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; Ruiz a1206 [introduction, and invariants of 3-manifolds, decomposition theorem].

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):= ∏en In R je(U(e, A)) .

* Intertwiners: Maps Iv: ⊗incoming e  je → ⊗outgoing e  je associated with vertices v of graphs.
* Properties: \(\langle\)ψS | ψS'\(\rangle\) = 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]; Freidel & Hnybida JMP(13)-a1201 [generating all SU(2) spin networks associated with a given graph]; Schroeren FP(13)-a1206 [decoherence functional, decoherent histories formulation]; Bitencourt et al LNCS-a1211-conf [asymptotic computations]; Bonzom et al CMP(16)-a1504 [spin network generating series and Ising models].
@ Intertwiners: Bianchi et al PRD(11)-a1009 [and quantum polyhedra]; Freidel & Hnybida CQG(14)-a1305 [new discrete and coherent basis]; Dittrich & Hnybida a1312 [2D Ising model and continuum limit with propagating degrees of freedom].
@ 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].
@ Coarse-graining: Dittrich et al NJP(12)-a1109, Dittrich NJP(12)-a1205 [and cylindrically consistent dynamics]; Dittrich et al NJP(13) [dynamics of intertwiners]; Livine CQG(14)-a1310 [and renormalization]; Dittrich et al PRD(16)-a1609 [coarse-graining flow]; Charles PhD-a1705.
@ 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].
@ Entanglement: Chirco et al a1703 [and separability]; Mele a1703-MS [quantum metric].
@ 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]; Långvik & Speziale PRD(16)-a1602 [twisted geometries, twistors and conformal transformations]; Charles & Livine GRG(16)-a1603 [Fock space], GRG(17) [closure constraint as a Bianchi identity]; > 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]; Dupuis et al GRG(14)-a1403 [hyperbolic twisted geometries]; > s.a. topological field theories.
@ Braided ribbon networks: Hackett & Wan JPCS(11)-a0811 [and degeneracy of states]; Hackett a1106 [invariants]; Bilson-Thompson et al Sigma(12)-a1109 [and emergent braided matter]; > s.a. Ribbons.
@ Other generalizations: 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 CQG(12)-a1111 [topspin network formalism]; Marcolli & van Suijlekom JGP(13)-a1301 [gauge networks in almost-commutative manifolds]; Feller & Livine CQG(16)-a1509 [Ising spin network states]; Perlov & Bukatin a1510 [without 3+1 slicing]; Zuo a1607 [generalized to Kac-Moody algebra].

Related Models and Topics > s.a. 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]; Amaral et al a1602 [quantum walk on a spin network]; Anzà & Chirco PRD(16)-a1605 [emergence of a typical state, and quantum geonetry].
> Related topics: see Fusion Coefficients; graph types; quantum information.

Online Resources > Greg Egan's page.


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