C-symmetry

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 maximally.

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 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. $\psi \rightarrow -i({\bar {\psi }}\gamma ^{0}\gamma ^{2})^{T}$
2. ${\bar {\psi }}\rightarrow -i(\gamma ^{0}\gamma ^{2}\psi )^{T}$
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 even this symmetry have now 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. $C\psi (q)=C\psi (-q)$, as opposed to odd C-parity which refers to antisymmetry under charge conjugation ex. $C\psi (q)=-C\psi (-q)$). Now reformulate things so that $\psi \ {\stackrel {\mathrm {def} }{=}}\ {\phi +i\chi \over {\sqrt {2}}}$. 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.

References

Sozzi, M.S. (2008). Discrete symmetries and CP violation. Oxford University Press. ISBN 978-0-19-929666-8.