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{{Short description|Diagram used to represent quantum field theory calculations}} {{More citations needed|date=June 2022}} [[File:Spin-network-after-penrose.svg|thumb|Spin network diagram, after Penrose]] In [[physics]], a '''spin network''' is a type of diagram which can be used to represent [[Quantum state|states]] and interactions between [[Particle physics|particles]] and [[Quantum field theory|fields]] in [[quantum mechanics]]. From a [[mathematical]] perspective, the diagrams are a concise way to represent [[multilinear function]]s and functions between [[representation theory|representations]] of [[matrix group]]s. The diagrammatic notation can thus greatly simplify calculations. [[Roger Penrose]] described spin networks in 1971.<ref name=Penrose71/> Spin networks have since been applied to the theory of [[quantum gravity]] by [[Carlo Rovelli]], [[Lee Smolin]], [[Jorge Pullin]], [[Rodolfo Gambini]] and others. Spin networks can also be used to construct a particular [[functional (mathematics)|functional]] on the space of [[connection (mathematics)|connections]] which is invariant under local [[gauge transformation]]s. == Definition == === Penrose's definition === A spin network, as described in Penrose (1971),<ref name=Penrose71>R. Penrose (1971a), "Angular momentum: an approach to combinatorial spacetime," in T. Bastin (ed.), ''Quantum Theory and Beyond'', Cambridge University Press (this paper can be found online on [[John C. Baez]]'s [http://math.ucr.edu/home/baez/penrose/ website]); and R. Penrose (1971b), "Applications of negative dimensional tensors," in D. J. A. Welsh (ed.), ''Combinatorial Mathematics and its Applications'' ([[Proceedings|Proc.]] [[Academic conference|Conf.]], 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, [https://web.archive.org/web/20210623190333/http://users.ox.ac.uk/~tweb/00001/ "On the Origins of Twistor Theory" (Archived June 23, 2021)] in: ''Gravitation and Geometry, a Volume in Honour of [[Ivor Robinson (physicist)|I. Robinson]]'', Biblipolis, Naples 1987).</ref> is a kind of diagram in which each line segment represents the [[world line]] of a "unit" (either an [[elementary particle]] 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 ''closed spin networks''. 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. Each line segment is labelled with an integer called a [[spin number]]. A unit with spin number ''n'' is called an ''n''-unit and has [[angular momentum]] ''nħ/2'', where ''ħ'' is the reduced [[Planck constant]]. For [[boson]]s, such as [[photon]]s and [[gluon]]s, ''n'' is an even number. For [[fermion]]s, such as [[electron]]s and [[quark]]s, ''n'' is odd. Given any closed spin network, a non-negative integer can be calculated which is called the ''norm'' of the spin network. Norms can be used to calculate the [[probabilities]] 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 vertex joins three units with spin numbers ''a'', ''b'', and ''c''. Then, these requirements are stated as: * [[Triangle inequality]]: ''a'' ≤ ''b'' + ''c'' and ''b'' ≤ ''a'' + ''c'' and ''c'' ≤ ''a'' + ''b''. * Fermion conservation: ''a'' + ''b'' + ''c'' must be an even number. For example, ''a'' = 3, ''b'' = 4, ''c'' = 6 is impossible since 3 + 4 + 6 = 13 is odd, and ''a'' = 3, ''b'' = 4, ''c'' = 9 is impossible since 9 > 3 + 4. However, ''a'' = 3, ''b'' = 4, ''c'' = 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 ''a'' + ''b'' + ''c'' must be a whole number. === Formal approach to definition === Formally, a spin network may be defined as a (directed) [[graph theory|graph]] whose [[graph theory|edges]] are associated with [[Irreducible representation|irreducible]] [[Representations of Lie groups/algebras|representations]] of a [[Compact group|compact]] [[Lie group]] and whose [[vertex (graph theory)|vertices]] are associated with [[intertwiner#Representation theory|intertwiner]]s of the edge representations adjacent to it. === Properties === A spin network, immersed into a manifold, can be used to define a [[functional (mathematics)|functional]] on the space of [[Connection (mathematics)|connections]] on this manifold. One computes [[holonomy|holonomies]] 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 local [[gauge transformation]]s. == Usage in physics == === In the context of loop quantum gravity === In [[loop quantum gravity]] (LQG), a spin network represents a "quantum state" of the [[gravitational field]] on a 3-dimensional [[hypersurface]]. The set of all possible spin networks (or, more accurately, "[[s-knot]]s"{{snd}}that is, equivalence classes of spin networks under [[diffeomorphisms]]) is [[countable]]<!-- inaccurate - diffeomorphism class? -->; it constitutes a [[basis (linear algebra)|basis]] of LQG [[Hilbert space]]. One of the key results of loop quantum gravity is [[quantization (physics)|quantization]] of areas: the operator of the area ''A'' of a two-dimensional surface Σ should have a discrete [[Spectrum of a matrix|spectrum]]. Every '''spin network''' is an [[eigenstate]] of each such operator, and the area eigenvalue equals :<math>A_{\Sigma} = 8\pi \ell_\text{PL}^2\gamma \sum_i \sqrt{j_i(j_i+1)}</math> where the sum goes over all intersections ''i'' of Σ with the spin network. In this formula, *{{ell}}<sub>PL</sub> is the [[Planck length]], *<math>\gamma</math> is the [[Immirzi parameter]] and *''j<sub>i</sub>'' = 0, 1/2, 1, 3/2, ... is the [[Spin (physics)|spin]] associated with the link ''i'' of the spin network. The two-dimensional area is therefore "concentrated" in the intersections with the spin network. 