S-matrix

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 (QFT). U ( Λ , a ) Ψ p 1 σ 1 n 1 ; p 2 σ 2 n 2 ⋯ = e − i a μ ( ( Λ p 1 ) μ + ( Λ p 2 ) μ + ⋯ ) ( Λ p 1 ) 0 ( Λ p 2 ) 0 ⋯ p 1 0 p 2 0 ⋯ ∑ σ 1 ′ σ 2 ′ ⋯ D σ 1 ′ σ 1 ( j 1 ) ( W ( Λ , p 1 ) ) D σ 2 ′ σ 2 ( j 2 ) ( W ( Λ , p 2 ) ) ⋯ Ψ Λ p 1 σ 1 ′ n 1 ; Λ p 2 σ 2 ′ n 2 ⋯ , {displaystyle U(Lambda ,a)Psi _{p_{1}sigma _{1}n_{1};p_{2}sigma _{2}n_{2}cdots }=e^{-ia_{mu }((Lambda p_{1})^{mu }+(Lambda p_{2})^{mu }+cdots )}{sqrt {frac {(Lambda p_{1})^{0}(Lambda p_{2})^{0}cdots }{p_{1}^{0}p_{2}^{0}cdots }}}sum _{sigma _{1}'sigma _{2}'cdots }D_{sigma _{1}'sigma _{1}}^{(j_{1})}(W(Lambda ,p_{1}))D_{sigma _{2}'sigma _{2}}^{(j_{2})}(W(Lambda ,p_{2}))cdots Psi _{Lambda p_{1}sigma _{1}'n_{1};Lambda p_{2}sigma _{2}'n_{2}cdots },}     (1) 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 (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 (non-interacting) if it transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation (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 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 Poincaré 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. 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'. In this paper Wheeler introduced a scattering matrix – a unitary matrix of coefficients connecting 'the asymptotic behaviour of an arbitrary particular solution with that of solutions of a standard form',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. Today, however, exact S-matrix results are a crowning achievement of conformal field theory, integrable systems, and several further areas of quantum field theory and string theory. S-matrices are not substitutes for a field-theoretic treatment, but rather, complement the end results of such. In high-energy particle physics we are interested in computing the probability for different outcomes in scattering experiments. These experiments can be broken down into three stages: The process by which the incoming particles are transformed (through their interaction) into the outgoing particles is called scattering. For particle physics, a physical theory of these processes must be able to compute the probability for different outgoing particles when different incoming particles collide with different energies.