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;
observables; time.
* Harmonic coordinates:
Ones such that 
2xa
= 0; > s.a.
harmonic functions.
* 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]; Hájícek
& Kijowski PRD(00)gq/99
[covariant]; Esposito & Stornaiolo gq/99-in,
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
FRW models].
@ 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]; Hájícek gq/99-in;
Mercuri & Montani
IJMPD(04)gq/03 [kinematical
action].
@ Quantum cosmology: Shestakova a0801-in [dependence on gauge and interpretation].
@ Various theories of gravity:
da Rocha & Rodrigues 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 – 
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. electromagnetism;
gauge theory; quantum
gauge thories; yang-mills
gauge theory.
* Idea: Gauge fixing corresponds to picking a cross section of the
appropriate fiber bundle; This can always be done locally, but there may
be problems with
global choices (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 
is
just the instantaneous
Coulomb
potential;
Also known as radiation gauge.
* Feynman gauge: The choice 
=
1 in the electromagnetic Lagrangian.
* Landau gauge: The choice → 0
in the gauge term 
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 
Aa
+ 
a 
,
with 2 =
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 
a =
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.
@ 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,
576;
Cronstrom ht/98 [Yang-Mills
theory]; Haller & Ren PRD(03)
[and Weyl, for QCD]; > s.a. yang-mills theory.
@ Lorentz gauge: Jackson AJP(08)-a0708 [attribution];
Rodrigues a0801-in
[and Killing vector fields].
@ Radial gauge:
Modanese & Toller JMP(90);
Magliaro et al PRD(07)-a0704 [compatibility
with others].
@ Other choices: Heras AP(06)
[Kirchhoff gauge]; Landshoff a0810-in
[non-covariant
gauges].
> In related theories: see dirac fields; Gauge
Theory of Gravity.
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