S-matrix

S-matrix
Name	S-matrix
Field	Quantum field theory, Scattering theory
Inventor	John Archibald Wheeler, Werner Heisenberg
Year	1937, 1943
Related concepts	LSZ reduction formula, Cross section (physics), Feynman diagram, Renormalization group

S-matrix. In theoretical physics, the S-matrix, or scattering matrix, is a fundamental object that encodes the probabilities for all possible outcomes of a scattering process between particles. It connects the initial state of non-interacting particles in the distant past to the final state of non-interacting particles in the distant future. The principles of unitarity and analyticity impose powerful constraints on its mathematical structure, making it a central tool in quantum field theory and high-energy physics.

Definition and basic properties

The S-matrix is formally defined as the unitary operator connecting the in states and out states in the interaction picture of quantum mechanics. Its matrix elements, denoted ⟨f|S|i⟩, give the probability amplitude for an initial state |i⟩ to evolve into a final state |f⟩. A fundamental property is that it can be decomposed as S = I + iT, where I is the identity matrix, representing no interaction, and the transition matrix T contains the nontrivial scattering information. Key constraints from quantum mechanics include unitarity, which ensures conservation of probability, and crossing symmetry, a property derived from the analytic structure of its elements. The squares of its magnitudes are directly related to measurable quantities like the differential cross section observed in experiments at facilities like CERN.

Historical development

The concept was first introduced in nuclear physics by John Archibald Wheeler in 1937 to describe compound nucleus reactions. It was developed independently and more fully by Werner Heisenberg in 1943 as part of his program to create a theory based solely on observable quantities, avoiding the divergences of quantum field theory. The modern formulation was profoundly advanced in the late 1940s and 1950s by figures like Julian Schwinger and Freeman Dyson, and particularly through the work of Murray Gell-Mann and Marvin L. Goldberger, who established the dispersion relations for the S-matrix. The S-matrix theory program, championed by Geoffrey Chew and the Berkeley group, sought to bypass quantum field theory entirely during the 1960s, emphasizing analyticity and bootstrap principles.

Mathematical formulation

In the framework of quantum field theory, the S-matrix is expressed using the Dyson series expansion in terms of the interaction Hamiltonian within the interaction picture. This expansion is intimately tied to Wick's theorem and is represented pictorially by Feynman diagrams, where each diagram corresponds to a specific integral contribution to the amplitude. The LSZ reduction formula, developed by Harry Lehmann, Kurt Symanzik, and Wolfhart Zimmermann, provides a rigorous link between the S-matrix elements and the correlation functions of the underlying quantum field. For theories like quantum chromodynamics, the evaluation of these matrix elements involves complex techniques of perturbation theory and renormalization.

Relation to scattering theory

In non-relativistic scattering theory, as formalized by Lippmann–Schwinger equation, the S-matrix is related to the scattering operator and the T-matrix. The central observable, the scattering amplitude, is directly extracted from the S-matrix and determines the angular distribution of scattered particles. The relationship is further clarified through the optical theorem, which connects the imaginary part of the forward scattering amplitude to the total cross section (physics). This formalism is essential for analyzing data from particle colliders like the Large Hadron Collider and earlier machines such as the Stanford Linear Accelerator Center.

Applications in particle physics

The primary application is calculating probabilities for processes like electron-proton scattering, annihilation events in positron-electron colliders, and particle production in hadron colliders. It is indispensable for determining Standard Model predictions for processes involving the Higgs boson, top quark, and gauge boson interactions. Computations using tools like the Mandelstam variables allow physicists to compare S-matrix predictions with experimental results from DESY, Fermilab, and Brookhaven National Laboratory. Furthermore, the study of its analytic properties has led to insights into fundamental principles like causality and the spin-statistics theorem.

Generalizations and related concepts

Generalizations include the notion of the S-matrix in string theory, where it describes the scattering of string (physics) excitations and is expected to be finite. In quantum gravity and studies of black hole thermodynamics, the concept of a gravitational S-matrix is explored, grappling with issues like the black hole information paradox. Related formal constructs include the transfer matrix used in statistical mechanics and the R-matrix in the context of Yang–Baxter equation and integrable systems. The modern amplituhedron geometry, introduced by Nima Arkani-Hamed, provides a novel geometric formulation for calculating S-matrix elements in certain supersymmetric theories.

Category:Scattering theory Category:Quantum field theory Category:Theoretical physics