Loop Space Representation of Quantum Gravity| 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.
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