Why does parity violation imply Lorentz violation?
Parity violation is Lorentz violation for massive fermon. Consider a massive fermion with left-handed helicity. Under parity, the massive fermion is supposed to be transformed into the right-handed helicity. However, we can find the exactly same transformation under the Lorentz transformations; the left-handed massive fermion can be observed as a right-handed helicity one as its momentum becomes reversed in the boosted frame. Therefore, what parity operator does to a massive fermion is the same as Lorentz transformationa and the parity violation means Lorentz violation for massive fermion.
