Connection Representation of Quantum Gravity ("Loop Quantum Gravity")  

In General > s.a. connections [generalized connections and fluxes]; semiclassical [including relation with continuum].
* Quantum configuration space: A distributional version of the classical space of connections modulo gauge transformations (the symbol denotes closure),

\(\overline{{\cal A}/{\cal G}} = \overline{\cal A}/\overline{\cal G}\) = generalized connections modulo generalized gauge transformations.

* Elementary operators: Heuristically, the Ashtekar-Barbero su(2) connection and its conjugate momentum, (Eai, Aai); In a rigorous approach, the Lie derivatives with respect to the left-invariant vector fields on the copy of SU(2) associated with each edge of a graph in the spatial manifold,

\[\def\ii{{\rm i}}\def\dd{{\rm d}} J{}_{x,e_J}^i\Psi_\gamma(\bar A) = \ii\,(\bar A(e_J)\,\tau^i)^A_B\,{\partial \psi\over\partial(\bar A(e_J)^A_B} = \ii\,{\dd\over\dd t}\,\psi(\bar A(e_1),\ldots,\bar A(e_J)\,{\rm e}^{t\tau^i},\ldots,\bar A(e_N))\;.\]

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-conf [quantum field theory]; Ita a0806/HJ [covariance]; Bianchi et al NPB(09)-a0905 [propagator, from new spin-foam models]; Thiemann a1010 [lessons from parametrized field theory]; Wieland a1105 [complex variables]; Gielen Sigma(11)-a1111 [and connection dynamics]; Bojowald AIP(12)-a1208 [as an effective theory]; Ben Achour & Noui PoS-a1501 [analytic continuation]; Ben Achour PhD(15)-a1511; Varadarajan CQG(19)-a1808 [from Euclidean to Lorentzian]; > s.a. connection formulation of gravity.
@ Bibliography: Brügmann gq/93; Schilling gq/94; Beetle & Corichi gq/97; Corichi & Hauser gq/05.
@ Basic algebra and representations: Ashtekar et al CQG(98)gq [no triad representation]; Sahlmann JMP(11)gq/02, JMP(11)gq/02; Sahlmann & Thiemann CQG(06)gq/03; Fleischhack PRL(06), CMP(09)mp/04; Varadarajan CQG(08) [alternatives]; Kaminski a1108, a1108, a1108, a1108, a1108, a1108 [different algebras]; Ashtekar & Campiglia CQG(12)-a1209 [and covariance under spatial diffeomorphisms]; Stottmeister & Thiemann a1312 [structural aspects]; Bahr et al a1506 [new representation and quantum geometry]; Chagas-Filho a1705.
@ Kinematical Hilbert space: Fairbairn & Rovelli JMP(04)gq [separability]; Okołów CQG(05)gq/04 [non-compact G]; Velhinho CQG(05)gq; Cianfrani CQG(11)-a1012 [from BF theory]; Fleischhack in(07)-a1505 [kinematical uniqueness]; Carvalho & Franco a1610 [separability]; Giesel a1707-in.
@ Projective state space: Lanéry & Thiemann JMP(16)-a1411 [states as projective families]; Lanéry & Thiemann a1510, a1510 [semiclassical states].
@ Discretized versions: Renteln & Smolin CQG(89); Loll NPB(95)gq, PRD(96)gq [real variables], NPPS(97)gq, 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]; Engle CQG(10)-a0812 [PL]; Bahr & Thiemann CQG(09) [combinatorial]; Aastrup & Grimstrup a0911 [and semiclassical states]; Bahr et al Sigma(12)-a1111 [constraints and diffeomorphisms]; > s.a. diffeomorphisms; discrete geometry.
@ Purely spinorial variables: Livine & Tambornino JMP(12)-a1105, JPCS(12)-a1109; Livine & Tambornino PRD(13)-a1302 [holonomy-flux operator algebra].
@ Reduced phase space quantization: Giesel & Thiemann CQG(10)-a0711; Alesci et al PRD(13)-a1309 [relationship with the full theory].
@ Other variants: Bianchi GRG(14)-a0907 [à la Aharonov-Bohm, topological field theory with network of defects]; Sahlmann CQG(10)-a1006 [with non-degenerate spatial background]; Bodendorfer et al CQG(13)-a1105 [higher-dimensional], Bodendorfer et al CQG(13)-a1203 [without the Hamiltonian constraint]; Dupuis et al a1201-proc [spinors and twistors]; Mäkelä a1905 [and Wheeler's 'it-from-bit' proposal]; > s.a. approaches to canonical qg [including covariant lqg]; higher-order theories; holonomy [quantum]; loop representation [including deformations]; modified theories [scalar-tensor]; other approaches [including group field theory, topos theory]; spin-foam models; teleparallel equivalent; twistors.

