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| In [[physics]], a '''spin network''' is a type of diagram which can be used to represent 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 often simplifies calculation because simple diagrams may be used to represent complicated [[Function (mathematics)|functions]]. [[Roger Penrose]] is credited with the invention of spin networks in 1971, although similar diagrammatic techniques existed before that time.
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| Spin networks have been applied to the theory of [[quantum gravity]] by [[Carlo Rovelli]], [[Lee Smolin]], [[Jorge Pullin]] and others. They 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.
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| == Definition ==
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| === Penrose's original definition ===
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| A spin network, as described in Penrose 1971, 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.
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| Each line segment is labeled with an integer called a [[spin number]]. A unit with spin number ''n'' is called an ''n''-unit and has [[angular momentum]] ''nħ'', 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.
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| 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:
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| * [[Triangle inequality]]: ''a'' must be less than or equal to ''b'' + ''c'', ''b'' less than or equal to ''a'' + ''c'', and ''c'' less than or equal to ''a'' + ''b''.
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| * Fermion conservation: ''a'' + ''b'' + ''c'' must be an even number.
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| 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 3 + 4 < 9. However, ''a'' = 3, ''b'' = 4, ''c'' = 5 is possible since 3 + 4 + 5 = 12 is even and the triangle inequality is satisfied.
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| Some conventions use labellings by half-integers, with the condition that the sum ''a'' + ''b'' + ''c'' must be a whole number.
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| === Formal definition ===
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| More formally, a '''spin network''' is a (directed) [[graph theory|graph]] whose [[graph theory|edges]] are associated with [[irreducible]] [[Representations of Lie groups/algebras|representations]] of a [[Compact group|compact]] [[Lie group]] and whose [[vertex (graph theory)|vertices]] are associated with [[intertwiner]]s of the edge representations adjacent to it.
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| A spin network, immersed into a manifold, can be used to define a [[functional (mathematics)|functional]] on the space of connections on this manifold. One computes [[holonomy|holonomies]] of the connection along every link 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.
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| == Usage in physics ==
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| === In the context of loop quantum gravity ===
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| 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" - that is, equivalence classes of spin networks under [[diffeomorphisms]]) is [[countable]]; it constitutes a [[basis (linear algebra)|basis]] of LQG [[Hilbert space]].
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| 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
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| :<math>A_{\Sigma} = 8\pi \ell_\text{PL}^2\gamma | |
| \sum_i \sqrt{j_i(j_i+1)}</math>
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| where the sum goes over all intersections ''i'' of Σ with the spin network. In this formula,
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| *{{ell}}<sub>PL</sub> is the Planck Length,
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| *γ is the [[Immirzi parameter]] and
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| *''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.
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| 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>.
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| The formula for area eigenvalues becomes somewhat more complicated if the surface is allowed to pass through the nodes (it is not yet clear if these situations are physically meaningful.)
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| Similar quantization applies to the volume operator. The volume of 3-d submanifold that contains part of 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.
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| === More general gauge theories ===
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| 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]].
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| [[Michael A. Levin]] and [[Xiao-Gang Wen]] have also defined [[string-net]]s using [[tensor 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.
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| == Usage in mathematics ==
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| 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]].
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| ==See also==
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| *[[Character variety]]
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| *[[Penrose graphical notation]]
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| *[[Spin foam]]
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| *[[String-net]]
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| *[[Trace diagram]]
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| ==References==
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| {{Commons category|Spin networks}}
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| {{No footnotes|date=July 2009}}
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| Early papers:
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| *Sum of Wigner coefficients and their graphical representation, I. B. Levinson, ``Proceed. Phys-Tech Inst. Acad Sci. Lithuanian SSR'' 2, 17-30 (1956)
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| *Applications of negative dimensional tensors, [[Roger Penrose]], in ''Combinatorial Mathematics and its Applications'', Academic Press (1971)
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| *Hamiltonian formulation of Wilson's lattice gauge theories, [[John Kogut]] and [[Leonard Susskind]], ''Phys. Rev. D'' 11, 395–408 (1975)
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| *The lattice gauge theory approach to quantum chromodynamics, [[John Kogut|John B. Kogut]], ''Rev. Mod. Phys.'' 55, 775–836 (1983) (see the Euclidean high temperature (strong coupling) section)
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| *Duality in field theory and statistical systems, [[Robert Savit]], ''Rev. Mod. Phys.'' 52, 453–487 (1980) (see the sections on Abelian gauge theories)
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| Modern papers:
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| * Spin Networks and Quantum Gravity, Carlo Rovelli and Lee Smolin, Physical Review D 53, 5743 (1995); gr-qc/9505006.
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| * The dual of non-Abelian lattice gauge theory, Hendryk Pfeiffer and Robert Oeckl, hep-lat/0110034.
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| * Exact duality transformations for sigma models and gauge theories, Hendryk Pfeiffer, hep-lat/0205013.
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| * Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants, Robert Oeckl, hep-th/0110259.
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| * Spin Networks in Gauge Theory, [[John C. Baez]], Advances in Mathematics, Volume 117, Number 2, February 1996, pp. 253–272. <!-- (20) -->
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| * Quantum Field Theory of Many-body Systems – from the Origin of Sound to an Origin of Light and Fermions, Xiao-Gang Wen, [http://dao.mit.edu/~wen/pub/chapter11.pdf]. (Dubbed ''string-nets'' here.)
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| * A Spin Network Primer, Seth A. Major, American Journal of Physics, Volume 67, 1999, gr-qc/9905020.
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| * Pre-geometry and Spin Networks. An introduction. [http://www.expressanimator.com/spin-foam.pdf].
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| Books:
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| * Diagram Techniques in Group Theory, G. E. Stedman, Cambridge University Press, 1990
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| * Group Theory: Birdtracks, Lie's, and Exceptional Groups, [[Predrag Cvitanović]], Princeton University Press, 2008, http://birdtracks.eu/
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| {{DEFAULTSORT:Spin Network}}
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| [[Category:Loop quantum gravity]]
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| [[Category:Mathematical physics]]
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| [[Category:Quantum field theory]]
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| [[Category:Diagram algebras]]
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| [[Category:Diagrams]]
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