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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 $|\phi \rangle \ne C_\alpha |\alpha\psi \rangle$?

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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.

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