Transition probability with Møller operators

Question

I’ve encountered the following arguments multiple times in notes and textbooks:

$$ \lim_{t\to -\infty} || U(t)^\dagger U_0(t) |\phi_{\text{in}}(0)\rangle - |\Psi(0)\rangle || = 0 \implies |\Psi(0)\rangle = \Omega_+ |\phi_{\text{in}}(0)\rangle $$

$$ \lim_{t\to \infty} || U(t)^\dagger U_0(t) |\phi_{\text{out}}(0)\rangle - |\Psi(0)\rangle || = 0 \implies |\Psi(0)\rangle = \Omega_- |\phi_{\text{out}}(0)\rangle $$

Here:

• $U(t)$ is the time evolution operator for the full interacting system.
• $U_0(t)$ is the time evolution operator for the free (non-interacting) system.
• $|\phi_{\text{in}}(0)\rangle$ is the "in" state at $t = 0$, representing the initial state of a particle in the free theory.
• $|\Psi(0)\rangle$ is the interacting state at $t = 0$, representing the actual state of the system.
• $\Omega_-$ is the Møller operator.

While I agree with this reasoning, there is something puzzling when computing the transition rate:

$$ \langle \phi_{\text{out}} | \Omega_-^\dagger \Omega_+ | \phi_{\text{in}} \rangle $$

From the above arguments, this expression should be equivalent to:

$$ \langle \Psi(0) | \Psi(0) \rangle $$

However, shouldn’t that always equal unity due to normalization ($\langle \Psi | \Psi \rangle = 1$)? This seems contradictory when interpreting it as a transition rate. How does one resolve this?

Answer 1

If you want to compute the transition rate (or scattering matrix) for an arbitrary initial state, described by the in-asymptote $|\phi \rangle $ to transition into an arbitrary final state, described by the out-asymptote $|\chi \rangle$, you need to use the Møller operators individually on both states to relate them to $t=0$ and take their inner product.

The actual state at $t=0$ corresponding to the in-asymptotic state is given by
$$ |\phi +\rangle = \Omega_+ |\phi \rangle,$$ and similarly for the out-asymptotic state is $$ |\chi -\rangle = \Omega_- |\chi \rangle,$$ and so the transition probability is given by the square of the inner product $$ w(\phi \rightarrow \chi) = |\langle \chi - | \psi + \rangle|^2 = |\langle \chi | \Omega^\dagger_- \Omega_+ |\phi \rangle|^2,$$ where $S= \Omega^\dagger_-\Omega_+$ is defined as the scattering operator.

Ref: Scattering Theory: The Quantum Theory of Nonrelativistic Collisions by John R. Taylor, page 34-35.