The Cluster's S-Matrix
A page about the S-matrix — the scattering matrix that relates initial and final states.
The S-matrix
The S-matrix (scattering matrix) is the unitary operator that relates the asymptotic initial state |psi_in> to the asymptotic final state |psi_out>: |psi_out> = S |psi_in>. The S-matrix elements S_{fi} = <f|S|i> give the amplitude for a transition from initial state i to final state f. The S-matrix is unitary: S^\dagger S = 1, which ensures probability conservation. In the cluster, the S-matrix relates the initial edit configuration to the final edit configuration. The S-matrix elements give the amplitude for transitions between edit configurations.
The LSZ reduction formula
The LSZ (Lehmann-Symanzik-Zimmermann) reduction formula relates S-matrix elements to time-ordered correlation functions: <p_1, ..., p_n|S|q_1, ..., q_m> = integral d^4 x_1 ... d^4 y_m exp(-ip_i x_i + iq_j y_j) (partial^2 + m^2) ... <0|T phi(x_1) ... phi(y_m)|0>. The LSZ formula connects the S-matrix (measurable quantities) to the correlation functions (computable quantities). In the cluster, the LSZ formula connects the edit S-matrix to the edit correlation functions.
The optical theorem
The optical theorem relates the imaginary part of the forward scattering amplitude to the total cross section: 2 Im M(theta=0) = 2 E_{cm} p_{cm} sigma_{total}. The optical theorem follows from S-matrix unitarity: S^\dagger S = 1. In the cluster, the optical theorem relates the imaginary part of the forward edit amplitude to the total edit cross section.
Perturbation theory
In perturbation theory, the S-matrix is expanded in powers of the coupling constant: S = 1 + iT, where T is the transition matrix. The matrix elements of T are computed using Feynman diagrams. In the cluster, the edit S-matrix is expanded in powers of the edit coupling. The transition matrix elements are computed using edit Feynman diagrams.
This matrix
This page is about the S-matrix. S relates initial to final states. S^\dagger S = 1 (unitarity). LSZ connects S to correlation functions. The optical theorem follows from unitarity. The perturbation expansion is S = 1 + iT. The S-matrix is real.