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| Lorenz gauge | |
|---|---|
| Name | Lorenz gauge |
| Field | Electrodynamics |
| Introduced | 1867 |
| Introduced by | Ludvig Lorenz |
| Related | Maxwell's equations, electromagnetic potentials, gauge theory |
Lorenz gauge is a condition imposed on electromagnetic potentials that simplifies Maxwell's equations and preserves manifest relativistic covariance. It constrains the four-potential so that the inhomogeneous wave equations decouple into simpler forms, facilitating solutions in classical electrodynamics, radiation theory, and quantum electrodynamics. The condition is central to formulations used by physicists working with James Clerk Maxwell's theory, and it appears in treatments by Hendrik Antoon Lorentz and Paul Dirac in modern field theory.
The Lorenz gauge is defined by the four-divergence condition on the electromagnetic four-potential A^μ used in formulations derived from Maxwell's equations, specifically requiring ∂_μ A^μ = 0, which in SI units reads ∇·A + (1/c^2) ∂φ/∂t = 0. In many texts the four-potential appears alongside the electromagnetic field tensor F^μν introduced in relativistic presentations by Hendrik Antoon Lorentz and formalized in treatments by Albert Einstein and Hermann Minkowski. Imposing the Lorenz gauge reduces the coupled system to inhomogeneous d'Alembert equations solved by Green's functions associated with the wave operator used in computations by Oliver Heaviside and analysts following Lord Rayleigh.
The condition bears the name of Ludvig Lorenz, who published a formulation in 1867 predating the widespread use of James Clerk Maxwell's potential formalism in some circles; later adoption and popularization in relativistic electrodynamics involved figures such as Hendrik Antoon Lorentz, whose similar surname sometimes causes misattribution. Debates about priority and attribution involved historians and scientists connected to institutions like the Royal Society and commentary by scholars influenced by editions of Maxwell's Treatise on Electricity and Magnetism and translations circulated at universities such as University of Copenhagen and University of Groningen. The terminology became standardized during the early 20th century as relativistic formulations by Albert Einstein and tensor methods by Hermann Minkowski entered mainstream physics curricula at institutions including University of Göttingen.
Mathematically, the Lorenz gauge condition ∂_μ A^μ = 0 defines an affine hyperplane in the space of four-potentials modulo gauge transformations generated by scalar functions satisfying the homogeneous wave equation. Solutions in the Lorenz gauge are obtained using retarded and advanced Green's functions developed in analyses by George Green and applied in contexts treated by Lord Rayleigh and J. D. Jackson in textbooks. The condition is Lorentz-covariant under transformations of the Poincaré group and is compatible with constraints used in formulations by Émile Picard and functional analytic methods promoted at institutions like École Normale Supérieure. Residual gauge freedom remains: scalar gauge functions that satisfy □χ = 0 preserve the Lorenz condition, a fact exploited in quantization approaches pioneered by Paul Dirac and later refined by researchers at CERN.
In classical electrodynamics, the Lorenz gauge enables explicit construction of retarded potentials used to describe radiation from sources treated in problems associated with Hertzian dipole models and scattering computations encountered in work at Bell Labs and research by Heinrich Hertz. It simplifies calculations of electromagnetic waves emitted by accelerating charges studied in analyses inspired by J. J. Thomson and experimental programs at facilities like MIT and Caltech. The gauge is widely used in computational electrodynamics software developed by groups at Lawrence Livermore National Laboratory and in theoretical treatments appearing in standard texts by Richard Feynman and John David Jackson.
The Lorenz gauge contrasts with other gauge choices such as the Coulomb gauge employed in nonrelativistic quantum mechanics contexts developed at institutions like Harvard University and in many-body treatments by Lev Landau. Unlike the axial gauges used in perturbative calculations at SLAC and the temporal gauge invoked in lattice studies at CERN, the Lorenz gauge maintains manifest covariance under Lorentz transformations emphasized in relativistic formulations by Paul Dirac and Werner Heisenberg. Transformations between the Lorenz gauge and the Coulomb gauge involve solving constraint equations similar to those addressed by researchers at Princeton University and in classical analyses by Oliver Heaviside.
In quantum field theory, the Lorenz gauge is employed in covariant quantization of the photon field and in path integral formulations developed by proponents at Institute for Advanced Study and in foundational work by Richard Feynman and Freeman Dyson. Gauge fixing in the Lorenz condition leads to propagators with well-defined Feynman prescriptions used in perturbative calculations at DESY and Fermilab, and requires introduction of ghost fields in nonabelian extensions treated in the Yang–Mills theory framework by Chen Ning Yang and Robert Mills. The Lorenz gauge remains a standard choice in calculations of radiative corrections explored by communities at SLAC National Accelerator Laboratory and in precision tests discussed at CERN.