Connection
Formulation of Canonical General Relativity| Connection
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 self-dual action; A complex SO(3) connection Aai(x)
and a densitized triad Eai(x)
(initially, a densitized SU(2) soldering form,
to be used
with coupled spinorial matter), with Poisson brackets {Aai(x), Ebj(y)}
= –i 
ab ij (x–y);
In a solution of the field equations, Aai = 
–1
(
ai +
i Kai),
with the connection
of E, and K the extrinsic curvature.
* Ashtekar-Barbero variables: Using the Immirzi parameter 
,
the connection can be generalized to
Aai = –1
(ai + Kai)
;
Notice however that for 
i
this spatial connection is not the pull-pack of a spacetime connection.
* Action:
S = 
dt Sigma d3x [–2i Eai Aai +
2i Na Ebi Fabi – 2i N 
a Eai
+ N Eai Ebj 
ijk Fabk]
+ boundary terms .
* Constraints and evolution:
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:= abc 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 involved might be SO(3) rather than SU(2).
General References > s.a. first-order
actions; higher-order
gravity.
@ Bibliography: Beetle & Corichi
gq/97, Corichi
& Hauser gq/05.
@ 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].
@ Real variables: Barbero PRD(94)gq/93,
PRD(95)gq/94;
Holst PRD(96)gq/95;
Loll gq/97-in; Samuel
CQG(00)gq,
PRD(01).
@ Immirzi parameter: Immirzi CQG(97)gq/96;
Rovelli & Thiemann
PRD(98)gq/97;
Krasnov
CQG(98)gq/97;
Corichi & Krasnov MPLA(98);
Barros e Sá IJMPD(01)gq/00;
Samuel
PRD(01);
Mena CQG(02)gq [not
local]; Garay & Mena PRD(02)gq [and
black-hole entropy];
Dreyer
PRL(03)gq/02 [and
black-hole entropy]; Oppenheim PRD(04)gq/03 [and
quasinormal modes]; Pérez & Rovelli PRD(06)gq/05 [physical
effects]; Chou et al PRD(05)gq [meaning,
scalar vs pseudo-scalar]; Fatibene et al CQG(07)-a0706 [action];
Date
et al PRD(09)-a0811,
Mercuri a0903-in
[topological
interpretation]; Schucker a0906 [without
spinors]; > s.a. Conformal
Gravity, models [Dirac
fields, Immirzi parameter as local field], regge calculus.
@ 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;
Alexandrov CQG(06)gq/05 [and
Lorentz-covariant formulation].
@ 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: Manojlovic & Mikovic NPB(92) [fixing]; Montesinos & Vergara
GRG(01)gq/00 [invariance].
@ Related topics: Giannopoulos & Daftardar CQG(92)
[algebraic evaluation]; Chang & Soo PRD(92)
[and 4-manifolds]; Rovelli PRD(93) [and surface areas]; > s.a. holonomy.
Variations and Generalizations > s.a. 3D
gravity; higher-dimensional gravity; loop
variables; models [including
matter]; Topological Gravity.
@ Linearized: Ashtekar & Lee IJMPD(94).
@ Covariant formulation: Fatibene et al CQG(07)gq;
Cianfrani & Montani PRL(09)-a0811,
a0904-in; Cianfrani
& Montani PRD(09)-a0904 [with
scalar field]; Fatibene & Francaviglia a0905.
@ 2+2 decomposition: d'Inverno & Vickers CQG(95);
d'Inverno et al CQG(06)
[double-null, Hamiltonian].
@ Based on threading: Fodor & Pejés in(91); > s.a. modified
canonical formalism.
@ As BF theory: Capovilla et al CQG(01)gq [and
Immirzi parameter].
@ Other similar variables: Rosas-Rodríguez IJMPA(08)
[Eai and Bai].
@ And torsion: Maluf JMP(92); Montesinos JMP(99).
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