How a quantum system (eg. particle physics QFT) governed by <PERSON> equation can break the $T$ symmetry (or $CP$ symmetry) microscopically at all?
Why and how a quantum system (like our quantum vacuum of our universe) governed by <PERSON> equation can break the time-reversal $T$ symmetry (or $CP$ symmetry) microscopically at all?
p.s. the $T$ symmetry (or $CP$ symmetry) breaking is known and verified in neutral kaons. See also the Wikipedia texts on $CP$ violation and strong $CP$ problems.
The <PERSON> equation governs the quantum system (regardless relativistic or not) looks: $$ i \frac{\partial}{\partial t} |\Psi{}(t) \rangle = H(t) |\Psi{}(t) \rangle. $$ It is easy to check the <PERSON> equation is time-reversal $T$ invariant if
1. 346
$H(t)$ has no explicit time dependent, or time-reversal symmetric, then $ H(-t)=H(t)=H$. 2. Since we do not want to make $t=0$ becomes a special fixed point, we also require the time translational symmetry $H(t)=H(t+\Delta)$ for some arbitrary time interval $\Delta$.
We can do a time reversal transformation on the whole equation, $$ T i T^{-1} T\frac{\partial}{\partial t} |\Psi{}(t) \rangle = T H(t) T^{-1} T|\Psi{}(t) \rangle $$ $$ =T i T^{-1} T\lim_{\Delta \to 0}\frac{ |\Psi{}(t+\Delta) \rangle- |\Psi{}(t) \rangle}{\Delta} = H(-t) |\Psi{}(-t) \rangle $$ $$=Ti T^{-1} \lim_{\Delta \to 0}\frac{ |\Psi{}(-t-\Delta) \rangle- |\Psi{}(-t) \rangle}{\Delta} = H(-t) |\Psi{}(-t) \rangle $$ $$ =T i T^{-1} (-)\frac{\partial}{\partial \tilde t} |\Psi{}(\tilde t) \rangle \Big \vert_{\tilde t = -t} = H(-t) |\Psi{}(-t) \rangle. $$
As we already knew, the <PERSON> equation is time-reversal $T$ invariant requiring that $$T i T^{-1} =-i$$
My question is that there seems to have no room to ask any quantum system governed by <PERSON> equation to break the $T$ symmetry (or $CP$ symmetry) microscopically, if we already demand a microscopically time-reversal symmetric Hamiltonian $ H(-t)=H(t)=H$.
Now my puzzle is, - our quantum universe does break $T$ symmetry (or $CP$ symmetry) microscopically. - our quantum universe is governed by <PERSON> equation (regardless relativistic or not). How can such a microscopically time-reversal symmetric Hamiltonian $ H(-t)=H(t)=H$ break the $T$ symmetry? 955