(1) I read that CPT theorem can be proved with Lorentz invariance. Also, CPT violation implies Lorentz violation.
Is CPT invariance equivalent with Lorentz invariance, or just one-side direction holds?
CPT transformation is discrete, in contrast, Lorentz transformation is continuous. It seems so quite different, but how do they related each other?
(2) In CPT violation case, particle mass and antiparticle mass can be different, like as:
$$ | m_{\phi} - m_{\bar{\phi}} | \neq 0 $$
I think:
$[ CPT, \hat{H} ] \neq 0$
$ CPT \hat{H} \phi - \hat{H} CPT \phi \neq 0$
$ ( m_{\phi} - m_{\bar{\phi}} ) CPT \phi \neq 0$
($H = E = \sqrt{m^2 + p^2} \sim m$)
However, such logic also holds for acting only C(charge conjugation). What is the correct explanation for mass difference of particle and antiparticle from CPT violation?
