Connection
Representation of Quantum Gravity| Connection
Representation of Quantum Gravity |
In General > s.a. canonical quantum
gravity [including covariant version];
loop quantum cosmology.
* Quantum configuration space:
A distributional version of the classical
one of connections modulo gauge transformations,
=
generalized connections modulo generalized gauge transformations.
* Elementary operators: Heuristically, the Ashtekar new variables (Eai, Aai); In a rigorous approach, the Lie derivatives wrt the left-invariant vector fields on the copy of SU(2) associated with each edge,
or the holonomies along edges and electric fluxes through surfaces.
* Kinematical Hilbert space:
The completion of the space Cyl of cylindrical functions, with the measure
induced by the Haar measure on SU(2); A nice basis to work with has spin
networks as elements; > s.a. projective
limit.
@ General references: Ashtekar PRL(86), PRD(87),
in(89); Fukuyama & Kamimura PRD(90);
Zegwaard CQG(91);
Kodama
IJMPD(92)gq;
Mena IJMPD(94)gq/93 [reality
conditions]; Thiemann ACosm(95)gq [transforms],
CQG(96)gq/95 [reality
conditions]; Rainer gq/99-in
[quantum field theory]; Ita a0806 [covariance];
Engle
a0812 [piecewise-linear
version]; Bianchi et al a0905 [propagator,
from new spin-foam models]; > s.a. approaches
to quantum gravity; connection
formulation of general relativity.
@ Bibliography: Brügmann gq/93; Schilling gq/94;
Beetle & Corichi gq/97;
Corichi
& Hauser gq/05.
@ Representation of basic algebra: Ashtekar et al CQG(98)gq [no
triad representation]; Sahlmann & Thiemann CQG(06)gq/03;
Fleischhack
PRL(06),
CMP(09)mp/04; Varadarajan CQG(08)
[alternatives].
@ Kinematical 
:
Fairbairn & Rovelli JMP(04)gq [separability];
Okolów CQG(05)gq/04 [non-compact G];
Velhinho CQG(05)gq.
@ Discretized: Renteln & Smolin CQG(89);
Loll NPB(95)gq, PRD(96)gq [real
variables], gq/97-in, PLB(97)gq [det E > 0], CQG(98)gq/97 [diffeo
constraints];
Zapata CQG(04)gq [and
lattice gauge theory]; Gambini & Pullin PRL(05)gq/04 [consistent]; > s.a. diffeomorphisms.
@ Other variants: Giesel & Thiemann a0711 [reduced
phase space quantization]; Bianchi a0907 [topological
field theory with network
of defects]; Bahr & Thiemann CQG(09) [towards a combinatorial formulation]; > s.a. loop
representation [including
deformations].
References > s.a. anomaly; connection; geometrical
operators; path-integral quantum gravity; philosophy; Wilson
Loops.
@ I: Sen & Butler ThSc(89)nov; Bartusiak disc(93)apr;
Vaas
bdw(03)phy/04 [and
strings]; Rovelli pw(03)nov;
Smolin SA(04)jan.
@ Intros, reviews: Ashtekar 88, 91, ht/92, gq/94,
in(95)gq/93, IJMPD(96)ht, gq/01-in;
Rovelli CQG(91), LRR(98)gq/97;
Smolin in(92), gq/92;
Ashtekar & Rovelli CQG(92);
Ashtekar & Lewandowski ht/93-in;
Pullin in(97)gq/96;
Gaul & Rovelli LNP(00)gq/99;
Thiemann LNP(03)gq/02;
Ashtekar & Lewandowski CQG(04)gq [intro];
Smolin ht/04/RMP;
Pérez gq/04-ln;
Nicolai et al CQG(05)ht
[outside
view]; Liko & Kauffman CQG(06)ht/05 [and
knot theory]; Corichi JPCS(05)gq [geometry];
Han et al IJMPD(07)gq/05;
Ashtekar NJP(05);
Nicolai & Peeters ht/06-in
[intro];
Ashtekar gq/06-in;
Thiemann ht/06 [inside
view]; Ashtekar NCB(07)gq [introduction
through quantum cosmology], a0705-in
[faq's];
Han
a0706-MS; Thiemann
07, IJMPA(08)-in;
Rovelli LRR(08).
@ Immirzi parameter: Immirzi NPPS(97)gq,
CQG(97)gq/96;
Rovelli & Thiemann
PRD(98)gq/97;
Gambini et al PRD(99)gq/98 [Yang-Mills
version]; Krasnov CQG(99)gq [rotating
black holes]; Rainer
G&C(00)gq/99 [and
black-hole entropy]; Samuel
PRD(01);
Garay & Mena PRD(02);
Mena CQG(02);
Dreyer PRL(03)gq/02,
Domagala & Lewandowski
CQG(04)gq [from
black-hole entropy]; Mercuri PRD(08)-a0708 [and
large gauge transformations]; Sengupta a0904 [topological
interpretation, wave
function rescaling]; > s.a. connection
formulation
of general relativity.
@ Quantum configuration space: Ashtekar & Isham CQG(92);
Ashtekar & Lewandowski
JMP(95)gq/94,
JGP(95)ht/94;
Marolf & Mourão CMP(95)ht/94;
Doering & de
Groote gq/01.
