Definition of the $S$-Matrix in Schwartz QFT-Book: Why is $\langle f, t_f | i, t_i \rangle$ in the Schroedinger picture, and not Heisenberg-picture?

Question

On page 51, (equation 5.1), Mathew Schwartz introduces the $S$-matrix as

\begin{align} \langle f| S | i \rangle_{Heisenberg} = \langle f, \infty | i, -\infty \rangle_{Schrödinger} \end{align}

Were $|i, t\rangle$ is a Schrödinger-state at time $t$ (at least if I understood correctly).

So now my question is: Shouldn't it be exactly the other way around?

I think that it should be the other way around because of several things:

• If I set up a schrödinger state at time $-\infty$ and at $\infty$ then the time evolution of the system (which is the interesting part) doesn't have an effect on $\langle f, \infty | i, -\infty \rangle$ at all.

• Schwartz later, when prooving the LSZ-formula starting at equation 6.6, uses the fields and annihilation/creation operators in the Heisenberg picture. Yet he just calculates the $S$-matrix simply as the scalar product between the defined in- and out-states, as this question points out.

Can somebody clear things up here? Especially, if an answer writes something like "$|i, t_i \rangle$ is a state at time $t$", or "state $|i, t_i \rangle$ fixed at time $t$" it would be great if additionally one says what is meant by that, because that phrases meaning is very ambiguous.

Answer 1

This is a common confusion, and I don't think Schwartz's description is clear at all.

An in state $|i, \text{in} \rangle$ is a Heisenberg state, fixed at time $t = 0$, which would look like several widely separated, incoming particles if you evolved it like a Schrodinger state to $t \to -\infty$. (Explicitly, if $U(t_f, t_i)$ is the time evolution operator in Schrodinger picture, then $|i, \text{in} \rangle = U(0, -\infty) |i \rangle$. That is, it's the Schrodinger state evaluated at time $t = 0$.)

Similarly, an out state $| f, \text{out} \rangle$ is a Heisenberg state which has widely separated, outgoing particles as $t \to \infty$. Here, $i$ and $f$ stand for a specification like ``two particles, of momenta $p^\mu$ and $q^\mu$''.

We want to know the transition amplitudes $\langle f, \text{out} | i, \text{in} \rangle$. The $S$-matrix is defined by $$|i, \text{in} \rangle = S | i, \text{out} \rangle.$$ Upon using the unitarity property $S^\dagger = S^{-1}$, this implies $$\langle f, \text{out} | i, \text{in} \rangle = \langle f, \text{in} | S | i, \text{in} \rangle.$$ Again, all of these are Heisenberg states. But "in" and "out" states are a weird kind of Heisenberg state defined by how they would look if they were time evolved like Schrodinger states, even though they actually aren't, which is why books can be unclear on this point.