S-matrix

In <a href="/facts/Physics/a7aH2y08">physics</a>, the S-matrix or scattering matrix is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> that relates the initial state and the final state of a physical system undergoing a <a href="/facts/Scattering/nMou7lEH">scattering process</a>. It is used in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, <a href="/facts/Scattering_theory/79Nx76Ub">scattering theory</a> and <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> (QFT).
More formally, in the context of QFT, the S-matrix is defined as the <a href="/facts/Unitary_matrix/8fsxv3KD">unitary matrix</a> connecting sets of asymptotically free particle states (the in-states and the out-states) in the <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> of physical states: a multi-particle state is said to be free (or non-interacting) if it <a href="/facts/Representation_theory_of_the_Lorentz_group/XJRuPVkL">transforms</a> under <a href="/facts/Lorentz_transformation/jotJL06G">Lorentz transformations</a> as a <a href="/facts/Tensor_product/7K3pR1ap">tensor product</a>, 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 (<a href="/facts/Spacetime/uRIN1x4b">spacetime</a>) that is asymptotically solvable and has no <a href="/facts/Event_horizon/KXH2ptnI">event horizons</a>, it has a simple form in the case of the <a href="/facts/Minkowski_space/vDkfkQJn">Minkowski space</a>. In this special case, the Hilbert space is a space of irreducible <a href="/facts/Unitary_representation/W1XRatG8">unitary representations</a> of the <a href="/facts/Inhomogeneous_space/YGebqcJA">inhomogeneous</a> <a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a> (the <a href="/facts/Poincar%25C3%25A9_group/3IerRYG1">Poincaré group</a>); the S-matrix is the <a href="/facts/Evolution_operator/8XxTc0fN">evolution operator</a> between 
 
 
 
 t
 =
 −
 ∞
 
 
 {\displaystyle t=-\infty }
 
 (the distant past), and 
 
 
 
 t
 =
 +
 ∞
 
 
 {\displaystyle t=+\infty }
 
 (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 <a href="/facts/Mass_gap/IhyTCS2v">mass gap</a>, the <a href="/facts/Quantum_state/7jbxd7Dx">state</a> in the asymptotic past and in the asymptotic future are both described by <a href="/facts/Fock_space/I2PYLlhl">Fock spaces</a>.

S-matrix open-in-new

S-matrix