Editing S-matrix (section)

{{short description|Matrix representing the effect of scattering on a physical system}}
{{redirect|Scattering matrix|the meaning in linear electrical networks|Scattering parameters}}
{{For|the 1960s approach to particle physics|S-matrix theory}}
{{Italic title|noerror|string=S}}

In [[physics]], the '''''S''-matrix''' or '''scattering matrix''' is a [[Matrix (mathematics)|matrix]] that relates the initial state and the final state of a physical system undergoing a [[scattering|scattering process]]. It is used in [[quantum mechanics]], [[scattering theory]] and [[quantum field theory]] (QFT).

More formally, in the context of QFT, the ''S''-matrix is defined as the [[unitary matrix]] connecting sets of asymptotically free particle states (the ''in-states'' and the ''out-states'') in the [[Hilbert space]] of physical states: a multi-particle state is said to be ''free'' (or non-interacting) if it [[Representation theory of the Lorentz group|transforms]] under [[Lorentz transformation]]s as a [[tensor product]], or ''direct product'' in physics parlance, of ''one-particle states'' as prescribed by equation {{EquationNote|1|(1)}} below. ''Asymptotically free'' then means that the state has this appearance in either the distant past or the distant future.

While the ''S''-matrix may be defined for any background ([[spacetime]]) that is asymptotically solvable and has no [[event horizon]]s, 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 representation]]s of the [[inhomogeneous space|inhomogeneous]] [[Lorentz group]] (the [[Poincaré group]]); the ''S''-matrix is the [[evolution operator]] between <math>t= - \infty </math> (the distant past), and <math>t= + \infty </math> (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 [[quantum state|state]] in the asymptotic past and in the asymptotic future are both described by [[Fock space]]s.