@ States: Jacobson & Smolin NPB(88);
Smolin in(88); Husain NPB(89);
Brügmann & Pullin NPB(91);
Ezawa PRP(97)gq/96;
Lewandowski & Marolf IJMPD(98)gq/97 ["vertex-smooth"];
Hari
Dass & Mathur CQG(07)gq/06;
Ita a0710; > s.a. spin
networks.
@ Inner product: Rendall CQG(93)gq;
Thiemann CQG(98)gq/97;
Bahr & Thiemann CQG(07)gq/06 [approximating].
@ Measure: Baez in(94)ht/93;
Baez & Sawin JFA(97)qa/95;
Mourão et al JMP(99)ht/97; > s.a. connection.
Constraints and Hamiltonian > s.a. classical
version [including reality conditions].
* Gauss law: Can be written 
vi = 
I
J iv, I ,
for all vertices v (I labels the edges at v) and
internal directions i.
* Solutions of constraints:
Heuristically, the quantum Gauss and scalar
constraints
have been solved for a large set of states which are concentrated
on
loops in a hypersurface, as well as for some "topological" ones.
@ Hamiltonian constraint: Blencowe NPB(90);
Borissov PRD(97)gq/94 [regularization,
algebra]; Gambini & Pullin CQG(96)gq [and
knot theory]; Thiemann PLB(96)gq,
CQG(98)gq/96,
CQG(98)gq/96 [operator];
Smolin gq/96 [and
long-range correlations]; Borissov et al CQG(97)gq [matrix
elements]; Gambini
et
al
IJMPD(98)gq/97 [algebra];
Neville PRD(99)gq/98 [correlations
and non-locality];
Di Bartolo et al CQG(00)gq/99 [algebra];
Rovelli PRD(99)gq/98 [projector];
Gaul & Rovelli CQG(01)
[all
irrep's of SU(2)]; Pérez PRD(06)gq/05 [regularization
ambiguities]; Ita a0706,
a0707 [general
solution].
@ Diffeomorphism constraints: Renteln CQG(90)
[lattice regularization];
Loll CQG(98)
[on a lattice]; Arnsdorf & García
CQG(99)gq/98 [vs
vector]; Koslowski gq/06 [stratified];
Ita a0806, a0806 [and
Kodama state, dimensional extension].
@ Master Constraint Programme: Thiemann CQG(06)gq/03,
CQG(06)gq/05;
Han & Ma PLB(06)gq/05; > s.a. dirac
quantization.
Special Solutions and Related Topics > s.a. FRW
models;
gowdy models; inflation; minisuperspace;
models; quantum
cosmology.
@ Vacuum: Mielke PLA(99) [teleparallel equivalent];
Varadarajan
PRD(02)gq [gravitons],
CQG(05)gq/04 [graviton
vacuum].
@ Symmetries: Thiemann in(94)gq/99,
NPB(95)
[spherical]; Alexandrov et al CQG(98)
[SU(2)-invariant];
Bojowald & Kastrup CQG(00)ht/99 [reduction];
Ma PRD(02)gq/01 [static 
2+1
Euclidean + Klein-Gordon].
@ Other topics: Torre CQG(88)
[propagator]; Arnsdorf & García CQG(99)gq/98 [spinorial
states from
topology]; Speziale a0810-ASL
[n-point functions]; Yang & Ma a0812 [quasilocal
energy]; Botelho a0902 [and fermionic string Ising models]; > s.a. Lemaître-Tolman-Bondi.
With Matter / Cosmological Constant > s.a. matter
phenomenology; non-commutative field theory; supergravity;
symmetry breaking.
* With cosmological constant: Need to deform
SU(2) to SU(2)q , with q =
exp{2
/k+2},
k:= 6/G2
.
@ Scalar: Kiefer PLB(89);
Matschull CQG(93)gq;
Han & Ma CQG(06)gq;
Ita gq/07,
a0710.
@ Fermions and Higgs: Baez & Krasnov JMP(98)ht/97;
Thiemann CQG(98)gq/97;
Montesinos & Rovelli CQG(98)gq;
Bojowald et al PRD(08)-a0710 [and
early-universe cosmology]; Ita a0805 [scalar
and fermion, and Kodama state]; Bojowald & Das PRD(08)
[fermions].
@ Other matter: Thiemann CQG(98)gq/97 [standard
model]; Lambiase & Singh PLB(03)
[matter/antimatter]; Gambini
et al GRG(06)gq/04-in
[Yang-Mills fields].
@ Cosmological constant: Alexander & Calcagni FP(08)-a0807 [as
a Fermi-liquid theory].
@ Chern-Simons-Kodama
state:
Brügmann et al NPB(92);
Crane ht/93-in;
Mena CQG(95)gq/94 [non-normalizable];
Gambini et al PLB(97)gq;
Soo CQG(02)gq/01;
Smolin ht/02 [overview];
Witten
gq/03; Freidel & Smolin
CQG(04)ht/03 [linearized];
Alexander et al gq/05 [fermionic
sectors]; Randono gq/05 [arbitrary
Immirzi parameter], gq/06, gq/06,
a0709-PhD
[real
Immirzi parameter]; Ita a0705,
a0705, a0706 [canonical
and path integral]; Ita a0805,
a0901 [Chang-Soo
variables], a0806,
a0904; > s.a. minisuperspace, quantum
gauge theory.
Online Resources > see Wikipedia page; Answers.com page; Dan Christensen's page; Seth Major's reading guide.
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 14
nov
2009