Scattering theory

Scattering theory
KCVelaga · CC BY-SA 4.0 · source
Name	Scattering theory
Field	Mathematical physics
Notable people	Isaac Newton, Erwin Schrödinger, Paul Dirac, Max Born, John von Neumann, Enrico Fermi, Lev Landau, Wolfgang Pauli, Werner Heisenberg, Eugene Wigner, Richard Feynman, Julian Schwinger, Martin Gutzwiller, David Bohm, Murray Gell-Mann, Victor Weisskopf, Hans Bethe, Stanley Mandelstam, Mark Kac, Gérard Mourre, Tullio Regge, Simon Donaldson, Michael Berry, L. D. Faddeev, Mikhail Birman, Israel Gel'fand, Paul Dirac, John Wheeler, Hendrik Casimir, Nicolaas Bloembergen, Peter Debye, Hendrik Lorentz, J. J. Thomson, Lord Rayleigh]

Scattering theory is the study of how waves, particles, and fields deviate from straight-line propagation due to interactions with targets, potentials, or media. It unifies concepts across Isaac Newton's particle collisions, Lord Rayleigh's acoustic scattering, Max Born's quantum perturbation, and modern treatments in Enrico Fermi's nuclear physics and Richard Feynman's diagrammatic methods. The subject connects mathematical analysis, experimental methods, and computational simulation across many institutions such as CERN, MIT, Caltech, and Stanford University.

Overview

Scattering theory encompasses frameworks developed by figures like Erwin Schrödinger, Paul Dirac, and John von Neumann and manifested in formal tools associated with Eugene Wigner, Julian Schwinger, and Martin Gutzwiller. It appears in contexts ranging from J. J. Thomson's atomic experiments to Hans Bethe's nuclear cross sections and from Michael Berry's phase phenomena to Tullio Regge's complex angular momentum. Core goals include predicting differential cross sections, phase shifts, S-matrix elements, and resonances studied at facilities such as Fermilab, SLAC National Accelerator Laboratory, and DESY.

Classical scattering

Classical scattering techniques trace to Isaac Newton and were advanced in optics by Lord Rayleigh and Hendrik Lorentz, with key experimental systems examined at Bell Labs and Bell Telephone Laboratories. Methods include ray tracing used in Royal Society reports, Mie theory developed by Gustav Mie for electromagnetic scattering, and acoustic approaches refined by Peter Debye and Nicolaas Bloembergen. Classical regimes analyze deflection angles, impact parameters, and scattering lengths in contexts like Battle of Trafalgar-era artillery modeling (historical simulations), atmospheric scattering probed by Voyager and Pioneer missions, and remote sensing performed by NASA.

Quantum scattering

Quantum scattering builds on the Schrödinger equation and formalisms by Paul Dirac and Wolfgang Pauli. Central objects include the S-matrix conceptualized by Eugene Wigner and perturbative series developed in Richard Feynman's quantum electrodynamics and Julian Schwinger's operator approach. Techniques like partial-wave analysis used by Lev Landau and Hans Bethe, Lippmann–Schwinger equations associated with L. D. Faddeev, and renormalization strategies from Murray Gell-Mann and Gerard 't Hooft are crucial. Quantum scattering underpins experiments at CERN, Brookhaven National Laboratory, and Los Alamos National Laboratory, where resonances, bound states, and scattering amplitudes are probed.

Mathematical formulations and methods

Rigorous foundations draw on operator theory from John von Neumann and spectral analysis by Israel Gel'fand and Mark Kac, with modern contributions by Gérard Mourre and Mikhail Birman. The S-matrix framework links to analytic continuation explored by Tullio Regge and to dispersion relations proven with techniques related to work by Stanley Mandelstam. Functional analysis, complex analysis methods used by Andrey Kolmogorov-era mathematics, and inverse scattering problems studied by Peter Lax and Boris Kayser provide structure. Integral equations like the Lippmann–Schwinger and Fredholm equations, Green's function constructions from Hendrik Casimir-type calculations, and spectral shift functions associated with Mark Krein are central.

Computational and numerical techniques

Numerical scattering leverages algorithms and software developed at institutions including Lawrence Berkeley National Laboratory, Argonne National Laboratory, Los Alamos National Laboratory, and computational groups at Princeton University and Harvard University. Methods include finite element approaches used in Stanford University applied programs, boundary element methods refined by C. B. Moler-style teams, and Monte Carlo techniques pioneered in Los Alamos research by Stanislaw Ulam and Nicholas Metropolis. High-performance computing projects at Oak Ridge National Laboratory and NERSC enable lattice scattering simulations, while optimization frameworks from IBM and Intel accelerate ab initio scattering codes. Machine learning methods from Google and DeepMind are increasingly applied to inverse scattering and phase retrieval.

Applications and experimental methods

Scattering is applied across fields: particle physics at CERN and Fermilab; nuclear physics at TRIUMF and Rutherford Appleton Laboratory; condensed matter at Bell Labs, Max Planck Institute for Solid State Research, and Argonne; astrophysical observations by Hubble Space Telescope and Chandra X-ray Observatory; and medical imaging developed at Mayo Clinic and Johns Hopkins University. Experimental tools include particle accelerators like Large Hadron Collider, synchrotrons such as European Synchrotron Radiation Facility, neutron sources like Institut Laue–Langevin, electron microscopes from Hitachi and JEOL, and laser facilities exemplified by Lawrence Livermore National Laboratory's National Ignition Facility. Measurement techniques range from Rutherford backscattering developed by Ernest Rutherford to angle-resolved photoemission spectroscopy advanced at Brookhaven.

Advanced topics and extensions

Advanced areas connect to semiclassical trace formulas by Martin Gutzwiller, topological scattering influenced by Michael Berry, integrable models studied by L. D. Faddeev, and non-Hermitian scattering linked to Niels Bohr-inspired complex potentials. Extensions include inverse scattering transforms used in soliton theory by Peter Lax and Gardner Greene Kruskal Miura-style groups, multichannel and coupled-channel methods employed in Hans Bethe-era nuclear theory, and quantum field theoretic scattering incorporating axiomatic results from Gerard 't Hooft and Alexander Polyakov. Current research involves collaborations across National Institutes of Health for biomedical scattering, European Organization for Nuclear Research for collider phenomenology, and interdisciplinary projects at California Institute of Technology and Imperial College London.

Category:Mathematical physics