133:
36:
501:
of the connection along every link (closed path) of the graph, determines representation matrices corresponding to every link, multiplies all matrices and intertwiners together, and contracts indices in a prescribed way. A remarkable feature of the resulting functional is that it is invariant under
762:
Similar quantization applies to the volume operator. The volume of a 3D submanifold that contains part of a spin network is given by a sum of contributions from each node inside it. One can think that every node in a spin network is an elementary "quantum of volume" and every link is a "quantum of
320:
of various spin values. A network whose norm is zero has zero probability of occurrence. The rules for calculating norms and probabilities are beyond the scope of this article. However, they imply that for a spin network to have nonzero norm, two requirements must be met at each vertex. Suppose a
242:
or a compound system of particles). Three line segments join at each vertex. A vertex may be interpreted as an event in which either a single unit splits into two or two units collide and join into a single unit. Diagrams whose line segments are all joined at vertices are called
675:
750:
The formula for area eigenvalues becomes somewhat more complicated if the surface is allowed to pass through the vertices, as with anomalous diffusion models. Also, the eigenvalues of the area operator
1468:
1655:
247:. Time may be viewed as going in one direction, such as from the bottom to the top of the diagram, but for closed spin networks the direction of time is irrelevant to calculations.
431:= 5 is possible since 3 + 4 + 5 = 12 is even, and the triangle inequality is satisfied. Some conventions use labellings by half-integers, with the condition that the sum
716:
580:
743:
According to this formula, the lowest possible non-zero eigenvalue of the area operator corresponds to a link that carries spin 1/2 representation. Assuming an
1601:
938:, Oxford, 1969), Academic Press, pp. 221–244, esp. p. 241 (the latter paper was presented in 1969 but published in 1971 according to Roger Penrose,
1502:
1412:
463:
1576:
119:
1646:
1309:
Xiao-Gang Wen, "Quantum Field Theory of Many-body
Systems – from the Origin of Sound to an Origin of Light and Fermions,"
1511:
57:
100:
1757:
1752:
1747:
1606:
53:
1591:
72:
1566:
1488:
1405:
739:
of the spin network. The two-dimensional area is therefore "concentrated" in the intersections with the spin network.
1650:
79:
1670:
1631:
873:
460:
1586:
821:
that are objects very similar to spin networks. However the exact connection with spin networks is not clear yet.
800:
1762:
1571:
1436:
945:
784:
939:
846:
494:
490:
234:
A spin network, as described in
Penrose (1971), is a kind of diagram in which each line segment represents the
214:
210:
86:
1551:
1269:
822:
46:
1710:
1581:
1398:
555:
544:
68:
776:
474:
173:
1380:
806:
1716:
1339:
1235:
1188:
1141:
1090:
1043:
1013:
984:
975:
Kogut, John; Susskind, Leonard (1975). "Hamiltonian formulation of Wilson's lattice gauge theories".
756:
563:
520:
503:
218:
169:
157:
927:); and R. Penrose (1971b), "Applications of negative dimensional tensors," in D. J. A. Welsh (ed.),
915:
R. Penrose (1971a), "Angular momentum: an approach to combinatorial spacetime," in T. Bastin (ed.),
1452:
1444:
935:
524:
337:
239:
792:
1355:
1329:
1298:
1278:
1251:
1225:
1204:
1178:
1157:
1131:
1114:
1080:
1216:
Oeckl, Robert (2003). "Generalized lattice gauge theory, spin foams and state sum invariants".
1169:
Pfeiffer, Hendryk (2003). "Exact duality transformations for sigma models and gauge theories".
1742:
1665:
1636:
1106:
868:
842:
826:
818:
771:
Similar constructions can be made for general gauge theories with a compact Lie group G and a
744:
719:
670:{\displaystyle A_{\Sigma }=8\pi \ell _{\text{PL}}^{2}\gamma \sum _{i}{\sqrt {j_{i}(j_{i}+1)}}}
161:
701:
1698:
1641:
1621:
1347:
1288:
1243:
1196:
1149:
1098:
1051:
1021:
992:
312:
Given any closed spin network, a non-negative integer can be calculated which is called the
263:
153:
132:
1704:
1660:
1611:
1596:
1556:
1523:
1493:
858:
772:
536:
274:
203:
187:
93:
1126:
Pfeiffer, Hendryk; Oeckl, Robert (2002). "The dual of non-Abelian
Lattice Gauge Theory".
1343:
1239:
1192:
1145:
1094:
1047:
1017:
988:
186:
described spin networks in 1971. Spin networks have since been applied to the theory of
1692:
1686:
1626:
1616:
1541:
1460:
893:
863:
732:
1247:
1153:
1004:
Kogut, John B. (1983). "The lattice gauge theory approach to quantum chromodynamics".
