I understand that the time reversal operator, $\Theta$, acting on a state has the effect $t\rightarrow -t$ (and also takes complex conjugate). In the Heisenberg representation the states kets $|x,t_0\rangle_H$ are independent of time, while the base kets evolve backwards in time $|x,t\rangle_H=e^{\frac{iH(t-t_0)}{\hbar}}|x,t_0\rangle_H$.
I am wondering what the effect of the time reversal operator on the base ket in the Heisenberg representation is.
I suspect it is something like
\begin{equation} \begin{aligned} \Theta|x,t\rangle_H&=e^{\frac{(-i)H(-t)}{\hbar}}\Theta|x,0\rangle_H\\ &=e^{\frac{iHt}{\hbar}}|x,0\rangle_H\\ &=|x,t\rangle_H \end{aligned} \end{equation}
Hence the time reversal operator has no effect on the Heisenberg picture states because they are independent on time. Is this true?
