Interacting Picture in QFT I'm having trouble understanding how the interaction picture describes scattering. In quantum theory, the probability amplitude for a system in state $|i(t_i) \rangle$ to be measured in state $|f(t_f) \rangle$ is $\langle f(t_f)|$$|i(t_i) \rangle$, and we call this amplitude $A$. My textbook says that "in experiments we do not measure  $\langle f(t_f)|$$|i(t_i) \rangle$, but  $\langle f(t_f)|$$S$$|i(t_i) \rangle$". This confused me, since in quantum mechanics, we calculate just that for the probability amplitude, and surely this amplitude is measurable in the sense that we can make repeated tests on the same initial state to find out if the theoretical probability matches with the experimental probability. Nonetheless, my textbook says we can write $\langle f(t_f)|$$S$$|i(t_i) \rangle$ as $\langle f_i(t_f)$$S_f(t)$$|i_i(t_i) \rangle$, where $\langle f_i(t_f)|$ = $\langle U_i(t)f_i(0)|$, $|i_i(t_i) \rangle$ = $|U_i(t)i_i(0) \rangle$, and $S_f(t)$ = $U_f^{\dagger}(t)OU_f(t)$. $U_f(t)$ is the part of the time evolution operator with a free Hamiltonian. $U_i(t)$ is the part with an interacting Hamiltonian. This brings me to 3 questions:

*

*Why do we use $\langle f(t_f)|$$S$$|i(t_i) \rangle$ instead of $\langle f(t_f)|$$|i(t_i) \rangle$?

*What operator are we using in place of $O$, and why are we using that operator?

*$U_f(t)$ is "free", yet why do we then use the interacting Hamiltonian when calculating scattering processes?

 A: I will be trying to answer your questions in order:

*

*We are time-evolving the initial state and calculate its overlap with the final state. It wouldn't make much sense to take the overlap of two states in different time moments... This would yield a meaningless result. Instead, we know how the initial state will evolve (deterministic way) and we also know the final state of the scattering. The latter two things are our input. So, we time evolve and take the overlap. I do not know if this is re-assuring or not, but if not, please let me know.


*The operator we are using is the time evolution operator, or also known as the S-matrix operator. This operator is roughly speaking the exponential of the Hamiltonian, integrated with respect to time.


*Again,roughly speaking, if the interacting Hamiltonian commutes with the free part of the Hamiltonian, then the exponential can be written as two exponentials. The one (free) is absorbed into the definition of the interaction picture, whereas the other is responsible for the time evolution of states in the interaction picture. The remaining part is usually proportional to a small coupling constant and therefore perturbation theory makes sense. This is the main advantage of using the interaction picture. Asymptotically, states are time-evolved according to the free Hamiltonian, whereas the interacting one dictates the time evolution describing the interaction.