1736:
1507:
1421:
888:
810:
796:
693:
548:
467:
317:
191:
183:
149:
1302:
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1208:
1161:
1118:
1721:
1359:
920:
838:
528:
456:
452:
199:
177:
165:
967:
I. B. Levinson, "Sum of Wigner coefficients and their graphical representation,"
932:
788:
478:
251:
35:
747:
on the order of 1, this gives the smallest possible measurable area of ~10 cm.
1055:
1025:
883:
814:
571:
235:
195:
1102:
17:
1561:
1310:
996:
878:
540:
470:
1293:
1264:
1110:
1071:
Rovelli, Carlo; Smolin, Lee (1995). "Spin networks and quantum gravity".
1034:
Savit, Robert (1980). "Duality in field theory and statistical systems".
924:
780:
498:
298:
1183:
1136:
1230:
535:" – that is, equivalence classes of spin networks under
294:
141:
1334:
1283:
1200:
1085:
532:
282:
1351:
180:. The diagrammatic notation can thus greatly simplify calculations.
489:
A spin network, immersed into a manifold, can be used to define a
302:
286:
278:
131:
1390:
919:, Cambridge University Press (this paper can be found online on
1394:
531:. The set of all possible spin networks (or, more accurately, "
1469:
Fashion, Faith, and
Fantasy in the New Physics of the Universe
1030:(see the Euclidean high temperature (strong coupling) section)
29:
940:"On the Origins of Twistor Theory" (Archived June 23, 2021)
27:
Diagram used to represent quantum field theory calculations
523:(LQG), a spin network represents a "quantum state" of the
209:
Spin networks can also be used to construct a particular
168:
perspective, the diagrams are a concise way to represent
451:
Formally, a spin network may be defined as a (directed)
316:
of the spin network. Norms can be used to calculate the
1385:
Group Theory: Birdtracks, Lie's, and
Exceptional Groups
250:
Each line segment is labelled with an integer called a
837:
In mathematics, spin networks have been used to study
574:
of each such operator, and the area eigenvalue equals
562:
of a two-dimensional surface ÎŁ should have a discrete
704:
583:
148:
is a type of diagram which can be used to represent
1679:
1534:
1479:
1428:
407:= 6 is impossible since 3 + 4 + 6 = 13 is odd, and
60:. Unsourced material may be challenged and removed.
710:
669:
554:One of the key results of loop quantum gravity is
969:Proceed. Phys-Tech Inst. Acad Sci. Lithuanian SSR
1320:Major, Seth A. (1999). "A spin network primer".
944:Gravitation and Geometry, a Volume in Honour of
787:are needed to make the duality exact (smearing
419:= 9 is impossible since 9 > 3 + 4. However,
929:Combinatorial Mathematics and its Applications
1406:
684:of ÎŁ with the spin network. In this formula,
8:
1060:(see the sections on Abelian gauge theories)
481:of the edge representations adjacent to it.
1582:Penrose interpretation of quantum mechanics
1128:Nuclear Physics B - Proceedings Supplements
911:
909:
321:vertex joins three units with spin numbers
1413:
1399:
1391:
680:where the sum goes over all intersections
333:. Then, these requirements are stated as:
1333:
1292:
1282:
1229:
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1135:
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791:is tricky). Later, it was generalized by
703:
650:
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631:
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588:
582:
120:Learn how and when to remove this message
1503:The Large, the Small and the Human Mind
905:
515:In the context of loop quantum gravity
7:
1602:Penrose–Hawking singularity theorems
58:adding citations to reliable sources
1387:, Princeton University Press, 2008.
1377:, Cambridge University Press, 1990.
558:of areas: the operator of the area
136:Spin network diagram, after Penrose
1375:Diagram Techniques in Group Theory
589:
25:
1647:Orchestrated objective reduction
1520:White Mars or, The Mind Set Free
845:, which correspond to spaces of
799:in 2 and 3 dimensions using the
34:
1265:"Spin Networks in Gauge Theory"
1218:Journal of Geometry and Physics
1171:Journal of Mathematical Physics
763:area" surrounding this volume.
497:on this manifold. One computes
217:which is invariant under local
45:needs additional citations for
662:
643:
1:
1607:Riemannian Penrose inequality
1248:10.1016/S0393-0440(02)00148-1
1154:10.1016/S0920-5632(01)01913-2
731:= 0, 1/2, 1, 3/2, ... is the
447:Formal approach to definition
1514:and Stephen Hawking) (1997)
1489:The Nature of Space and Time
829:states in condensed matter.
