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 on 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 
,
from the diffeomorphism constraint.
* Remark: Because the whole construction is gauge-invariant, operators
corresponding to Eai cannot be constructed.
* Action of operators:
| T[
]:=
| or 
( # | – # –1|)
;
| T a[](s)
:= 
G 
dt (da(t)/dt) 
3((s);
(t))
( #s | – #s –1|)
.
* Solutions of the scalar
constraint: The second coefficient of the Conway polynomial,
a2 (for 
=
0; probably not a3); The Kauffman
bracket knot polynomial (for 
0).
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.
@ Reviews: Rovelli in(90); Smolin in(91); Brügmann gq/93-in; Gambini & Pullin
96; Rovelli & Upadhya gq/98;
Pullin IJTP(99)gq/98-in;
Gaul & Rovelli
LNP(00)gq/99-in.
@ 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 [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).
@ 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.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008