The following paragraph is from pag. 55 of Peskin and Schroeder's An Introduction to Quantum Field Theory:
First consider the propagation amplitude $\langle 0|\psi(x) \bar{\psi}(y)|0 \rangle$, which is to represent a positive-energy particle propagating from $y$ to $x$. In this case we want the (Heisenberg) state $\bar{\psi}(y)|0 \rangle$ to be made up of only positive-energy, or negative-frequency components (since a Heisenberg state $\Psi_H=e^{+iHt}\Psi_S$).
Now, my doubts:
(1) This has nothing to do with the above paragraph, but with my understanding of the mathematical construction of the theory: before in the same chapter, to construct an invariant quadratic on the fields, we made the 'replacement' $u^\dagger \to \bar{u}=u^\dagger\gamma^0$, where $u$ is the spinor part in the solution to Dirac's equation. My question is: why is this replacement physically acceptable? That is, doesn't it change the postulated probabilistic quantities, like the propagation amplitude from the part I quoted (since $\psi$ is proportional to $u$)?
(2) Regarding the quoted passage, why do we want the state $\bar{\psi}(y)|0 \rangle$ to be made up of only positive-energy or negative-frequency components? And why are these two requirements 'equivalent' here?
(3) Why the consideration which caused my second question follows from the fact that $\Psi_H=e^{+iHt}\Psi_S$ or, if it doesn't, what the does the author mean with '(since [...])'?
