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| Banks–Casher relation | |
|---|---|
| Name | Banks–Casher relation |
| Era | 20th century |
| Field | Theoretical physics |
| Notable works | Quantum chromodynamics |
Banks–Casher relation The Banks–Casher relation is a fundamental statement connecting the spectral density of the Dirac operator to spontaneous chiral symmetry breaking in quantum chromodynamics. It links properties of the Dirac spectrum in finite-volume Euclidean formulations to order parameters used across particle physics, lattice gauge theory, and random matrix theory, influencing studies at institutions such as CERN, Fermilab, SLAC National Accelerator Laboratory, Brookhaven National Laboratory.
The relation was introduced in the context of nonperturbative studies by authors working within frameworks associated with University of Oxford, Harvard University, Princeton University, and collaborative projects involving Institut des Hautes Études Scientifiques and Max Planck Society. It provided a bridge between earlier analytical approaches from researchers linked to Murray Gell-Mann, Steven Weinberg, Gerard 't Hooft, and numerical investigations at Los Alamos National Laboratory, shaping approaches used by groups at Yale University, MIT, University of Tokyo, University of California, Berkeley, and University of Cambridge. The idea rapidly influenced work on lattice formulations pioneered by scientists at Brookhaven National Laboratory and CERN, and fed into developments in Random matrix theory communities around Institute for Advanced Study and École Normale Supérieure.
The Banks–Casher statement expresses the chiral condensate in terms of the limit of the spectral density of the Euclidean Dirac operator at zero eigenvalue. This formulation is used in mathematical analyses found in monographs from Princeton University Press and lecture series delivered at Perimeter Institute, Kavli Institute for Theoretical Physics, Institute for Nuclear Theory, and International Centre for Theoretical Physics. It connects to spectral problems studied at Los Alamos National Laboratory and mathematical methods developed at Courant Institute and Cambridge University Press-published texts. The precise equality is often invoked in works associated with Stanford University and University of Chicago research groups.
Physically, the relation ties a macroscopic order parameter, the chiral condensate, to microscopic eigenvalue accumulation tied to instanton-like configurations studied by researchers at CERN and Institute for Advanced Study. It underlies phenomenology discussed in reviews from American Physical Society meetings and conferences at Niels Bohr Institute, KITP, and European Physical Society sessions. The interpretation influenced studies of hadron structure at Jefferson Lab, heavy-ion programs at RHIC associated with Brookhaven National Laboratory, and high-energy experiments at Large Hadron Collider.
Derivations employ methods from Euclidean functional integrals and spectral theory, leveraging techniques developed at IHES and formalism created by theorists affiliated with Stanford Linear Accelerator Center and University of California, San Diego. They often use ensembles and limits considered in Random matrix theory seminars at Institute for Advanced Study and proofs discussed in schools organized by Perimeter Institute and CERN summer programs. The derivation steps appear in theses from University of Oxford, University of Cambridge, and lecture notes connected to Caltech.
The Banks–Casher relation is applied on lattice simulations run on supercomputing centers like Oak Ridge National Laboratory, National Energy Research Scientific Computing Center, and in algorithm development at Los Alamos National Laboratory. It informs phenomenological models used by groups at Brookhaven National Laboratory and Lawrence Berkeley National Laboratory, and constrains effective theories referenced in textbooks from Cambridge University Press and Oxford University Press. Implications extend to studies of phase transitions explored in programs at CERN and GSI Helmholtz Centre for Heavy Ion Research.
Generalizations consider finite-volume, finite-temperature, and finite-density settings investigated at RIKEN, RIKEN BNL Research Center, and in collaborations between University of Tokyo and RIKEN. Extensions incorporate chemical potential studies associated with Institut Pasteur and adaptations in supersymmetric contexts developed by researchers at ETH Zurich and Max Planck Institute for Physics. Connections to Random matrix theory and universality classes have been pursued in workshops at Institute for Advanced Study and Perimeter Institute, while adaptations for other gauge groups appear in joint projects involving University of Illinois Urbana-Champaign and University of Washington.