775:. This is actually an exact
1322:American Journal of Physics
949:, Biblipolis, Naples 1987).
767:More general gauge theories
1779:
1671:Conformal cyclic cosmology
1632:Penrose graphical notation
874:Penrose graphical notation
783:however, assumptions like
254:. A unit with spin number
229:
1572:Weyl curvature hypothesis
1056:10.1103/RevModPhys.52.453
1036:Reviews of Modern Physics
1026:10.1103/RevModPhys.55.775
1006:Reviews of Modern Physics
917:Quantum Theory and Beyond
785:diffeomorphism invariance
735:associated with the link
152:and interactions between
1592:Newman–Penrose formalism
1103:10.1103/PhysRevD.52.5743
443:must be a whole number.
1552:Abstract index notation
1270:Advances in Mathematics
997:10.1103/PhysRevD.11.395
823:String-net condensation
779:over a lattice. Over a
711:{\displaystyle \gamma }
391:must be an even number.
293:is an even number. For
238:of a "unit" (either an
1711:John Beresford Leathes
1651:Penrose–Lucas argument
1642:Penrose–Terrell effect
1437:The Emperor's New Mind
1294:10.1006/aima.1996.0012
1263:Baez, John C. (1996).
1130:. 106–107: 1010–1012.
795:to representations of
712:
671:
379:Fermion conservation:
172:and functions between
137:
1587:Moore–Penrose inverse
1562:Geometry of spacetime
827:topologically ordered
801:Tannaka–Krein duality
713:
672:
504:gauge transformations
219:gauge transformations
170:multilinear functions
135:
1758:Mathematical physics
1753:Loop quantum gravity
1748:Quantum field theory
1717:Illumination problem
1577:Penrose inequalities
833:Usage in mathematics
702:
581:
521:loop quantum gravity
477:are associated with
459:are associated with
245:closed spin networks
230:Penrose's definition
54:improve this article
1453:The Road to Reality
1445:Shadows of the Mind
1344:1999AmJPh..67..972M
1240:2003JGP....46..308O
1193:2003JMP....44.2891P
1146:2002NuPhS.106.1010P
1095:1995PhRvD..52.5743R
1048:1980RvMP...52..453S
1018:1983RvMP...55..775K
989:1975PhRvD..11..395K
843:character varieties
755:are constrained by
617:
543:; it constitutes a
527:on a 3-dimensional
525:gravitational field
338:Triangle inequality
240:elementary particle
1381:Predrag Cvitanović
813:have also defined
708:
667:
630:
603:
138:
1730:
1729:
1666:Andromeda paradox
1637:Penrose transform
1567:Cosmic censorship
1201:10.1063/1.1580071
1079:(10): 5743–5759.
977:Physical Review D
869:Character variety
819:tensor categories
745:Immirzi parameter
720:Immirzi parameter
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162:quantum mechanics
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16:(Redirected from
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1763:Diagram algebras
1699:Jonathan Penrose
1656:FELIX experiment
1622:Penrose triangle
1527:
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1512:Nancy Cartwright
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1480:Coauthored books
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213:on the space of
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1661:Trapped surface
1612:Penrose process
1597:Penrose diagram
1557:Black hole bomb
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1524:Brian W. Aldiss
1518:
1500:
1494:Stephen Hawking
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1424:
1419:
1373:G. E. Stedman,
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1352:10.1119/1.19175
1328:(11): 972–980.
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1184:hep-lat/0205013
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1137:hep-lat/0110034
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773:connection form
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275:Planck constant
273:is the reduced
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204:Rodolfo Gambini
188:quantum gravity
174:representations
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43:This article
41:
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18:Spin networks
1722:Quantum mind
1547:Spin network
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962:Early papers
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928:
921:John C. Baez
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793:Robert Oeckl
789:Wilson loops
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568:spin network
567:
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200:Jorge Pullin
182:
166:mathematical
146:spin network
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64:
52:Please help
47:verification
44:
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946:I. Robinson
847:connections
815:string-nets
495:connections
461:irreducible
252:spin number
215:connections
1737:Categories
1312:. (Dubbed
900:References
884:String-net
572:eigenstate
499:holonomies
491:functional
485:Properties
473:and whose
297:, such as
281:, such as
236:world line
225:Definition
211:functional
196:Lee Smolin
80:newspapers
1701:(brother)
1695:(brother)
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1496:) (1996)
879:Spin foam
825:produces
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623:∑
619:γ
605:ℓ
601:π
590:Σ
541:countable
471:Lie group
299:electrons
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154:particles
110:June 2022
1743:Diagrams
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1535:Concepts
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564:spectrum
475:vertices
309:is odd.
295:fermions
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