Time reversal symmetry operator T^2 can be either +1 or -1, in which +1 is for spin less particles and -1 is for spin-half particles. Similarly, charge conjugation symmetry operator can also take either +1 or -1, in this case what is the physical meaning of C = +1 or -1? Do both (C = +1 and -1) represent the same CHC^-1 = - H ?
Classically under charge conjugation Aμ → -Aμ and Jμ → -Jμ so that H=∫d3x Aμ Jμ is invariant.
In quantum theory C is an operator CAμ → -Aμ and CJμ → -Jμ such that the eigenvalue of this operator for photon and current is -1. Because C does not change space-time coordinates it is an internal symmetry operation.
If you need to know how this definition of charge conjugation is implemented in quantum field theory please let me know here.
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