S matrix

The S matrix or scatter matrix describes the scatter amplitudes in the scattering theory of quantum mechanics and quantum field theory . It was introduced in 1937 by John Archibald Wheeler in nuclear physics and independently of Wheeler in 1943 by Werner Heisenberg in the quantum theories of elementary particles.

The squares of the absolute values of the elements of the S-matrix give the probability for an initial and a final state that the initial state changes into the final state when the scattering occurs . The nontrivial part of the S-matrix is called the T-matrix or transfer matrix .

The axiomatic S-matrix theory, a sub-area of axiomatic quantum field theory , attempts central properties of the S-matrix, such as B. their unitarity , to be axiomatic . An early success of axiomatic considerations is the LSZ reduction formula found by Harry Lehmann , Kurt Symanzik and Wolfhart Zimmermann and named after the first letters of their surnames . This states that the S matrix of a quantum field theory can be calculated from the time-ordered n-point functions .

In the 1960s, people distrusted the applicability of conventional quantum field theory in the theory of strong interaction . Here the S-matrix theory was considered an alternative and was therefore a very active field of research, especially in Geoffrey Chew's school .

The formulation of an S matrix is only possible if the existence of non-interacting asymptotic states or fields is assumed before and after the scattering process .

Because in quantum field theory on curved spacetime the formulation of a Fock space of asymptotic states at very early and late times is generally not possible, alternatives to the formulation of an S matrix are being researched for these cases.

The definition of an S matrix is also impossible in conformal quantum field theories , because here the definition of asymptotic fields and states is impossible; This is because points that are far away can be mapped into nearby points by dilation .

Individual evidence

↑ The observable quantities in the theory of elementary particles 1,2 , Zeitschrift für Physik Vol. 120, 1943, pp. 513, 673, Vol. 123, 1944, p. 93
^ Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 60 (English).

[1] The observable quantities in the theory of elementary particles 1,2 , Zeitschrift für Physik Vol. 120, 1943, pp. 513, 673, Vol. 123, 1944, p. 93

[2] Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 60 (English).