Definition of the S-matrix

Question

When I think about scattering process, I reach to slightly another definition to the S-matrix. because I understand my reasoning I hope someone could refine it to a correct one so that I can have a clear idea about the S-matrix.

In QM if $A$ be an observable with eigenstates $|A_\alpha\rangle$ you can set system at time $t_1$ to state $|A_1\rangle$. system at time $t_2$ will be at $e^{-ih(t_2-t_1)}|A_1\rangle$. now if you measure $A$ the probability of collapsing to $A_2$ is $\left<A_2|e^{-iH(t_2-t_1)}|A_1\right>$.

Now in scattering experiment $t_1=-\infty$, $t_2=+\infty$ and $A=H_0$. So $$ S_{21}= \lim_{t_i=\mp} \left<\Phi_2|e^{-iH(t_2-t_1)}|\Phi_1\right> $$ where $\Phi_\alpha$s are eigenstates of $H_0$. but conventional S-matrix reads: (Weinberg Vol.1 P.114) $$ S_{21}= \lim_{t_i=\mp}\left<\Phi_2|e^{iH_0t_2}e^{-iH(t_2-t_1)} e^{-iH_0t_1} |\Phi_1\right>. $$ Is my reasoning refinable?

Answer 1

I consider a related question but where I consider a pair of Schrodinger equations for unperturbed and perturbed states. The occurrence of time is an inconvenience, so instead work with the Hamiltonians and Green' function. This is much the same, in that the S-matrix is concerned with the asymptotic scattering states and their differences in eigenspectrum.

The ordered S-matrix is constructed to that a set of vertices, or particles, transition. Suppose we have the state $$ |\phi\rangle = |p_1, p_2, \dots, p_n\rangle $$ that transitions to the state $$ |\phi'\rangle = |q_1, q_2, \dots, q_n\rangle. $$ The evaluation of the overlap of this state with the S-matrix $$ \langle\phi |S|\phi'\rangle = \langle p_1, p_2, \dots, p_n|S|q_1, q_2, \dots, q_n\rangle. $$ This operation is unitary and a part of this is that $S = 2\pi iT$ for $T$ the transition matrix

The transition matrix can be evaluated explicitly. The S-matrix describes the transition of a state that obeys $H|\phi\rangle = E\phi\rangle$ to $H'|\phi'\rangle = E\phi'\rangle$. The state $|\phi'\rangle = |\phi\rangle + V|\phi'\rangle$, where $V$ is the perturbing or potential that induces scattering. We are looking at a process that propagates a scattering influence, and propose the Green's function $(E - H)G = 1$ for $G = G(x,x')$. We can then see $$ (H' - E)|\phi'\rangle = (H' - E - V)|\phi'\rangle + (E - H)GV|\phi'\rangle $$ $$ (E - E)(|\phi'\rangle - GV|\phi'\rangle) = (H - E)|\phi\rangle = 0 $$ makes this consistent. The relevant term is $ |\phi'\rangle - GV|\phi'\rangle$ that gives the scattered portion of the wave $|psi\rangle = |\phi'\rangle - |\phi\rangle$ $ = GV|\phi'\rangle$. The transition matrix is then found by the expansion of the state $|\phi'\rangle$ $$ |\phi'\rangle = (1 + GV + GVGV + \dots)|\phi\rangle = (1 - GV)^{-1}|\phi\rangle $$ so the transition matrix $$ T = \langle\phi|T|\phi\rangle = \langle\phi|V|\phi'\rangle = \langle\phi|V(1 - GV)^{-1}|\phi\rangle. $$ So the transition matrix is $T = V(1 - GV)^{-1}$.

The unitarity of the S-matrix can be verified by computing $SS^\dagger$. $$ SS^\dagger = (1 + 2\pi iT)(1 - 2\pi iT^\dagger) = 1 + 2\pi i(T - T^\dagger) + 4\pi^2TT^\dagger + O(T^3) $$ The first term $T - T^\dagger = T^\dagger(G^\dagger - G)T$ where for $G = (E - H - i\epsilon)^{-1}$ gives $T - T^\dagger = 2\pi TT^\dagger$ and the unitarity of the S-matrix is proven.