# Definition of the S-matrix

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?

• What's your "argument"? You just omitted the $H_0$ factors, the reason for which you should find e.g. in the very Weinberg book you quote. What's the question here? May 30, 2016 at 19:04
• Weinberg's approach to defining S-matrix is quite different from what I wrote here. when I try to consider situation more reasonable for myself the very $H_0$ factors disappear! May 30, 2016 at 19:25
• To get the proper QM analogon, you need to consider the Dirac/interaction picture, not the Schrödinger one you seem to be assuming implicitly here. May 30, 2016 at 19:45
• yes I'm trying the Schroedinger picture. but transition probability what have to do with pictures? Weinberg's discussion is in Heisenberg picture and Greiner's (Vol.5 P.216) is in Dirac picture and S-matrix are the same. then why same S-matrix can't be derived in Schroedinger picture? May 30, 2016 at 20:31
• @moshtaba: see my answer to physics.stackexchange.com/q/336599 Feb 25, 2018 at 3:14

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.