Gell-Mann–Low

Gell-Mann–Low
Name	Murray Gell-Mann–Low
Field	Theoretical physics
Notable work	Gell-Mann–Low theorem; Gell-Mann–Low function

Gell-Mann–Low

The term denotes a foundational set of results in quantum field theory associated with Murray Gell-Mann and Kenneth A. Olive that relate interaction picture evolution, vacuum structure, and coupling-flow under scale changes in models studied at Institute for Advanced Study, Caltech, and Princeton University. The results influenced research at CERN, Fermilab, Imperial College London, Harvard University, and Stanford University and intersect with work by Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga, Lev Landau, and Kenneth Wilson on renormalization, perturbation theory, and scaling.

Overview and Historical Context

The development followed parallel advances during the postwar era in quantum electrodynamics at Columbia University, Cornell University, Yale University, and University of Cambridge and in many-body theory at Princeton University, Bell Labs, and Los Alamos National Laboratory. Influences include techniques from Wolfgang Pauli, Enrico Fermi, Hans Bethe, Erwin Schrödinger, and John von Neumann and responded to puzzles raised by the Landau pole problem and results of the Kadanoff blocking picture and early renormalization group ideas advanced by Kenneth Wilson and Leo Kadanoff. The formalism was rapidly adopted in calculations at Brookhaven National Laboratory, DESY, and within the Soviet Academy of Sciences community including work by Lev Petrovich Pitaevskii.

Gell-Mann–Low Theorem

The theorem establishes a mapping between the interacting vacuum and the noninteracting vacuum via an adiabatic switching operator first explored in papers circulated between Cambridge University Press and Physical Review. It employs asymptotic conditions similar to those used by Hans Bethe and Freeman Dyson in perturbation expansions, and it underpins scattering formulations developed at SLAC National Accelerator Laboratory, Rutherford Appleton Laboratory, and Max Planck Institute. The statement interrelates Dyson series, LSZ reduction as used by Heinz Lehmann, and the S-matrix methods of Mandelstam and Gell-Mann colleagues.

Gell-Mann–Low Function and Renormalization Group

The Gell-Mann–Low function encodes the beta-function flow of coupling constants under scale transformations central to Wilsonian renormalization and modern treatments in QCD, QED, and models studied at CERN and SLAC. It connects with computations by Gerard 't Hooft, Martinus Veltman, David Gross, Frank Wilczek, and David Politzer on asymptotic freedom and infrared behavior observed at LHC experiments and advocated in theoretical frameworks at Perimeter Institute. The formal beta-function links to fixed-point analyses used in Kosterlitz–Thouless transition studies and conformal methods associated with Alexander Belavin and Paul Ginsparg.

Applications in Quantum Field Theory

The construct is applied in renormalized perturbation theory in Scalar field theory computations, gauge-theory beta-function derivations as in QCD phenomenology at Fermilab and CERN, and many-body analyses relevant to condensed-matter problems studied at Bell Labs and Argonne National Laboratory. Its implications appear in effective-field-theory work pursued at Institute for Advanced Study and in lattice studies at CERN and Fermilab inspired by Kenneth Wilson and implemented on clusters curated by Los Alamos National Laboratory. The theorem informs treatments of anomalous dimensions used by Alexander Polyakov and flow-equation approaches favored by Franz Wegner.

Mathematical Formulation and Derivation

The derivation employs adiabatic switching functions, interaction-picture time evolution, and operator-valued distributions consistent with frameworks developed by Arthur Wightman and Eugene Wigner. It uses contour-deformation techniques reminiscent of methods by Henri Poincaré and Augustin-Louis Cauchy and algebraic structures developed by Israel Gelfand and John von Neumann. Treatments often reference renormalization prescriptions of Bogoliubov and Hepp and regularization schemes advanced by Gerard 't Hooft and Martinus Veltman, and canonical constructions used by Paul Dirac and Pascual Jordan.

Examples and Solvable Models

Concrete demonstrations appear in exactly solvable contexts such as the Schwinger model, the Thirring model, and certain two-dimensional conformal field theories examined by Alexander Zamolodchikov and John Cardy. Perturbative examples include one-loop beta-function calculations in QED by Hans Bethe-style renormalization and multiloop computations driven by techniques from Stanley Mandelstam and Gerard 't Hooft. Lattice implementations testing flow predictions have been carried out in collaborations at CERN and Fermilab and in numerical projects associated with Riken and RIKEN research centers.

Category:Quantum field theory