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In the answer given by knzhou to the post What distinguishes the behaviour of particle from its antiparticle: C violation or CP violation? it is said that

"but the reaction $i \rightarrow f$ will run at the same rate as its $CP$ conjugate $\bar{f}_P \rightarrow \bar{i}_P$."

My questions is: is not $\bar{f}_P \rightarrow \bar{i}_P$ the $CT$ of $i \rightarrow f$ instead of its $CP$ conjugate since you have the anti-$f$ particles in the initial state?

EDIT I found something else that I do not understand. Why does knzhou say that C is not enough? If C distingishes particles and antiparticles and our theory violates C (not necessarily CP), the reactions $i\rightarrow f$ and its C conjugate $\bar{i} \rightarrow \bar{f}$ will have different rate since $i, f$ and its counterparts $\bar{i}, \bar{f}$ are different, so why do we need CP and not just C?

Moreover, if CPT is preserved but we look for CP violation, does it imply that T is not preserved? But is it not T a symmetry usually?

Vicky
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    The f->i reaction should be the other way around to be CP conjugated... that shown reaction would be the CPT conjugated. – Mr Puh Apr 04 '19 at 07:58
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    Sorry, I just made a mistake there, thanks for the catch! – knzhou Apr 04 '19 at 08:47
  • @knzhou I have another question related to that post so I edited mine. If you could answer it I'd appreciate it – Vicky Apr 04 '19 at 14:48
  • It's simply that if $i \to f$ converts, say, matter to antimatter at some rate, then $\tilde{i}_P \to \tilde{f}_P$ would convert it right back, giving no net change. – knzhou Apr 04 '19 at 14:55
  • @knzhou Yeah, but why do you need to include parity? With C you have already that unbalance effect since your theory would not be C-even – Vicky Apr 04 '19 at 14:57
  • @Vicky Unfortunately I'm not really sure what you mean, can you edit your post to explain in more detail? – knzhou Apr 04 '19 at 14:59
  • @knzhou already done, better that way? – Vicky Apr 04 '19 at 18:47
  • Possibly useful mnemonic: for one generation of left handed leptons and right-handed antileptons, the term $W_\mu^+\overline{\nu_L}\gamma^\mu e_L + W_\mu^-\overline{e_L}\gamma^\mu \nu_L$ is CPT , CP, and T invariant, but not P or C invariant. – Cosmas Zachos Apr 05 '19 at 16:56

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