Spin-Foam Models  

In General > s.a. 2D and 3D quantum gravity; BF theory; lattice field theory [generalized gauge theory] / path-integral quantum gravity.
* Idea: The path-integral counterpart of loop quantum gravity, spacetime versions of spin networks used either as tools to calculate amplitudes or considered as histories; It uses 2D complexes with faces labelled by representations of SU(2), edges labelled by intertwiners; States are combinations of spin networks, and amplitudes are often calculated as sums over spin foams σ bounded by fixed spin networks s and s',

W(s, s') = ∑spin foams bounded by s and s' measure(σ) Πvertices v Av(σ) ;

The difficult part is coming up with a good proposal for the vertex amplitudes Av.
@ Intros, reviews: Baez LNP(00)gq/99; Oriti RPP(01)gq; Livine PhD(03)gq; Oriti PhD(03)gq; Pérez CQG(03)gq; Miković ht/04-conf; Pérez in(09)gq/06; Alexandrov & Roche PRP(11)-a1009; Bonzom PhD(10)-a1009; Livine Hab(10)-a1101; Banerjee et al Sigma(12)-a1109; Pérez PiP-a1205, LRR(13)-a1205; Långvik a1303; Engle ch(14)-a1303; Rovelli & Vidotto 14.
@ General references: Freidel & Louapre NPB(03)gq/02 [diffeomorphisms]; Bojowald & Pérez GRG(10)gq/03 [anomalies and criteria]; Oeckl gq/03-proc [general boundary approach]; Bahr et al CQG(11)-a1010 [systematic approach, operator spin foams]; Rovelli & Smerlak CQG(12)-a1010 [combinatorial structure, summing = refining]; Kisielowski et al CQG(12)-a1107 [Feynman diagrammatic approach]; Wieland CQG(14)-a1301 [Hamiltonian formulation]; Bodendorfer & Neiman CQG(13)-a1303 [effective action, and trans-planckian regime]; Shirazi & Engle CQG(14)-a1308 [purely geometric path integral]; Smolin CQG(14) [general relativity as the equation of state]; Hnybida a1411-PhD [generating functionals]; Finocchiaro & Oriti a1812; Belov a1905-PhD [geometry]; Asante et al PRL(20)-a2004 [effective models].
@ Simplicity constraints: Alexandrov PRD(08)-a0802, Han & Thiemann CQG(13)-a1010 [and closure constraints]; Banburski & Chen PRD(16)-a1512 [simpler approach]; Han & Huang PRD(17)-a1702 [as surface defect in SL(2,C) Chern-Simons theory].
@ Euclidean: Reisenberger & Rovelli PRD(97)gq/96 [sum-over-surfaces lqg], gq/00, CQG(01)gq/00; Reisenberger gq/97; Iwasaki gq/00; Pérez NPB(01)gq/00 [finiteness]; Bianchi et al PRD(10)-a1004 [holomorphic representation]; Han & Zhang CQG(12)-a1109 [on a 4D simplicial complex].
@ EPRL model: Perini a1211 [on arbitrary 2-complexes, holomorphic representation]; Han & Krajewski CQG(14)-a1304 [path integral representation]; Bahr & Belov PRD(18)-a1710 [volume simplicity constraint]; Donà et al PRL(19)-a1903 [numerical study].
