Smatrices

An S-matrix, or scattering matrix, is an operator that encodes the probabilities of transitions between physical states in scattering experiments. In relativistic quantum mechanics and quantum field theory, it relates asymptotic in-states, defined long before an interaction, to asymptotic out-states, after the interaction. Its elements, called S-matrix elements or scattering amplitudes, determine the likelihood of observing specific final particles given a particular initial configuration.

The S-matrix is unitary, reflecting probability conservation, S^† S = S S^† = I. In practice one constructs in- and out-states from free or asymptotic eigenstates of the Hamiltonian, and the S-matrix connects them. The LSZ reduction formula relates S-matrix elements to time-ordered correlation functions of quantum fields, providing a bridge to field theory calculations.

Key properties include Lorentz invariance, analyticity in complex energy and momentum variables, unitarity constraints such as the optical theorem, and crossing symmetry, which relates amplitudes for different processes by analytic continuation. In many models, especially two-dimensional integrable theories, the S-matrix factorizes and satisfies the Yang-Baxter equation.

Historical context: the S-matrix program, developed in the 1950s–1960s, aimed to reconstruct strong-interaction physics from scattering data without assuming a detailed Lagrangian. With the rise of quantum field theory, the S-matrix became a central observable object rather than the sole focus of theory. Today, S-matrix calculations are essential in collider phenomenology, and exact S-matrices are known in some exactly solvable models; string theory also yields perturbative S-matrices from worldsheet analyses.

Limitations include reliance on well-defined asymptotic states and short-range interactions; long-range forces and certain massless theories complicate the construction. Nevertheless, the S-matrix remains a compact and widely used framework for summarizing and predicting outcomes of scattering experiments across particle and nuclear physics.