In Peskin & Schroeder's book (Page 104, equation 4.70) localized in and out states of different momenta have been defined. The authors state that they are working in the Heisenberg picture. However, in equation 4.70, to compute the scalar product between the in and the out state, the authors put in a factor $e^{-i2HT}$ in between them, to relate the states to be defined at a "common reference time". As was explained in this: question, the authors do so because that way the states $|k \rangle_{in}$ and $|k \rangle_{out}$ are both drawn from the same Hilbert space, belonging to an arbitrary time slice.
However, I didn't exactly understand why one state labeled by $k_a$ or $k_b$ differs for different times in the first place. Peskin says that those states are named by the same name, but are different states, because they should recreate the same eigenvalues for (for example) the momentum operator.
My question now is: When Peskin writes a $|k_{a}\rangle$ at a "common reference time", what is meant by that? Is it a state in which the (field- and time-dependent) momentum operator at time $t$ has the expectation value $k_{a}$ ? What would it be for a wave package with some finite spatial extent, like the ones that were defined earlier in the chapter?
