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Inphysics, aspin network is a type of diagram which can be used to representstates and interactions betweenparticles andfields inquantum mechanics. From amathematical perspective, the diagrams are a concise way to representmultilinear functions and functions betweenrepresentations ofmatrix groups. The diagrammatic notation can thus greatly simplify calculations.
Roger Penrose described spin networks in 1971.[1] Spin networks have since been applied to the theory ofquantum gravity byCarlo Rovelli,Lee Smolin,Jorge Pullin,Rodolfo Gambini and others.
Spin networks can also be used to construct a particularfunctional on the space ofconnections which is invariant under localgauge transformations.
A spin network, as described in Penrose (1971),[1] is a kind of diagram in which each line segment represents theworld line of a "unit" (either anelementary 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 calledclosed 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 aspin number. A unit with spin numbern is called ann-unit and hasangular momentumnħ/2, whereħ is the reducedPlanck constant. Forbosons, such asphotons andgluons,n is an even number. Forfermions, such aselectrons andquarks,n is odd.
Given any closed spin network, a non-negative integer can be calculated which is called thenorm of the spin network. Norms can be used to calculate theprobabilities 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 numbersa,b, andc. Then, these requirements are stated as:
For example,a = 3,b = 4,c = 6 is impossible since 3 + 4 + 6 = 13 is odd, anda = 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 suma +b +c must be a whole number.
Formally, a spin network may be defined as a (directed)graph whoseedges are associated withirreduciblerepresentations of acompactLie group and whosevertices are associated withintertwiners of the edge representations adjacent to it.
A spin network, immersed into a manifold, can be used to define afunctional on the space ofconnections on this manifold. One computesholonomies 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 localgauge transformations.
Inloop quantum gravity (LQG), a spin network represents a "quantum state" of thegravitational field on a 3-dimensionalhypersurface. The set of all possible spin networks (or, more accurately, "s-knots" – that is, equivalence classes of spin networks underdiffeomorphisms) iscountable; it constitutes abasis of LQGHilbert space.
One of the key results of loop quantum gravity isquantization of areas: the operator of the areaA of a two-dimensional surface Σ should have a discretespectrum. Everyspin network is aneigenstate of each such operator, and the area eigenvalue equals
where the sum goes over all intersectionsi of Σ with the spin network. In this formula,
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 anImmirzi parameter on the order of 1, this gives the smallest possible measurable area of ~10−66 cm2.
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 operatorA are constrained byladder 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.
Similar constructions can be made for general gauge theories with a compact Lie group G and aconnection form. This is actually an exactduality over a lattice. Over amanifold however, assumptions likediffeomorphism invariance are needed to make the duality exact (smearingWilson loops is tricky). Later, it was generalized byRobert Oeckl to representations ofquantum groups in 2 and 3 dimensions using theTannaka–Krein duality.
Michael A. Levin andXiao-Gang Wen have also definedstring-nets usingtensor categories that are objects very similar to spin networks. However the exact connection with spin networks is not clear yet.String-net condensation producestopologically ordered states in condensed matter.
In mathematics, spin networks have been used to studyskein modules andcharacter varieties, which correspond to spaces ofconnections.