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 [[Immirzi parameter]] on the order of 1, this gives the smallest possible measurable area of ~10<sup>−66</sup> cm<sup>2</sup>. 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 ''A'' are constrained by [[ladder symmetry]]. 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 area" surrounding this volume. === More general gauge theories === Similar constructions can be made for general gauge theories with a compact Lie group G and a [[connection form]]. This is actually an exact [[duality (mathematics)|duality]] over a lattice. Over a [[manifold]] however, assumptions like [[diffeomorphism invariance]] are needed to make the duality exact (smearing [[Wilson loop]]s is tricky). Later, it was generalized by [[Robert Oeckl]] to representations of [[quantum group]]s in 2 and 3 dimensions using the [[Tannaka–Krein duality]]. [[Michael A. Levin]] and [[Xiao-Gang Wen]] have also defined [[string-net]]s using [[monoidal category|tensor categories]] that are objects very similar to spin networks. However the exact connection with spin networks is not clear yet. [[String-net condensation]] produces [[topological order|topologically ordered]] states in condensed matter. == Usage in mathematics == In mathematics, spin networks have been used to study [[skein module]]s and [[character variety|character varieties]], which correspond to spaces of [[Connection (mathematics)|connections]]. ==See also== {{Commons category|Spin networks}} *[[Spin connection]] *[[Spin structure]] *[[Character variety]] *[[Penrose graphical notation]] *[[Spin foam]] *[[String-net]] *[[Trace diagram]] *[[Tensor network]] ==References== {{reflist}} ==Further reading== === Early papers === *I. B. Levinson, "Sum of Wigner coefficients and their graphical representation," ''Proceed. Phys-Tech Inst. Acad Sci. Lithuanian SSR'' 2, 17-30 (1956) *{{cite journal|doi=10.1103/PhysRevD.11.395|title=Hamiltonian formulation of Wilson's lattice gauge theories|journal=Physical Review D|volume=11|issue=2|pages=395–408|year=1975|last1=Kogut|first1=John|last2=Susskind|first2=Leonard|bibcode = 1975PhRvD..11..395K }} *{{cite journal|doi=10.1103/RevModPhys.55.775|title=The lattice gauge theory approach to quantum chromodynamics|journal=Reviews of Modern Physics|volume=55|issue=3|pages=775–836|year=1983|last1=Kogut|first1=John B.|bibcode=1983RvMP...55..775K}} (see the Euclidean high temperature (strong coupling) section) *{{cite journal|doi=10.1103/RevModPhys.52.453|title=Duality in field theory and statistical systems|journal=Reviews of Modern Physics|volume=52|issue=2|pages=453–487|year=1980|last1=Savit|first1=Robert|bibcode=1980RvMP...52..453S}} (see the sections on Abelian gauge theories) === Modern papers === * {{cite journal|title=Spin networks and quantum gravity|last1=Rovelli|first1=Carlo|last2=Smolin|first2=Lee|journal=Phys. Rev. D|volume=52|issue=10|pages=5743–5759|doi=10.1103/PhysRevD.52.5743|arxiv=gr-qc/9505006|year=1995|pmid=10019107|bibcode = 1995PhRvD..52.5743R |s2cid=16116269}} * {{cite journal|doi=10.1016/S0920-5632(01)01913-2|arxiv=hep-lat/0110034|title=The dual of non-Abelian Lattice Gauge Theory|journal=Nuclear Physics B - Proceedings Supplements|volume=106-107|pages=1010–1012|year=2002|last1=Pfeiffer|first1=Hendryk|last2=Oeckl|first2=Robert|bibcode = 2002NuPhS.106.1010P |s2cid=14925121}} * {{cite journal|doi=10.1063/1.1580071|arxiv=hep-lat/0205013|title=Exact duality transformations for sigma models and gauge theories|journal=Journal of Mathematical Physics|volume=44|issue=7|pages=2891–2938|year=2003|last1=Pfeiffer|first1=Hendryk|bibcode = 2003JMP....44.2891P |s2cid=15580641}} * {{cite journal|doi=10.1016/S0393-0440(02)00148-1|arxiv=hep-th/0110259|title=Generalized lattice gauge theory, spin foams and state sum invariants|journal=Journal of Geometry and Physics|volume=46|issue=3–4|pages=308–354|year=2003|last1=Oeckl|first1=Robert|bibcode = 2003JGP....46..308O |s2cid=13226932}} * {{cite journal|doi=10.1006/aima.1996.0012|doi-access=free|title=Spin Networks in Gauge Theory|journal=[[Advances in Mathematics]]|volume=117|issue=2|pages=253–272|year=1996|last1=Baez|first1=John C.|arxiv=gr-qc/9411007|s2cid=17050932}} * Xiao-Gang Wen, "Quantum Field Theory of Many-body Systems – from the Origin of Sound to an Origin of Light and Fermions," [http://dao.mit.edu/~wen/pub/chapter11.pdf]. (Dubbed ''string-nets'' here.) * {{cite journal|doi=10.1119/1.19175|arxiv=gr-qc/9905020|title=A spin network primer|journal=American Journal of Physics|volume=67|issue=11|pages=972–980|year=1999|last1=Major|first1=Seth A.|bibcode = 1999AmJPh..67..972M |s2cid=9188101}} === Books === * G. E. Stedman, ''Diagram Techniques in Group Theory'', Cambridge University Press, 1990. * [[Predrag Cvitanović]], ''Group Theory: Birdtracks, Lie's, and Exceptional Groups'', Princeton University Press, 2008. {{Roger Penrose|state=collapsed}} {{DEFAULTSORT:Spin Network}} [[Category:Diagrams]] [[Category:Quantum field theory]] [[Category:Loop quantum gravity]] [[Category:Mathematical physics]] [[Category:Diagram algebras]] [[Category:Roger Penrose]]
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