scattering theory In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunli ...

, a scattering channel is a

quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...

of the colliding system before or after the collision (

t \to \pm \infty

). The

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

spanned by the states before collision (in states) is equal to the space spanned by the states after collision (out states) which are both Fock spaces if there is a mass gap. This is the reason why the S matrix which maps the in states onto the out states must be

unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation In mathematics, a unitary representation of a grou ...

. The scattering channel are also called scattering

asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...

s. The Møller operators are mapping the scattering channels onto the corresponding states which are solution of the

Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...

taking the

interaction Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to: Science * Interaction hypothesis, a theory of second language acquisition * Interaction (statistics) * Interaction ...

Hamiltonian into account. The Møller operators are isometric.

See also