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| Gauge fixing | |
|---|---|
| Name | Gauge fixing |
| Field | Theoretical physics |
| Related | Electromagnetism, Yang–Mills theory, General relativity |
| Introduced | 20th century |
Gauge fixing is the process of selecting a specific representative from an equivalence class of field configurations related by gauge transformations in theories with local symmetries. It appears in the formulation and quantization of classical and quantum field theories, influencing calculations in James Clerk Maxwell-based electromagnetism, Yang–Mills theory as developed by Chen Ning Yang and Robert Mills, and the canonical treatments related to Albert Einstein's general relativity. Practical implementations of gauge fixing play central roles in work by researchers at institutions such as CERN, Princeton University, and the Institute for Advanced Study.
Gauge fixing arises when a field theory possesses redundant variables because of invariance under a group of local transformations like the U(1) symmetry of Paul Dirac-inspired quantum electrodynamics or the non-Abelian groups underlying SU(2) and SU(3) in the Standard Model. Early formal recognition occurred in the lineage from Hendrik Lorentz and Ludwig Lorenz constraints in classical electromagnetism to modern path integral techniques formulated by Richard Feynman and Julian Schwinger. The procedure reduces degrees of freedom to enable well-defined initial-value problems and functional integrals used by practitioners at laboratories such as Fermilab.
Gauge fixing is motivated by the need to remove redundancy introduced by local symmetries so that operators, propagators, and observables become invertible and calculable. For example, in computations inspired by Paul Dirac's constrained Hamiltonian methods and the BRST quantization program advanced by Igor Tyutin and Claude Becchi, gauge choices permit construction of Feynman rules used in perturbative expansions developed by Gerard 't Hooft and Mikhail Veltman. In numerical relativity projects associated with Kip Thorne and groups at Max Planck Institute for Gravitational Physics, fixing coordinate gauges stabilizes evolution schemes. Gauge fixing also clarifies relationships between conserved currents discussed by Emmy Noether and symmetry breaking treated by Yoichiro Nambu.
Common gauges include the Coulomb gauge (used in analyses tracing back to Charles-Augustin de Coulomb), the Lorenz gauge (associated historically with Ludwig Lorenz), and the family of linear covariant gauges like the Feynman gauge introduced by Richard Feynman. Non-Abelian work often employs the Landau gauge favored in lattice studies by collaborations at Brookhaven National Laboratory and Rutherford Appleton Laboratory. In gravity, coordinate conditions such as the harmonic gauge used in post-Newtonian approximations by teams connected to Albert Einstein and Lev Landau are prevalent. Specialized choices like the axial gauge and the Coulomb–London gauge appear in canonical treatments influenced by Paul Dirac and John Bell.
Mathematically, gauge fixing imposes constraints that slice the space of connections or metrics into representatives that intersect each gauge orbit once locally. The method uses tools from functional analysis and differential geometry developed by Henri Poincaré and Elie Cartan, employing gauge conditions expressed as differential operators. In the path integral approach formalized by Richard Feynman, the insertion of delta-function constraints and determinants (the Faddeev–Popov procedure by Ludvig Faddeev and Victor Popov) yields a well-defined measure. Canonical approaches follow Paul Dirac's theory of constrained Hamiltonian systems and utilize Dirac brackets, while cohomological treatments invoke BRST symmetry introduced in works by Bruno Zumino and collaborators.
In quantum field theory, gauge fixing underpins renormalization programs executed by Gerard 't Hooft and Steven Weinberg and affects perturbative expansions applied in calculations performed at CERN and SLAC National Accelerator Laboratory. Ghost fields from the Faddeev–Popov ghost construction play roles in unitarity proofs and anomaly analyses by John Bell and Roman Jackiw. Lattice gauge theory implementations by groups at MIT and University of Edinburgh rely on discrete gauge fixing algorithms for measurements of confinement related to ideas from Kenneth Wilson. In effective field theory and scattering amplitudes research by teams led by Nima Arkani-Hamed and Edward Witten, judicious gauge choices simplify computations and reveal hidden symmetries.
Global properties of gauge orbits produce complications such as the Gribov ambiguity identified by Vladimir Gribov, showing that some gauge conditions intersect orbits multiple times. This affects nonperturbative analyses in Yang–Mills theory and confinement studies inspired by Alexander Polyakov and Ken Wilson. Resolving global issues draws on results from Michael Atiyah and Isadore Singer in index theory and topology, and on advances in moduli space studies by researchers at Mathematical Sciences Research Institute and Princeton University. In gravity, analogous global diffeomorphism issues connect to work by Roger Penrose and influence approaches to quantum gravity pursued at Perimeter Institute.
Practical applications include perturbative calculations of radiative corrections in processes studied at Large Hadron Collider collaborations such as ATLAS and CMS, lattice determinations of hadron spectra by HLPC-affiliated groups, and numerical relativity simulations for gravitational waves by teams associated with LIGO and Virgo. In condensed matter physics, gauge-fixing techniques enter analyses of emergent gauge fields in Philip W. Anderson-inspired theories and topological phases examined at Princeton University and Harvard University. The choice of gauge affects computational efficiency and conceptual clarity in a wide range of projects across institutions including Caltech, Yale University, and University of Cambridge.