C-symmetry
From Academic Kids
C-symmetry means the symmetry of physical laws over a charge-conjugation transformation. Electromagnetism, gravity and the strong interaction all obey C-symmetry, but weak interactions violate C-symmetry maximally. (Some postulated extensions of the Standard Model, like left-right models, restore this symmetry.)
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:
- <math>\psi \rightarrow -i(\bar\psi \gamma^0 \gamma^2)^T<math>
- <math>\bar\psi \rightarrow -i(\gamma^0 \gamma^2 \psi)^T<math>
- <math>A^\mu \rightarrow -A^\mu<math>
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.
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.
There is really a lot of ambiguity and arbitrariness in the definition of charge conjugation. To give an example, take two real scalar fields, φ and χ. Formulated as it is, both fields have even C-parity. Now reformulate things so that <math>\psi\equiv {\phi + i \chi\over \sqrt{2}}<math>. Now, φ has an even C-parity wheareas χ has an odd C-parity. But let's redefine <math>\psi\equiv {\chi + i\phi\over\sqrt{2}}<math>. Now it's the other way around. Similarly, a complex Weyl spinor can be reexpressed as a real Majorana spinor and vice versa. This arbitrariness allows physicists to define C the way it is in left-right models.
