Landau gauge

Landau gauge
Joel Holdsworth (Joelholdsworth) · Public domain · source
Name	Landau gauge
Field	Theoretical physics
Introduced	1950s
Introduced by	Lev Landau

Landau gauge is a specific choice of gauge condition used in gauge field theories, notably in quantum electrodynamics and quantum chromodynamics, that imposes a divergence-free constraint on the gauge potential. It is widely used in analytic calculations, perturbative renormalization, and nonperturbative lattice studies because of its covariance and simplified ghost sector. The condition plays a central role in studies involving the Faddeev–Popov procedure, the Becchi–Rouet–Stora–Tyutin formalism, and Dyson–Schwinger equations.

Definition and properties

The Landau gauge is defined by the linear condition ∂^μ A_μ = 0 for a gauge potential A_μ in relativistic gauge theories such as Quantum electrodynamics and Quantum chromodynamics. In abelian settings like Maxwell's equations the constraint reduces redundancies analogous to the Coulomb gauge used in Werner Heisenberg-era formulations, while in nonabelian contexts it affects the ghost sector introduced by the Faddeev–Popov method. The gauge is covariant under Lorentz transformations similar to constructions in Paul Dirac's relativistic framework and is a fixed point of a family of covariant gauges parameterized by the 'ξ' parameter in Gupta-Bleuler-type quantizations. Important properties include transversality of the gauge propagator in momentum space, the absence of explicit gauge parameter dependence at ξ = 0 in perturbative Green's functions, and simplifications in the Slavnov–Taylor identities related to Vladimir Gribov's observations about gauge copies.

Fixing procedure and gauge condition

Gauge fixing to Landau gauge is accomplished via the Faddeev–Popov procedure introduced by Ludvig Faddeev and Victor Popov and implemented in path integral quantization frameworks developed alongside work by Richard Feynman and Julian Schwinger. One imposes the delta-functional constraint ∂^μ A_μ = 0 and introduces ghost fields (c, c̄) whose dynamics are governed by a quadratic ghost action in perturbation theory, as formalized by the BRST symmetry originally named for Claude Becchi, Alberto Rouet, Raymond Stora, and later extended by Igor Tyutin. The condition is also analyzed in canonical quantization approaches influenced by Paul Dirac's constrained dynamics and in covariant operator formalisms developed by Res Jost-style axiomatic programs. Gribov ambiguities, first highlighted by Vladimir Gribov, reveal that global gauge fixing in nonabelian theories is obstructed by multiple gauge-equivalent configurations, prompting approaches by Daniel Zwanziger and investigations linked to the Kugo–Ojima confinement criterion by Taichiro Kugo and Izumi Ojima.

Applications in quantum field theory

Landau gauge is extensively used in perturbative computations of beta functions and anomalous dimensions in Quantum chromodynamics and in precision tests in Quantum electrodynamics, often alongside schemes such as Minimal subtraction and Modified minimal subtraction. It is favored in Dyson–Schwinger studies pursued by groups around Craig Roberts and Christian Fischer and in functional renormalization group analyses associated with Joaquim G. Ramos-type investigations. Lattice gauge theory collaborations like MILC Collaboration, CP-PACS, and groups at CERN use Landau gauge for nonperturbative determinations of propagators and vertex functions, informing confinement scenarios debated by proponents of the Gribov–Zwanziger framework and the Kugo–Ojima mechanism. In phenomenology, Landau gauge inputs feed into hadron structure studies in approaches developed by Stanley Brodsky, George Sterman, and groups applying Dyson–Schwinger equations to meson physics such as teams led by Craig Roberts.

Mathematical formulations and examples

Mathematically, Landau gauge corresponds to choosing a representative in each gauge orbit satisfying the linear elliptic condition ∂^μ A_μ = 0 on manifolds like Minkowski space or Euclidean space used in Wick-rotated treatments pioneered in contexts associated with Oskar Klein-era field theory. Examples include the free photon propagator in momentum space taking the transverse projector form used in computations by Freeman Dyson and applications in perturbative expansions first systematized in Gerard 't Hooft's renormalization studies. On curved backgrounds related to work by Stephen Hawking and Roger Penrose, similar divergence conditions require careful treatment of covariant derivatives and are connected to the BRST cohomology techniques developed in studies by Edward Witten and Boris Tsygan-style mathematical physics. Functional determinants arising from the Faddeev–Popov operator link to spectral theory studied by mathematicians like Michael Atiyah and Isadore Singer.

Renormalization and Green's functions

In Landau gauge renormalization, the ghost and gluon propagators exhibit characteristic ultraviolet and infrared behaviors analyzed in perturbation theory and nonperturbative frameworks; these analyses draw on renormalization group methods from Kenneth Wilson and scheme choices such as MS-bar. Slavnov–Taylor identities constrained by BRST symmetry, connecting to work by John C. Taylor, restrict the form of Green's functions and vertex renormalization constants used in calculations by David Gross and Frank Wilczek. Infrared studies of the gluon propagator and the running coupling in Landau gauge have informed confinement criteria discussed by Vladimir Gribov and Daniel Zwanziger, and are compared with lattice results from collaborations at Fermilab and Jülich Research Centre.

Lattice implementations and numerical studies

Lattice Landau gauge fixing is implemented via iterative minimization algorithms such as stochastic overrelaxation and Fourier-accelerated methods developed and used by collaborations like MILC Collaboration and groups at CERN and Brookhaven National Laboratory. Numerical studies address Gribov copy issues with techniques inspired by Vladimir Gribov and practical strategies by Daniel Zwanziger and Antonio Cucchieri; large-scale simulations have been reported by teams from Yukawa Institute for Theoretical Physics and University of Adelaide. Lattice determinations of propagators, vertex functions, and running couplings in Landau gauge play a crucial role in comparing continuum approaches like Dyson–Schwinger equations and functional renormalization group methods advanced by Jens Braun and Jan Pawlowski.

Category:Gauge fixing