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 \(\nabla^2 x^a = 0\); Alternatively, the densitized inverse metric
is divergenceless, (|*g*|^{1/2}
*g*^{ab})_{,a} = 0;
> s.a. harmonic functions;
Wikipedia page.

* __Synchronous gauge__:
Defined by \(h_{0a} = 0\), 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 \(h^{*a}{}^{}_{b,a} = 0\), where
\(h^*_{ab}:= h^{~}_{ab} - {1\over2}\,\eta^{~}_{ab} h\) (transverse?);
The gravitational counterpart to the Lorenz gauge; not conformally invariant.

* __Radiation gauge__: In addition,
*h* = 0 and *h*_{0i} = 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; This 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 \(\cal L_{\rm G}\) of the
electromagnetic Lagrangian.

* __Lorenz gauge__: A gauge in which
∇_{a} *A*^{a}
= 0, or = (any *g*-valued function on spacetime),
so ∇^{2}*A*_{a}
= *J*_{a}; The residual freedom is
*A*_{a} \(\mapsto\) *A*_{a}
+ ∂_{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,
*x*^{a} *A*_{a}
= 0; In Poincaré gauge theory, *x*^{a}
Γ_{a} = 0,
*x*^{a} *e*_{a} = 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 JPB(17)-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 PRD(16)-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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