A spin network is a (directed) graph whose edges are associated with irreducible representations of a compact Lie group, G and vertices are associated with intertwiners of the edge reps adjacent to it. It was invented by Roger Penrose in 1971. Spin networks were applied to the physics problem of quantum gravity by Carlo Rovelli, Lee Smolin, Fotini Markopoulou-Kalamara, and others to reformulate loop quantum gravity in the canonical approach. There, they chop off the Lorentz gauge group Spin(3,1), which is noncompact to SU(2), which is compact. Later, it was generalized to gauge theories with connections in general. A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... Irreducible can refer to: irreducible (mathematics) irreducible (philosophy) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... 1971 (MCMLXXI) was a common year starting on Friday (the link is to a full 1971 calendar). ... Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Lee Smolin at Harvard Lee Smolin is a theoretical physicist who has made major contributions to the quantum theory of gravity. ... Fotini Markopoulou-Kalamara is a theoretical physicist interested in foundational mathematics and quantum mechanics. ... Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... To meet Wikipedias quality standards, this article or section may require cleanup. ... In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ...

A spin network, immersed into a manifold, can be used to define a functional on the space of connections on this manifold. One simply computes 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 transformations. Generally, functional refers to something with and able to fulfill its purpose or function. ... For the television show Connections, go here. ... In differential geometry, the holonomy group of a connection on a vector bundle over a smooth manifold M is the group of linear transformations induced by parallel transport around closed loops in M. There is an analogous notion for connections on principal bundles over M. The holonomy group of a... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

In the context of loop quantum gravity

In loop quantum gravity, 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-knots" - that is, equivalence classes of spin networks under diffeomorphisms) is countable; it constitutes a basis of LQG Hilbert space. Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ... The gravitational field is a field (physics), generated by massive objects, that determines the magnitude and direction of gravitation experienced by other massive objects. ... In mathematics, a hypersurface is some kind of submanifold. ... In loop quantum gravity, an s-knot is an equivalence class of spin networks under diffeomorphisms. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...

One of the key results of loop quantum gravity is quantization of areas: the operator of the area A of a two-dimensional surface Σ should have a discrete spectrum. Every spin network is an eigenstate of each such operator, and the area eigenvalue equals The word spectrum (plural, spectra) has many uses: // Common nouns The Spectrum article explains why so many things are called by this name The spectrum of activity of a drug The political spectrum of opinion The economic spectrum The bipolar spectrum, in psychology The autistic spectrum, in psychology In the... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

where the sum goes over all intersections i of Σ with the spin network. In this formula,

• G_Newton is the gravitational constant,
• γ is the Immirzi parameter and
• is the 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 Immirzi parameter on the order of 1, this gives the smallest possible measurable area of ~10^-66 cm². According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. ...

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. )

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.

More general gauge theories

(Outside the context of LQG, the name spin networks is a bit of a misnomer...)

As mentioned, it was noticed that analogous constructions can be made for general gauge theories with a compact Lie group G and a connection form. This is actually an exact duality over a lattice. Over a manifold however, assumptions like diffeomorphism invariance are needed to make the duality exact (smearing Wilson loops is tricky). Later, it was generalized by Robert Oeckl to representations of quantum groups in 2 and 3 dimensions using the Tannaka-Krein duality. Michael A. Levin and Xiao-Gang Wen have also defined another generalization of spin networks which they call string-nets using tensor categories. String-net condensation produces topologically ordered states in condensed matter. In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In theoretical physics, general covariance (also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. ... In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... Tannaka-Krein duality theory concerns the interaction with a group and the category of its representations. ... A spin network is a (directed) graph whose edges are associated with irreducible representations of a compact Lie group, G and vertices are associated with intertwiners of the edge reps adjacent to it. ... In mathematics, a monoidal category (or tensor category) is a category equipped with a binary tensor functor and a unit object . ... In physics, topological order is a new kind of order (a new kind of organization of particles) in a quantum state that is beyond the Landau symmetry-breaking description. ...

Publications

Some random early papers (none of them actually called them spin networks; that is Penrose's name for them): Penrose may refer to: Penrose, New Zealand, a suburb of Auckland Penrose, Philadelphia, a neighborhood in Philadelphia Penrose, Colorado, a town in the United States Lionel Penrose (1898-1972), English geneticist, father of Roger and Jonathan Penrose Roger Penrose (born 1931), English mathematical physicist, son of Lionel Penrose Jonathan Penrose...

• Hamiltonian formulation of Wilson's lattice gauge theories, John Kogut and Leonard Susskind, Phys. Rev. D 11, 395–408 (1975)
• The lattice gauge theory approach to quantum chromodynamics, John B. Kogut, Rev. Mod. Phys. 55, 775–836 (1983) (see the Euclidean high temperature (strong coupling) section)
• Duality in field theory and statistical systems, Robert Savit, Rev. Mod. Phys. 52, 453–487 (1980) (see the sections on Abelian gauge theories)

Modern papers: Lenny Susskind at Stanford University Leonard Susskind is a theoretical physics professor at Stanford University in the field of string theory and quantum field theory. ...

• The dual of non-Abelian lattice gauge theory, Hendryk Pfeiffer and Robert Oeckl, hep-lat/0110034.
• Exact duality transformations for sigma models and gauge theories, Hendryk Pfeiffer, hep-lat/0205013.
• Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants, Robert Oeckl, hep-th/0110259.
• Spin Networks in Gauge Theory, John C. Baez, Advances in Mathematics, Volume 117, Number 2, February 1996, pp. 253–272.
• Quantum Field Theory of Many-body Systems – from the Origin of Sound to an Origin of Light and Fermions, Xiao-Gang Wen, [1]. (Dubbed string-nets here.)