Gauge Choices or Fixing  

In General > s.a. gauge group / symmetry.
* Motivation: In classical theories with gauge freedom, fixing the gauge is a useful way to do calculations keeping only physical degrees of freedom; Many approaches to the quantization of a field theory require gauge fixing.
@ References: Pons IJMPA(96) [and singular Lagrangians].
> Types of theories: see constrained systems.

In Gravity > s.a. coordinate systems; embedding; initial-value formulation; models [radial gauge]; observables; time.
* Harmonic coordinates / gauge: Ones such that ∇2xa = 0; Alternatively, the densitized inverse metric is divergenceless, (|g|1/2 gab),a = 0; > s.a. harmonic functions; Wikipedia page.
* Synchronous gauge: Defined by h0a = 0, for metric perturbations.
@ Various choices: Rovelli CQG(89) [fixed spatial volume element]; Bartnik CQG(97)gq/96 [null quasi-spherical]; Alcubierre & Massó PRD(98) [hyperbolic, pathologies]; Hájíček & Kijowski PRD(00)gq/99 [covariant]; Esposito & Stornaiolo gq/99-conf, NPPS(00)gq/99, CQG(00)gq/98 [family of gauges]; Pons et al JMP(00) [Einstein-Yang-Mills, transformations]; Pons CQG(01)gq [special relativity limit]; Salisbury MPLA(03); Pretorius CQG(05)gq/04 [harmonic, numerical relativity]; Leclerc CQG(07)gq [and FLRW models]; Chen & Zhu PRD(11)-a1006 [true radiation gauge]; Reiterer & Trubowitz a1104 [vielbein formalism].
@ Dirac gauge: Bonazzola et al PRD(04)gq/03 [spherical coordinates]; Cordero-Carrión et al PRD(08)-a0802.
@ 2+1 dimensions: Menotti & Seminara AP(91) [radial].
@ Quantum gravity: Avramidi et al NPPS(97) [axial]; Hájíček gq/99-TX19; Mercuri & Montani IJMPD(04)gq/03 [kinematical action].
@ Quantum cosmology: Shestakova in(07)-a0801 [dependence on gauge and interpretation].
@ Various theories of gravity: da Rocha & Rodrigues AIP(10)-a0806 [as field theory in Minkowski space]; > s.a. canonical general relativity and models in canonical general relativity; higher-order gravity; numerical relativity.

In Linearized Gravity > s.a. cosmological perturbations [longitudinal, comoving].
* Einstein / de Donder / Hilbert / Fock gauge: Defined by h* ab, a = 0, where h*ab:= hab – \(1\over2\)ηab h (transverse?); The gravitational counterpart to the Lorenz gauge; not conformally invariant.
* Radiation gauge: In addition, h = 0 and h0i = 0 with i = 1, 2, 3.
* Remark: The TT condition can only be imposed on the constraint surface–so it is not, strictly speaking, a gauge.
@ References: Esposito & Stornaiolo CQG(00)gq/98; Scaria & Chakraborty CQG(02)ht [Wigner's little group]; Leclerc CQG(07)gq; Price & Wang AJP(08)oct [transverse traceless gauge].

Electromagnetism and Other Gauge Theories > s.a. Gribov Problem; gauge theory; quantum gauge thories.
* Idea: Gauge fixing corresponds to picking a cross section of the appropriate fiber bundle; This can always be done locally, but a global gauge choice in the non-Abelian case beyond perturbation theory is a non-trivial problem, and it may be impossible (Gribov ambiguity).
* Axial gauge: Given a 4-vector u, impose u · A = 0, or u · A = any g-valued function on spacetime; Fixes the gauge everywhere if gauge transformations have to go to the identity at infinity.
* Coulomb gauge: Defined by ∇ · A = 0 (in 3D); Implies that the scalar potential Φ is just the instantaneous Coulomb potential; Also known as radiation gauge.
* Feynman gauge: The choice ζ = 1 in the electromagnetic Lagrangian.
* Landau gauge: The choice ζ → 0 in the gauge term \(\cal L\)G of the electromagnetic Lagrangian.
* Lorenz gauge: A gauge in which ∇a Aa = 0, or = (any g-valued function on spacetime), so ∇2Aa = Ja; The residual freedom is Aa \(\mapsto\) Aa + ∂a χ, with ∇2 χ = 0, and can be used to impose the Coulomb gauge; Note: It is named after Ludwig Valentin Lorenz, and not after the Hendrik Antoon Lorentz of the Lorentz transformations [@ see Iliev a0803].
* Radial gauge: In Maxwell theory, xa Aa = 0; In Poincaré gauge theory, xa Γa = 0, xa ea = 0.
@ General references: Itzykson & Zuber 80, p567; Jackson AJP(02)sep [general transformations, and causality]; Castellani IJTP(04) [Dirac's views]; Capri et al PRD(06) [interpolating]; Heras AJP(07)feb [different gauges and retarded electric and magnetic fields]; Leclerc CQG(07)gq; Frenkel & Rácz EJP(15)-a1407 [use of transverse projection operator for transformation between gauges]; Reiss a1609 [restrictions and practical consequences].
@ Axial gauge: in Itzykson & Zuber 80, p566; in Cheng & Li 84, p254; Krasnansky a0806 [for QCD].
@ Coulomb gauge: Brill & Goodman AJP(67)sep [causality]; in Itzykson & Zuber 80, p576; Cronstrom ht/98 [Yang-Mills theory]; Haller & Ren PRD(03) [and Weyl, for QCD]; > s.a. yang-mills theory.
@ Lorenz gauge: Jackson AJP(08)-a0708 [attribution]; Rodrigues AACA(10)-a0801-conf [and Killing vector fields]; Heras & Fernández-Anaya EJP(10) [potentials as physical quantities].
@ Radial gauge: Modanese & Toller JMP(90); Magliaro et al PRD(07)-a0704 [compatibility with others].
@ Other choices: Heras AP(06) [Kirchhoff gauge]; Landshoff APA-a0810-in [non-covariant gauges]; Maas a1510 [Landau gauge, first Gribov region].
> In specific theories: see dirac fields; electromagnetism; Gauge Theory of Gravity; yang-mills gauge theory.


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