Gauge Choices or Fixing| 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
gab),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 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; 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 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 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.
main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 9 jan 2019