Spin
Networks in Quantum
Gravity| Spin
Networks in Quantum
Gravity |
Original Penrose Version > s.a. quantum
spacetime and discrete
models; quantum
technology.
* Idea: A trivalent graph, with edges labelled by half-integers j corresponding
to representations of G = SO(3), subject to consistency conditions
coming from spin composition rules (Clebsch-Gordan coefficients).
@ References: Penrose in(70); Kauffman & Lins 94.
In Loop Quantum Gravity > s.a. 3D quantum gravity;
quantum gauge theories; quantum spacetime;
semiclassical quantum gravity.
* Motivation: A complete,
but not overcomplete, set of orthonormal states in the kinematical gauge theory Hilbert space.
* Idea: "Colored
graphs" for a group G, taken to be SU(2),
i.e., triples S = (γ, j, I)
of graphs γ with edges e labeled
by irreducible representations je of
SU(2), and vertices n by intertwiners; In the connection
representation, spin network states are
ψS(A):= ∏e ∏n In R je(U(e, A)) .
* Intertwiners: Maps Iv:
⊗incoming
e je →
⊗outgoing
e je associated
with vertices v of graphs.
* Properties: 
ψS |
ψS'
= δγ,
γ' δjj' δII',
and the eigenvalues of area and volume operators are discrete.
@ Precursors: Loll NPB(92), NPB(93); Smolin in(92).
@ General references: Baez AiM(96)gq/94,
in(96)gq/95;
Rovelli & Smolin PRD(95)gq;
Foxon CQG(95)gq/94;
Borissov et al CQG(96)gq/95;
Barbieri gq/97 [vertices];
Barrett & Crane JMP(98)gq/97;
Smolin gq/97;
Reisenberger JMP(99)gq/98;
Major AJP(99)nov-gq [primer];
Barrett & Steele
CQG(03)gq/02 [asymptotics];
Miković CQG(03)gq [and
vacuum]; Lorente gq/05-proc
[rev]; Conrady & Freidel JMP(09)-a0902 [and
reduced phase space of tetrahedra]; Dupuis & Livine PRD(10)-a1008 [lifting
to projected spin networks]; Bianchi et al PRD(11)-a1009 [intertwiners and quantum polyhedra]; Freidel & Hnybida a1201 [generating all SU(2) spin networks associated with a given graph].
@ Evolution: Markopoulou gq/97, & Smolin
NPB(97)gq,
PRD(98)gq/97,
PRD(98)ht/97;
Borissov PRD(97)gq/96, & Gupta
PRD(99)gq/98 [including
dual triangulations]; Miković CQG(01)gq [quantum
field theory]; Smolin & Wan NPB(08)
[braid states]; > s.a. spin-foam models.
@ Spin webs: Lewandowski & Thiemann
CQG(99)gq [all piecewise smooth].
@ Invariants: Gambini IJTP(99), et al NPB(98)gq,
Di Bartolo et al PRL(00)gq/99 & CQG(00)gq/99,
CQG(00)gq/99 [Vassiliev
knot invariants]; Carbone et al gq/99.
@ Braid excitations: Wan a0710;
Smolin & Wan NPB(08)-a0710; Wan NPB(09)
[effective theory in terms of Feynman diagrams].
@ Related topics: Freidel & Krasnov JMP(00)ht/99 [Feynman
graphs]; Lewandowski & Marolf IJMPD(98)
[T* states];
Zizzi Ent(00)gq/99 [holography];
Ma & Ling PRD(00)gq [Q];
Baez & Barrett
CQG(01)gq [integrability];
Pfeiffer ATMP(02)gq [positivity
of evaluations]; Miković a0706 [and
graviton propagator]; Rovelli & Vidotto PRD(10)-a0905 [BGS
entropy]; Borja et al CQG(11)-a1010 [U(N) framework]; Dittrich et al a1109 [coarse-graining]; > s.a. gravitational
thermodynamics; string
theory.
Modifications > s.a. supergravity.
* Extended: Non gauge-invariant
spin network states, given by quintuplets N = (γ, j, I, ρ,
M).
* Deformed: Edges of spin
networks are enlarged to ribbons or tubes, so
the network becomes a tubular, genus-g manifold, decomposable into trinions,
separated by circles; Each circle is labeled by a representation of SU(2)q,
each trinion by an intertwiner; Motivation are inclusion of a cosmological constant,
symmetries.
* Topspin networks: An extension of loop quantum gravity which allows topological information to be encoded in spin networks; It requires only minimal changes to the phase space, C*-algebra and Hilbert space of cylindrical functions.
@ Deformed: Markopoulou & Smolin PRD(98)gq/97 [(p, q)
string evolution]; Barrett & Crane
CQG(00)gq/99 [Lorentzian];
> s.a. topological field theories.
@ Braided
ribbon networks: Hackett & Wan JPCS(11)-a0811 [and degeneracy of states]; Hackett a1106 [invariants]; Bilson-Thompson et al a1109 [and emergent braided matter]; > s.a. Ribbons.
@ Other: Ashtekar & Lewandowski CQG(97)gq/96 [extended];
Baez & Sawin JFA(98)qa/97 [diffeomorphism-invariant];
Ling JMP(02)
[supersymmetric]; Freidel & Livine
JMP(03)ht/02 [non-compact G];
He & Wan NPB(08)-a0805,
NPB(08)-a0805 [framed,
braid excitations and C, P, T]; Duston a1111 [topspin network formalism].
Related Models and Topics > s.a. ising
model; lattice
field theory; spin
foam;
spin models; SU(2).
@ References: Martins & Miković CMP(08)gq/06 [and
3-manifold invariants]; Chen & Zhu IJMPA(08)-gq/07 [evolution
of spin labels and self-organized criticality]; Aquilanti et al PS(08)-a0901 [angular momentum recoupling in general].
> Related topics:
see Fusion Coefficients; graph
types; quantum information.
Online Resources > Greg Egan's page.
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mar 2012