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| Coulomb gauge | |
|---|---|
| Name | Coulomb gauge |
| Field | Electromagnetism |
| Introduced | 19th century |
Coulomb gauge is a specific choice of gauge fixing in classical and quantum electrodynamics that imposes a condition on the vector potential to simplify the equations of motion for the electromagnetic field. It is widely used in analyses of static and quasistatic problems, radiation theory, and canonical quantization, and connects to formulations used by many physicists and institutions involved in theoretical physics. The gauge is notable for its instantaneous scalar potential behavior and its role in canonical commutation relations in quantization approaches developed at several universities and laboratories.
The Coulomb gauge is defined by the transversality condition ∇·A = 0, where A is the vector potential appearing in Maxwell's equations and in the Lagrangians used by researchers at University of Cambridge, Massachusetts Institute of Technology, Princeton University, and École Normale Supérieure. In the presence of a scalar potential φ, the electromagnetic fields E and B are expressed as E = −∇φ − ∂A/∂t and B = ∇×A, tying the condition to formulations used by practitioners at Harvard University, California Institute of Technology, and Stanford University. The condition leads to a Poisson equation for φ, ∇^2φ = −ρ/ε0, where the charge density ρ appears in contexts studied at Imperial College London, University of Oxford, and Max Planck Institute for Physics.
Imposing ∇·A = 0 eliminates longitudinal components of A, leaving only transverse degrees of freedom analogous to the transverse photon polarizations emphasized by scholars at CERN, Brookhaven National Laboratory, and Fermilab. The Coulomb gauge is a partial gauge fixing that still admits residual gauge transformations satisfying Laplace’s equation, a property discussed in lectures and seminars at Yale University, University of Chicago, and University of Tokyo. Because φ is determined instantaneously by ρ via a Poisson equation, the Coulomb gauge manifests an action-at-a-distance feature reminiscent of treatments by theorists at Rutherford Appleton Laboratory, Los Alamos National Laboratory, and RIKEN. The condition affects canonical momenta and constraints in Hamiltonian formulations explored at University of California, Berkeley, Columbia University, and University of Illinois Urbana-Champaign.
In static and quasistatic problems, practitioners at General Electric Research Laboratory, Bell Labs, and Siemens often adopt the Coulomb gauge to simplify boundary-value problems involving conductors and dielectrics. Radiation reaction, multipole expansions, and antenna theory treated in textbooks authored by professors at University of Michigan, University of Pennsylvania, and Brown University frequently exploit the Coulomb gauge to separate near-field and far-field contributions. Computational electromagnetics groups at Lawrence Berkeley National Laboratory and National Institute of Standards and Technology use the gauge when solving Poisson and Helmholtz equations for φ and A in simulations related to accelerator physics at SLAC National Accelerator Laboratory and plasma studies at Princeton Plasma Physics Laboratory.
Canonical quantization in the Coulomb gauge isolates the two physical photon polarizations, a procedure developed in contexts associated with University of Cambridge, ETH Zurich, and Moscow State University. The instantaneous Coulomb potential appears as an interaction term in nonrelativistic reductions used in atomic and molecular physics at University of Toronto, University of British Columbia, and University of Sydney. Treatment of infrared divergences, asymptotic fields, and bound-state problems in quantum electrodynamics has been pursued by research groups at Institute for Advanced Study, Perimeter Institute, and Niels Bohr Institute. In non-Abelian gauge theories the analogues of the Coulomb gauge raise Gribov ambiguity issues studied by researchers at Bogolyubov Institute for Theoretical Physics, Ludwig Maximilian University of Munich, and SISSA.
Comparisons between the Coulomb gauge and gauges such as the Lorenz gauge, radiation gauge, and axial gauge are common in courses and reviews originating from University of Cambridge, King's College London, and University of Paris. Unlike the Lorenz gauge condition used by practitioners at Cornell University and Johns Hopkins University, which preserves manifest Lorentz covariance, the Coulomb gauge sacrifices manifest covariance to simplify the separation of transverse modes—an approach echoed in works from University of Göttingen and University of Bonn. Connections to temporal gauge and light-cone gauge formulations studied at Rutgers University and University of Maryland illuminate trade-offs between locality, causality, and computational convenience.
Numerical implementations of the Coulomb gauge require solving Poisson equations for φ and ensuring ∇·A = 0 via projection methods used by computational groups at Argonne National Laboratory, Sandia National Laboratories, and Oak Ridge National Laboratory. Fast Fourier transform techniques and finite-element methods for enforcing transversality are applied in software developed at Lawrence Livermore National Laboratory, National Renewable Energy Laboratory, and industrial R&D labs such as Siemens and Schlumberger. Gauge-fixing algorithms and preconditioning strategies used in lattice gauge theory simulations by teams at Brookhaven National Laboratory and RIKEN adapt Coulomb-like conditions to discrete meshes and periodic boundary conditions.
The gauge bears conceptual roots in electrostatics and in the work of scientists linked to institutions such as École Polytechnique and Sorbonne University. Important contributors to formalizing gauge conditions and canonical quantization include researchers affiliated with University of Chicago, Princeton University, Harvard University, and Moscow State University. Later developments addressing quantization, gauge ambiguities, and computational approaches involved participants from CERN, Institute for Advanced Study, Max Planck Institute for Physics, and Perimeter Institute.