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 {A_aⁱ(x), E^b_j(y)} = −i δ_a^b δⁱ_j δ(x−y); In a solution of the field equations, A_aⁱ = κ⁻¹ (Γ_aⁱ + i K_aⁱ), 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

A_aⁱ = κ⁻¹ (Γ_aⁱ + γ K_aⁱ) ;

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 E^a_i A_aⁱ + 2i N^a E^b_i F_abⁱ − 2i N ⁱ \(\cal D\)_a E^a_i + N E^a_i E^b_j ε^ijk F_abk] + boundary terms .

* Constraints and evolution:

\(\cal D\)_a E^a = 0 ,   tr E^a F_ab = 0 ,   tr E^a E^b F_ab = 0 .

* And geometry: The surface element of a 2-surface x^a(r, s) is (E^a_i E^ib f_a f_b )^1/2 dr ∧ ds, where f_a:= ε_abc x^bx^c.
* 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) [E^a_i and B^a_i]; 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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