@ Other Lorentzian: Baez CQG(98)gq/97; De Pietri NPPS(97)gq, gq/99-proc; De Pietri & Freidel CQG(99)gq/98; Freidel & Krasnov ATMP(98)ht; Iwasaki gq/99; Pérez & Rovelli PRD(01)gq/00, PRD(01)gq/00; Crane et al PRL(01)gq [finiteness of state sum]; Gambini & Pullin PRD(02)gq/01 [finite theory]; Maran gq/03, PRD(04)gq [and canonical quantum gravity]; Pereira CQG(08)-a0710; Barrett et al CQG(10)-a0907, GRG(11) [graphical calculus, asymptotics]; Conrady & Hnybida CQG(10)-a1002; Ding & Rovelli CQG(10)-a1006 [boundary Hilbert space and volume operator]; Han & Zhang CQG(13)-a1109 [on a 4D simplicial complex]; Bianchi & Ding PRD(12)-a1109 [propagator]; Perlov a1312; Liu & Han a1810 [with timelike triangles]; Asante et al a2104 [effective models].
@ And causality: Gupta PRD(00)gq/99; Oriti BJP(05)gq/04-proc, PRL(05)gq/04 [Feynman propagator]; Immirzi a1610 [and the Regge action]; > s.a. causal sets [energetic causal sets].
@ And lqg: Pérez & Rovelli in(11)gq/01 [transition amplitudes]; Arnsdorf CQG(02)gq/01; Livine CQG(02)gq, Livine & Oriti NPB(03)gq/02; Alesci et al PRD(08)-a0807 [and physical inner product]; Kamiński et al CQG(10)-a0909, Ding et al PRD(11)-a1011 [arbitrary 2-cell spin-foams]; Marin a1003; Rovelli & Speziale PRD(11)-a1012; Bonzom PRD(11)-a1101 [and Hamiltonian constraint]; Dupuis PhD(10)-a1104 [and semiclassical limit]; Alexandrov et al Sigma(12)-a1112; Speziale & Wieland PRD(12)-a1209 [and twistorial structure]; Engle CQG(13)-a1301 [without piecewise linearity]; Thiemann & Zipfel CQG(14)-a1307 [spin-foam projector]; Yang et al a2102; > s.a. canonical quantum gravity [covariant].
@ Vertex amplitudes: Livine & Speziale PRD(07)-a0705, EPL(08)-a0708; Engle et al PRL(07)-a0705 [from relationship with lqg], NPB(08)-a0708, NPB(08)-a0711 [flipped, EPRL]; Freidel & Krasnov CQG(08)-a0708; Engle & Pereira CQG(08)-a0710, PRD(09)-a0805; Conrady & Freidel CQG(08)-a0806 [path-integral representation]; Khavkine a0810-wd; Bonzom et al CQG(10)-a0911 [recurrence relations]; Alexandrov PRD(10)-a1004 [and canonical quantization]; Engle PRD(13)-a1111; Bianchi & Hellmann Sigma(13); Engle & Zipfel PRD(16)-a1502 [Lorentzian]; Mielczarek a1810 [using quantum algorithms]; > s.a. geometry.
@ Face amplitudes: Bianchi et al CQG(10)-a1005; Regoli PhD-a1104.
@ Quantum tetrahedra: Barbieri NPB(98)gq/97; Baez & Barrett ATMP(99)gq; Livine & Speziale CQG(08)-a0711 [boundary state]; Terno CQG(09)-a0808 [classical limit]; Freidel et al CMP(10)-a0905 [holomorphic factorization]; Carfora et al JPCA(09)-a1001 [6j symbol]; Schliemann CQG(13)-a1307 [large-volume limit].

