  in physics, the s-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. it is used in quantum mechanics, scattering theory and quantum field theory. more formally, the s-matrix is defined as the unitary matrix connecting asymptotic particle states in the hilbert space of physical states (scattering channels). while the s-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no event horizons, it has a simple form in the case of the minkowski space. in this special case, the hilbert space is a space of irreducible unitary representations of the inhomogeneous lorentz group (the poincarÃ© group); the s-matrix is the evolution operator between time equal to minus infinity (the distant past), and time equal to plus infinity (the distant future). it is defined only in the limit of zero energy density (or infinite particle separation distance). it can be shown that if a quantum field theory in minkowski space has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by fock spaces. ==history== the s-matrix was first introduced by john archibald wheeler in the 1937 paper "'on the mathematical description of light nuclei by the method of resonating group structure'".john archibald wheeler, 'on the mathematical description of light nuclei by the method. of resonating group structure' phys. rev. 52, 1107â1122 (1937) in this paper wheeler introduced a scattering matrix â a unitary matrix of coefficients connecting "the asymptotic behaviour of an arbitrary particular solution [of the integral equations] with that of solutions of a standard form",jagdish mehra, helmut rechenberg, the historical development of quantum theory (pages 990 and 1031) springer, 2001 isbn 0-387-95086-9, isbn 978-0-387-95086-0 but did not develop it fully. in the 1940s, werner heisenberg developed, independently, and substantiated the idea of the s-matrix. because of the problematic divergences present in quantum field theory at that time, heisenberg was motivated to isolate the essential features of the theory that would not be affected by future changes as the theory developed. in doing so, he was led to introduce a unitary "characteristic" s-matrix. after world war ii, the clout of heisenberg and his attachment to the s-matrix approach may well have slowed down development of alternative approaches, such as quantum field theory and the closer study of sub-hadronic physics for a decade or more, at least in europe: "pre
Mathematical_physics Matrices Quantum_field_theory Scattering_theory