On the interpretation of a correlator with different definite states

[Israel.cma]

Hello everyone, I was reading Ashos Das book on field theory, chapter 4.3, and I had this question.

This expression:

\begin{equation} \left< \psi_f | \psi_i \right> \end{equation}

is the transition amplitude of two states.

This expression:

\begin{equation} \frac{ \left< \psi|T(U_1...U_n)|\psi \right>}{\left< \psi | \psi \right>} =\left< T(U_1...U_n) \right> \end{equation}

is the expectation value of the operators U when used in that state (for example, the ground state).

But what does it would mean?

\begin{equation} \frac{ \left< \psi_f|T(U_1...U_n)|\psi_i \right>}{\left< \psi_f | \psi_i \right>} \end{equation}

I though that maybe is the expectation value of the operators U IF after the measure the state changes from the state 1 to the state 2.

Does anyone have seen that formula in another books or knows that does it means?

 

[Demystifier]

In general,
$$\frac{ \left< \psi_f|A|\psi_i \right>}{\left< \psi_f | \psi_i \right>}$$
is a weak value of the operator $A$. For more details see https://en.wikipedia.org/wiki/Weak_value

I answered this old question because it was suggested by PF as an unanswered thread and because I believe it may be interest to many readers here, even if the OP is not active any more.