With Matter / For Other Theories
@ General references: Steinhaus PRD(15)-a1509 [coupled intertwiner dynamics toy model].
@ With Yang-Mills fields: Miković CQG(02)ht/01, CQG(03)ht/02; Speziale CQG(07)-a0706 [3D]; Alexander et al PLB(12)-a1105 [and matter, unification].
@ With fermions: Miković AIP(06)gq/05; Fairbairn GRG(07)gq/06 [3D]; Dowdall & Fairbairn GRG(11)-a1003 [3D, observables]; Bianchi et al CQG(13)-a1012 [and Yang-Mills fields]; Han & Rovelli CQG(13)-a1101 [PCT symmetry, etc]; Crane a1105 [extension including the standard model].
@ With other matter: Smilga ht/04 [SO(3, 2)]; Fairbairn & Livine CQG(07)gq [3D]; Baccetti et al CQG(10)-a1004 [N = 1 supersymmetric]; Fani & Kaviani IJGMP(14)-a1102 [dimensional reduction and emergence of non-gravitational fields]; Kisielowski & Lewandowski CQG(19)-a1807 [scalar field].
@ And strings / M-theory: Grosse & Schlesinger PLB(02) [topological quantum field theories of 3-forms].

Related Topics and Other Variations > s.a. acceleration [maximal]; Pachner Moves; semiclassical quantum gravity.
@ Holonomy formulation: Magliaro & Perini IJMPD(12)-a1010; Bahr JPCS(12)-a1112 [and coarse graining]; Bahr et al PRD(13)-a1208 [new holonomy formulation]; Dittrich et al CQG(13)-a1209; Hellmann & Kamiński a1210 [on arbitrary triangulations, geometric asymptotics], JHEP(13)-a1307.
@ Spin nets: Dittrich et al NJP(13)-a1306 [coarse graining and continuum phases]; Dittrich et al PRD(14)-a1312 [quantum group spin nets].
@ Classical limit: Magliaro & Perini EPL(11)-a1108; Engle PLB(13)-a1201 [correct semiclassical limit]; Vojinović GRG(13)-a1307 [issue arising from vertex amplitudes]; Steinhaus & Thürigen a1803 [spectral dimension]; Donà et al CQG-a1909 [numerical].
@ Analytic continuation: Maran gq/05 [complex general relativity and various signatures]; Han & Liu a2104 [complex critical points].
@ Other variations: Zapata JMP(02) [continuum model]; Miković IJMPA(03)gq/02-conf [quantum field theory of spin networks]; García-Islas gq/04 [p-adic]; Baratin & Freidel CQG(07) [and 3D Feynman diagrams for quantum field theory], CQG(07)ht/06 [and 4D Feynman diagrams for quantum field theory]; Christensen et al PLB(09)-a0710 [area correlations]; Mamone & Rovelli CQG(09)-a0904 [second-order amplitudes]; Conrady CQG(10)-a1003 [with timelike surfaces]; Denicola et al CQG(10)-a1005 [with topological data]; Han JMP(11)-a1012, Fairbairn & Meusburger JMP(12)-a1012, PoS-a1112 [deformed versions]; Bonzom & Smerlak AHP(12)-a1103 [bubble divergences]; Magliaro & Perini CQG(11)-a1103 [curvature and discrete Einstein equations]; Bahr et al a1103 [with finite groups]; Magliaro & Perini IJMPD(13)-a1105 [and Regge gravity]; Perini JPCS(12)-a1110; Dupuis et al a1201-proc [spinors and twistors]; Bonzom & Smerlak PRL(12)-a1201 [gauge symmetries and "cellular quantization"]; Puchta JPCS(12) [graphs and characterization of foams]; García-Islas IJMPA(12)-a1206 [measurement and information]; Immirzi CQG(14)-a1311 [spinor construction of amplitudes]; Perlov a1407; Hnybida CQG(16)-a1508 [without spins, using generating functions instead of recoupling theory].
@ Coarse-graining: Markopoulou CQG(03)gq/02; Dittrich et al NJP(12)-a1109; Zapata JPCS(12)-a1203; Dittrich & Steinhaus NJP(14)-a1311 [refining, entangling operators]; Dittrich et al PRD(16)-a1609 [intertwiners]; Delcamp & Dittrich a1612 [systematic scheme for 3D lattice gauge models based on decorated tensor networks]; Steinhaus a2007 [rev].
@ Renormalization: Oeckl in(06)gq/04; Livine & Oriti JHEP(07)gq/05; Smerlak PhD(11)-a1201 [and divergences]; Bahr a1407 [background-independent]; Bahr & Steinhaus PRL(16)-a1605 [evidence for a phase transition], PRD(17)-a1701 [hypercuboidal renormalization].
@ Perturbations: Baez gq/99; Martins & Miković Sigma(11)-a0911.
@ Other types of spacetimes: Han & Zhang PRD(16)-a1606 [near a classical curvature singularity]; > s.a. 3D black holes.
@ Computational / numerical work: Khavkine CQG(09)-a0809; Dittrich & Eckert JPCS(12)-a1111.
@ Other related topics: Bahr a1812 [non-convex 4D polytopes]; Ansel GRG(21) [open quantum system theory]; > s.a. black-hole entropy; dimensionality of spacetime; FLRW models; graviton; linearized quantum gravity [propagator]; quantum cosmology.

Other State Sum Models
@ General features: Barrett gq/00-conf.
@ Categorical generalizations: Baratin & Wise AIP(09)-a0910 [based on 2-group representations]; Miković & Vojinović CQG(12)-a1110 [BFCG formulation]; Miković RVMP(13)-a1302 [spin-cube state sum models, representations of the Poincaré 2-group]; Miković & Vojinović JPCS(14)-a1512; Miković et al CQG(18)-a1807 [BFCG formulation, Hamiltonian analysis].
@ Cube-based: Baratin et al NJP(12)-a0812 [cubations and Holst action]; Vojinović PRD(16)-a1506 [spincube models and causal dynamical triangulations].
@ Other models: Davids gq/01 [Lorentzian, SU(1,1)]; Pérez ATMP(01)gq/02 [Plebański SO(4) model]; Miković MPLA(05)gq-conf; Bonzom PRD(09)-a0905 [from lattice path integrals]; Geiller & Noui CQG(12)-a1112 [3D Plebański SO(4) model].
> Quantum-gravity related: see Barrett-Crane Model; Crane-Yetter Model; Ponzano-Regge Model; Turaev-Viro Theory.
> In gauge theory and other theories: > see path-integral quantization for gauge theories; string theory.
> Online resources: see Dan Christensen's page.

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 29 apr 2021