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;
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):= ∏e ∏n 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];
Shoshany a1912-PhD [and piecewise-flat geometries].
@ 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];
Long et al PRD(19)-a1906 [coherent intertwiner solution].
@ 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 PRD(18)-a1703 [and separability];
Mele a1703-MS [quantum metric];
Livine PRD(18)-a1709 [intertwiner entanglement];
Baytaş et al PRD(18)-a1805 [guing polyhedra];
Ling et al ChPC(19)-a1811 [with boundary];
> s.a. entanglement entropy.
@ With matter:
Shoshany CQG(20)-a1911 [coupled to cosmic strings].
@ 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];
Kocik JKTR(18)-a1807 [modified skein relations];
Livine CQG(19)-a1902 [area propagator];
Freidel & Livine GRG(19)-a1810 [bubble networks];
> 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];
Freidel et al CQG(19)-a1906 [tube networks carrying Virasoro representations].
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-proc [quantum walk on a spin network];
Anzà & Chirco PRD(16)-a1605 [emergence of a typical state, and quantum geometry];
Mäkinen a1910-PhD
[SU(2) recoupling theory and graphical methods].
@ Numerical:
Mielczarek a1801 [on a D-Wave machine];
Li et al nComm(19)-a1712 [on a quantum simulator];
Czelusta & Mielczarek PRD(21)-a2003 [qubit of space].
> Related topics:
see Fusion Coefficients; graph types;
gravity and information.
Online Resources > see Greg Egan's page.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 17 feb 2021