References > s.a. anomaly; geometrical operators; Immirzi Parameter; path-integral quantum gravity; phenomenology; 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.
@ Books: Ashtekar 88, 91; Thiemann 07; Gambini & Pullin 11; Vaid & Bilson-Thompson 17; Ashtekar & Pullin ed-17 [intro and status]; Gambini & Pullin 20.
@ Intros, reviews: Ashtekar ht/92, gq/94, in(95)gq/93, IJMPD(96)ht, gq/01-GR16; Rovelli CQG(91), LRR(98)gq/97; Smolin in(92), gq/92; Ashtekar & Rovelli CQG(92); Ashtekar & Lewandowski ht/93-proc; 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 LNP(07)ht/06 [intro]; Ashtekar AIP(06)gq; Thiemann LNP(07)ht/06 [inside view]; Ashtekar NCB(07)gq [introduction through quantum cosmology], a0705-MGXI [faq's]; Han MSc(07)-a0706; Thiemann IJMPA(08)-proc; Rovelli LRR(08); Mercuri PoS-a1001; Sahlmann a1001-conf; Rovelli CQG(11)-a1004; Date a1004-ln; Doná & Speziale a1007-ln; Alexandrov & Roche PRP(11)-a1009; Rovelli CQG(11)-a1012 [25 years], a1102-ln; Bojowald a1101-conf [dynamical introduction]; Ashtekar LNP(13)-a1201; Giesel & Sahlmann PoS-a1203; Pullin & Singh a1301-MG13 [lqg session]; Bojowald PT(13)mar; Långvik a1303; Ashtekar proc(14)-a1303 [and cosmology]; Bilson-Thompson & Vaid book(17)-a1402 [pedagogical]; Chiou IJMPD(15)-a1412; Bodendorfer a1607-ln; Ashtekar & Pullin book(17)-a1703; Moreno a1808-MS; Belov a1905-PhD [geometry]; Ashtekar & Bianchi RPP(21)-a2104.
@ Quantum configuration space: Ashtekar & Isham CQG(92); Ashtekar & Lewandowski JMP(95)gq/94, JGP(95)ht/94; Marolf & Mourão CMP(95)ht/94; Döring & de Groote gq/01; Freidel et al CQG(13)-a1110 [relationship between holonomy-flux phase space and continuum phase space]; Freidel & Ziprick CQG(14)-a1308 [twisted geometry]; Yang & Ma ChPC(19)-a1908 [flux operators].
@ 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; Borja et al JPCS(12)-a1110, Sigma(12)-a1202 [simple graph with two modes, U(N)]; Bianchi et al a1605 [squeezed vacua]; Bianchi et al PRD(16)-a1609 [loop expansion]; Bianchi et al PRD(19)-a1812 [Bell-network states]; > 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.
@ Conceptual: Vidotto a1309-conf [atomism and relationalism]; Wüthrich in(17) [classical spacetime recovery]; Gilbert & Loveridge a2004 [interviews and analysis]; & Vera Matarese.

Constraints and Hamiltonian > s.a. classical version [including reality conditions].
* Gauss law: Can be written \(\cal G\)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: Blencowe NPB(90); 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)]; Ita a0706, a0707 [general solution]; Alesci et al PRD(12)-a1109 [spin-foam models and Euclidean solutions]; Bonzom & Laddha Sigma(12)-a1110 [lessons from toy models]; Alesci et al PRD(13)-a1306 [matrix elements]; Laddha a1401 [search for an off-shell anomaly free-version]; Lewandowski & Sahlmann PRD(15)-a1410 [symmetric]; Yang & Ma PLB(15)-a1507 [new proposal]; Assanioussi et al PRD(15)-a1506 [new proposal]; Lewandowski & Lin PRD(17)-a1606 [anomaly-free constraints and Minkowski condition]; Zhang et al PRD(18)-a1805; Mäkinen a1910-PhD; Varadarajan a2101 [Euclidean dynamics in terms of Electric Shift]; Zhang et al a2012, a2102 [coherent state expectation value].
@ Hamiltonian, regularization: Borissov PRD(97)gq/94 [and algebra]; Pérez PRD(06)gq/05 [ambiguities]; Alesci & Rovelli PRD(10)-a1005 [and spin-foam dynamics]; Alesci JPCS(12)-a1110 [regularized proposal].
@ Hamiltonian, approaches: Gambini & Pullin CQG(96)gq [and knot theory]; Ita CQG(14)-a0901v5 [affine group formalism]; Yang & Ma a1505 [graphical method]; Alesci et al a1606 [projections of intertwiners on spin coherent states]; Livine a1704 [coarse graining and holographic dynamics].
@ 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 HJ-a0806, a0806 [and Kodama state, dimensional extension]; Laddha & Varadarajan CQG(11)-a1105; Varadarajan JPCS(12), CQG(13)-a1306.
@ Master Constraint Programme: Thiemann CQG(06)gq/03, CQG(06)gq/05; Han & Ma PLB(06)gq/05; Han CQG(10) [path integral]; > s.a. dirac quantization.
@ Simplicity constraints: Bodendorfer et al CQG(13)-a1105 [quantum]; Anzà & Speziale CGQ(15)-a1409 [secondary].
@ Simplified theories: Henderson et al PRD(13)-a1204, PRD(13)-a1210 [U(1)\(^3\) toy model]; Lewandowski & Lin PRD(17)-a1606 [U(1)\(^3\) toy model].

