 Loop Space Representation of Quantum Gravity

In General > s.a. canonical quantum gravity / 3D quantum gravity; holography in field theory; loops; quantum black holes; symplectic structures.
* Idea: Originally formulated in terms of non-intersecting closed loops, now accomodates intersecting ones, and can be seen as equivalent to the (gauge-invariant) spin network formulation of the connection representation of canonical quantum gravity (note that what is usually meant by "loop quantum gravity" is the connection representation, not the loop representation).
* Elementary variables: The small T-algebra (> see loop formulation of canonical general relativity).
* Configuration space: Heuristically, diffeomorphism equivalence classes of multiloops.
* States: Heuristically, they must be (generalized) knot/link invariants of Σ, from the diffeomorphism constraint.
* Remark: Because the whole construction is gauge-invariant, operators corresponding to $$E^a{}_i$$ cannot be constructed.
* Action of operators:

$$\langle$$β| T[α]:= $$\langle$$β ∪ α|   or    $$1\over2$$($$\langle$$β # α| − $$\langle$$β # α−1|) ;

$$\langle$$β| T a[α](s) := $${1\over2}\hbar G \displaystyle\oint$$dt (dβa(t)/dt) δ3(α(s); β(t)) ($$\langle$$β #s α| −$$\langle$$β #s α−1|) .

* Solutions of the scalar constraint: The second coefficient of the Conway polynomial, a2 (for Λ = 0; probably not a3); The Kauffman bracket knot polynomial (for Λ ≠ 0).
> Online resources: see Wikipedia page [loop representation in gauge theories and quantum gravity].

References > s.a. angular momentum; lattice gauge theories; models; quantum cosmology; quantum gauge theories [precursors].
@ General: Rovelli in(88); Rovelli & Smolin PRL(88), NPB(90); Waldrop Sci(90)dec; Bezerra AP(90); Rayner CQG(90) [for (qab, pab) and for scalar field], CQG(90) [inner product and operators]; Rovelli in(91); Gambini PLB(91); Loll NPB(91), ht/93; Baez in(94)ht/93; Pullin AIP(94)ht/93; Gambini & Pullin CQG(18)-a1802; Lim a2105.
@ Reviews: Rovelli in(90); Smolin in(91); Brügmann LNP(94)gq/93; Gambini & Pullin 96; Rovelli & Upadhya gq/98; Pullin IJTP(99)gq/98-conf; Gaul & Rovelli LNP(00)gq/99-ln.
@ And connection representation: De Pietri CQG(97)gq/96, gq/97-MG8; Thiemann JMP(98)ht/96 [loop transform].
@ Constraints: Brügmann & Pullin NPB(93); Brügmann NPB(96)gq/95 [algebra]; Gambini et al IJMPD(95)gq/94 [algebra].
@ Hamiltonian: Rovelli & Smolin PRL(94)gq/93; Gaul & Rovelli CQG(01)gq/00.
@ Solutions: Aldaya & Navarro-Salas PLB(91); Brügmann et al PRL(92), NPB(92), GRG(93); Gambini et al in(92); Gambini & Pullin gq/93-in [Gauss linking number]; Di Bartolo et al JMP(95)gq; Hayashi CMP(96)qa/95 [Vassiliev invariants]; Gambini & Pullin PRD(96)gq/95, CQG(96)gq; Griego NPB(96)gq/95, PRD(96)gq/95 [Jones polynomial].
@ For non-compact spaces: Arnsdorf & Gupta NPB(00)gq/99; Arnsdorf gq/00-MG9 [asymptotically flat].
@ On a lattice: Loll CQG(95); Ezawa MPLA(96)gq/95; Fort et al PRD(97)gq/96 [lattice knot theory].
@ Related topics: Baez CQG(93), gq/94, ed-94 [knots, tangles]; Griego NPB(96)gq [extended knots]; Krasnov PRD(97)gq/96 [boundary states].

With Matter > s.a. matter phenomenology in quantum gravity; supergravity.
@ Einstein-Maxwell: Gambini & Pullin PRD(93); Krasnov PRD(96)gq/95 [+ fermions].
@ Fermions: Morales & Rovelli PRL(94)gq, NPB(95); Smolin gq/94 [and topology]; Baez & Krasnov JMP(98)ht/97; Vlasov mp/99.

Variations, Generalizations
@ Extended loop representation: Di Bartolo PRL(94)gq/93, PLB(96)gq [Gauß constraint], et al PRD(95)gq/94, JMP(95)gq; Shao et al IJMPA(02); Gambini et al a1907 [covariance].
@ Related variables: Schilling JMP(96)gq/95 [generalized holonomies]; Varadarajan & Zapata CQG(00)gq [fluxes].
@ With torsion: Mullick & Bandyopadhyay IJMPA(96).
@ Q-deformed: Major & Smolin NPB(96)gq/95; Borissov et al CQG(96)gq/95; Antonsen gq/97; Major CQG(08)-a0708; > s.a. spin networks.