Connection Representation of Quantum Gravity ("Loop Quantum Gravity")| 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;
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 22 apr 2020