Representations, Special Solutions and Related Topics > s.a. minisuperspace; models [with symmetries]; quantum cosmology and lqc.
* Holonomy representation: The Ashtekar-Lewandowski vacuum is independent from any classical background; It is maximally peaked on the configuration describing a totally degenerate spatial geometry, and maximally spread in the canonically conjugate variables encoding the extrinsic geometry.
* Flux representation: A representation dual to the Ashtekar-Lewandowski one, based on the Dittrich-Geiller vacuum which is diffeomorphism-invariant, peaked on flat connections and maximally spread in spatial geometry; Appears to be more natural for discussing semiclassical states and spin foams.
* Koslowski-Sahlmann representation: A generalization of the representation underlying lqg; The vacuum is peaked on a certain backgound geometry and not invariant under spatial diffeomorphisms, and state labels include a background electric field which describes 3D excitations of the triad.
@ Vacuum: Varadarajan PRD(02)gq [gravitons], CQG(05)gq/04 [graviton vacuum]; Dittrich & Geiller CQG(15)-a1401 + CQG+ [vacuum state].
@ Holonomy representation: Bilski a2012 [lattice regularization method].
@ Flux representation: Baratin et al CQG(11)-a1004; Dittrich & Geiller CQG(15)-a1412 [classical framework]; Cattaneo & Pérez a1611 [Poisson brackets of 2D smeared fluxes].
@ Koslowski-Sahlmann representation: Koslowski & Sahlmann Sigma(12) [vacuum with non-degenerate geometry]; Campiglia & Varadarajan CQG(14)-a1311 [diffeomorphism constraint], CQG(14)-a1406 [configuration space], CQG(15)-a1412 [asymptotically flat spacetimes].
@ Special solutions: Borja et al JPCS(11)-a1012 [simple model of 2 vertices linked by edges]; Beetle et al IJMPD(16)-a1603, a1706 [homogeneous and isotropic cosmologies]; Mäkinen a2004 [quantum-reduced, operators]; > s.a. anti-de sitter spacetime [asymptotically AdS]; FLRW models; gowdy models; inflation; Lemaître-Tolman-Bondi Solutions.
@ Other topics: Torre CQG(88) [propagator]; Arnsdorf & García CQG(99)gq/98 [spinorial states from topology]; Speziale a0810-ASL [n-point functions]; Yang & Ma PRD(09)-a0812 [quasilocal energy]; Botelho GRG(12)-a0902 [and fermionic string Ising models]; Bahr CQG(11)-a1006 [the EPRL model and knottings in the physical Hilbert space]; Borja et al CQG(11)-a1010, a1110-proc, AIP(12)-a1201 [U(N) tools]; Rovelli & Zhang CQG(11) [3-point functions]; Yamashita et al PTEP(14)-a1312 [generalized BF state]; Guo JMP(18)-a1611 [transition probability spaces]; > s.a. group theory; M-theory [duality]; quantum simulations.

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{2π/k+2}, k:= 6π/G2Λ.
@ Scalar fields: Kiefer PLB(89); Matschull CQG(93)gq; Han & Ma CQG(06)gq; Ita gq/07v1, a0710v1; Domagała et al PRD(10)-a1009; Alesci et al PRD(15)-a1504 [Hamiltonian operator]; Lewandowski & Sahlmann a1507 [Hilbert space and constraint].
@ Einstein-Maxwell theory: Gambini & Pullin PRD(93)ht/92 [and loop representation]; Krasnov PRD(96)gq/95 [with fermions].
@ 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]; Gambini & Pullin PLB(15)-a1506 [no fermion doubling]; Barnett & Smolin a1507; Mansuroglu & Sahlmann PRD(21)-a2011 [fermion spins]; > s.a. lattice fermions.
@ Connection with string theory: Gambini & Pullin IJMPD(14)-a1406-GRF; Vaid a1711 [via quantum geometry].
@ 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]; Date & Hossain Sigma(12)-a1110 [rev]; Husain & Pawlowski a1305-MG13 [computable framework]; Okołów JMP(17)-a1601 [arbitrary tensor fields, projective quantum states]; Liegener & Thiemann PRD(16)-a1605 [Einstein-Yang-Mills theory, fundamental spectrum]; Mansuroglu & Sahlmann a2011 [arbitrary spin].
@ Cosmological constant: Alexander & Calcagni FP(08)-a0807 [as a Fermi-liquid theory]; Dupuis & Girelli PRD(13)-a1307 [and quantum groups], PRD(14)-a1311 [observables]; > s.a. cosmological constant.
@ 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, PhD(07)-a0709 [real Immirzi parameter]; Ita a0705, a0705v1, a0706 [canonical and path integral]; Ita a0805/Sigma, HJ-a0901 [Chang-Soo variables], a0806, a0904; > s.a. minisuperspace; quantum gauge theory.

Online Resources
> Online seminars, blogs, videos: International Loop Quantum Gravity Seminar talks, portal and blog; YouTube 2019 video.
> Reference pages: see Wikipedia page; page; Dan Christensen's page; Seth Major's reading guide.

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