# Why is charge conjugation $\hat C | \alpha \psi \rangle = C_\alpha |\alpha\psi \rangle$?

Charge conjugation replaces all particles by antiparticles in the same state, so that momenta, positions, etc are unchanged. It can be represented by $$\hat C | \alpha \psi \rangle = C_\alpha |\alpha\psi \rangle = |\bar \alpha \psi \rangle$$

where $$| \alpha \psi \rangle$$ is the state of a particle $$\alpha$$ and $$C_\alpha$$ is a phase factor and $$|\bar \alpha \psi \rangle$$ is the stater of the antiparticle.

Why is it that the state of an antiparticle has the same form as a particle with the same position, momemtum etc? Why can't the state of the antiparticle be a general $$|\phi \rangle$$ where $$|\phi \rangle \ne C_\alpha |\alpha\psi \rangle$$?

Given any transformation $$C$$ that deserves to be called charge conjugation, any other transformation obtained by composing $$C$$ with a proper Lorentz transformation is equally deserving of the name charge conjugation, and likewise for any transformation obtained by composing $$C$$ with some other internal symmetry. It's a matter of convention which one we single out to call "the" charge conjugation transformation.