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Charge conjugation is a transformation that switches all particles with their corresponding antiparticles, and thus changes the sign of all charges: not only electric charge but also the charges relevant to other forces. 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 electroweak theory

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]

$$\psi \rightarrow -i({\bar \psi }\gamma ^{0}\gamma ^{2})^{T}$$
$${\bar \psi }\rightarrow -i(\gamma ^{0}\gamma ^{2}\psi )^{T}$$
$$A^{\mu }\rightarrow -A^{\mu }$$

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
Main article: C parity

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 e.g., $$C\psi (q)=C\psi (-q)$$, as opposed to odd C-parity which refers to antisymmetry under charge conjugation, e.g., $$C\psi (q)=-C\psi (-q)).$$

Define $$\psi \ {\stackrel {{\mathrm {def}}}{=}}\ {\phi +i\chi \over {\sqrt {2}}}$$. Now, φ and χ have even C-parities, and the imaginary number i has an odd C-parity (C is anti-unitary). Under C, ψ goes to ψ*.

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

C parity
Anti-particle
Antimatter
Truly neutral particle

References

Peskin, M.E.; Schroeder, D.V. (1997). An Introduction to Quantum Field Theory. Addison Wesley. ISBN 0-201-50397-2.

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

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