Ashtekar-Variables Formulation of Canonical General Relativity |
In General
> s.a. BRST; initial-value formulation;
numerical general relativity / connection
formulation of quantum gravity.
* Ashtekar variables: Based
on a self-dual form of the action; A complex SO(3) connection \(A_a{}^i(x)\)
and a densitized triad \(E^a{}_i(x)\) (initially, a densitized SU(2) soldering
form, to be used with coupled spinorial matter), with Poisson brackets
{Aai(x),
Ebj(y)}
= −i δab
δij
δ(x−y); In a solution of the field equations,
Aai
= κ−1
(Γai
+ i Kai),
with Γ the connection of E, and K the extrinsic curvature.
* Ashtekar-Barbero variables: Using
the Immirzi parameter γ, the connection can be generalized to
Aai = κ−1 (Γai + γ Kai) ;
Notice however that, for γ ≠ i, this spatial
connection is not the pull-pack of a spacetime connection.
* Action: For the
original, complex Ashtekar variables,
S = ∫ dt ∫Σ d3x [−2i Eai Aai + 2i Na Ebi Fabi − 2i N i \(\cal D\)a Eai + N Eai Ebj εijk Fabk] + boundary terms .
* Constraints and evolution:
\(\cal D\)a Ea = 0 , tr Ea Fab = 0 , tr Ea Eb Fab = 0 .
* And geometry: The surface
element of a 2-surface xa(r, s)
is (Eai
Eib fa
fb )1/2 dr ∧ ds,
where fa:= εabc
xbxc.
* SU(2) vs SO(3): The idea
that the contribution from j = 1 edges of spin networks dominates
black-hole areas, as opposed to j = 1/2, suggests (but does not
imply – an exclusion-principle argument might apply) that the true
gauge group might be SO(3) rather than SU(2).
> Online resources:
see scholarpedia article.
References > s.a. first-order actions;
higher-order gravity; Immirzi Parameter.
@ General: Beetle & Corichi gq/97,
Corichi & Hauser gq/05 [bibliography];
Fleischhack JPCS(12) [rev].
@ Complex variables: Sen PLB(82);
Ashtekar PRL(86),
PRD(87),
in(87), in(90);
Jacobson & Smolin PLB(87);
Dolan PLB(89);
Herdegen CQG(89);
Bergmann & Smith PRD(91);
Soloviev PLB(92);
Wallner PRD(92);
Chang & Soo IJMPD(93)ht;
Romano GRG(93)gq [vs geometrodynamics];
Khatsymovsky gq/93,
PLB(97)gq/96 [and self-duality];
Kerrick PRL(95);
Nieto MPLA(05) [form of the action];
Wieland AHP(11)-a1012 [and Holst action],
CQG(11) [twistorial phase space];
Rosales-Quintero IJMPA(16)-a1505 [pure-connection self-dual formulation, and supergravity];
Ashtekar & Varadarajan Univ(21)-a2012 [geometrical interpretation of Hamiltonian evolution].
@ Real variables: Barbero PRD(94)gq/93,
PRD(95)gq/94;
Holst PRD(96)gq/95;
Loll in(97)gq;
Samuel CQG(00)gq,
PRD(01).
@ SU(2) vs SO(3): Swain IJMPD(03)gq-GRF
and gq/04,
gq/04-MGX;
Chou et al PLB(06)gq/05.
@ Compared to metric variables: Anandan gq/95;
Zagermann CQG(98)gq/97 [2 Killing vectors].
@ Nature of equations: Iriondo et al PRL(97)gq,
ATMP(98)gq;
Yoneda & Shinkai PRL(99)gq/98;
Shinkai & Yoneda PRD(99)gq [stable form].
@ Reality conditions: Bengtsson TMP(93);
Mena IJMPD(94)gq/93 [and quantization];
Immirzi CQG(93);
Barbero PRD(95)gq/94,
PRD(95)gq/94;
Morales-Técotl et al CQG(96)gq [as Dirac constraints];
Yoneda & Shinkai CQG(96)gq [with cosmological constant];
Pons et al PRD(00)gq/99.
@ Euclidean / Lorentzian: Ashtekar PRD(96)gq/95;
Barbero PRD(96)gq [2-parameter action];
Barnich & Husain CQG(97)gq/96;
Mena G&C(98)gq/97 [generalized Wick transform];
Garay & Mena CQG(98)gq.
@ Solving the constraints: Thiemann CQG(93)gq;
Barbero CQG(95)gq/94;
Goldberg PRD(96) [gauge and diffeomorphism].
@ Initial-value problem: Saraykar & Wagh pr(89);
Robinson & Soteriou CQG(90);
Capovilla et al gq/93.
@ Gauge issues:
Manojlović & Miković NPB(92) [fixing];
Montesinos & Vergara GRG(01)gq/00 [invariance].
@ Holonomy of Ashtekar-Barbero connection: Charles & Livine PRD(15)-a1507;
Bilski a2012.
@ Related topics: Giannopoulos & Daftardar CQG(92) [algebraic evaluation];
Chang & Soo PRD(92) [and 4-manifolds];
Rovelli PRD(93) [and surface areas];
Fleischhack & Levermann a1112 [fiber-bundle perspective];
Freidel et al PRD(17)-a1611 [with a spatial boundary, auxiliary strings];
> s.a. holonomy.
Variations and Generalizations
> s.a. 3D general relativity; canonical gravity
[asymptotically flat] and models [including matter].
* Covariant formulation:
A two-parameter family of covariant connections has been obtained by Alexandrov
using Dirac brackets (generically these connections are not commutative), and by
Geiller et al solving explicitly the second-class constraints obtained from the
Holst action; The latter procedure hides the explicit Lorentz covariance, which
can be restored by suitably redefining the variables.
@ Linearized:
Ashtekar & Lee IJMPD(94).
@ Covariant formulation: Alexandrov CQG(06)gq/05 [and reality conditions];
Fatibene et al CQG(07)gq;
Cianfrani & Montani PRL(09)-a0811,
a0904-proc;
Cianfrani & Montani PRD(09)-a0904 [with scalar field];
Fatibene & Francaviglia a0905;
Rovelli & Speziale PRD(11)-a1012;
Geiller et al Sigma(11)-a1103,
PRD(11)-a1105;
> s.a. canonical quantum gravity [covariant lqg].
@ 2+2 decomposition:
d'Inverno & Vickers CQG(95);
d'Inverno et al CQG(06) [double-null, Hamiltonian].
@ Different foliations / decompositions:
Fodor & Pejés in(91) [based on threading];
Maran gq/03-wd [based on timelike foliations];
Gielen & Wise PRD(12)-a1111,
Gielen a1210-proc
[with a field of observers, Lorentz-covariant];
Perlov a2001 [for timelike 3+1 foliations];
> s.a. modified formalisms.
@ As BF theory:
Capovilla et al CQG(01)gq,
Celada et al CQG(16)-a1605 [and the Immirzi parameter].
@ Other similar variables:
Rosas-Rodríguez IJMPA(08)
[Eai
and Bai];
Dittrich & Geiller CQG(15)-a1412 [flux formulation];
Ziprick & Gegenberg PRD(16)-a1507 [discrete phase space and Hamiltonian];
Cattaneo & Pérez CQG(17)-a1611 [Poisson brackets of 2D smeared fluxes];
> s.a. loop variables.
@ And torsion: Maluf JMP(92);
Montesinos JMP(99).
> Other versions and theories: see
discretized gravity; higher-dimensional gravity;
higher-order theories; regge calculus;
Topological Gravity.
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