.

## Charge reversal in electromagnetism

The laws of electromagnetism (both classical and quantum) are invariant under this transformation: if each charge q were to be replaced with a charge −q, and thus the directions of the electric and magnetic fields were reversed, the dynamics would preserve the same form. In the language of quantum field theory, charge conjugation transforms:

1. $\psi \rightarrow -i\left(\bar\psi \gamma^0 \gamma^2\right)^T$
2. $\bar\psi \rightarrow -i\left(\gamma^0 \gamma^2 \psi\right)^T$
3. $A^\mu \rightarrow -A^\mu$

Notice that these transformations do not alter the chirality of particles. A left-handed neutrino would be taken by charge conjugation into a left-handed antineutrino, which does not interact in the Standard Model. This property is what is meant by the "maximal violation" of C-symmetry in the weak interaction.

(Some postulated extensions of the Standard Model, like left-right models, restore this C-symmetry.)

## Combination of charge and parity reversal

It was believed for some time that C-symmetry could be combined with the parity-inversion transformation (see P-symmetry) to preserve a combined CP-symmetry. However, violations of this symmetry have been identified in the weak interactions (particularly in the kaons and B mesons). In the Standard Model, this CP violation is due to a single phase in the CKM matrix. If CP is combined with time reversal (T-symmetry), the resulting CPT-symmetry can be shown using only the Wightman axioms to be universally obeyed.

## Charge definition

To give an example, take two real scalar fields, φ and χ. Suppose both fields have even C-parity (even C-parity refers to even symmetry under charge conjugation ex. $C\psi\left(q\right) = C\psi\left(-q\right)$, as opposed to odd C-parity which refers to antisymmetry under charge conjugation ex. $C\psi\left(q\right)=-C\psi\left(-q\right)$). Now reformulate things so that $\psi\ \stackrel\left\{\mathrm\left\{def\right\}\right\}\left\{=\right\}\ \left\{\phi + i \chi\over \sqrt\left\{2\right\}\right\}$. Now, φ and χ have even C-parities because the imaginary number i has an odd C-parity (C is antiunitary).

In other models, it is possible for both φ and χ to have odd C-parities.