Spin network

Spin network diagram, after Penrose

Inphysics,aspin networkis a type of diagram which can be used to representstatesand interactions betweenparticlesandfieldsinquantum mechanics.From amathematicalperspective, the diagrams are a concise way to representmultilinear functionsand functions betweenrepresentationsofmatrix groups.The diagrammatic notation can thus greatly simplify calculations.

Roger Penrosedescribed spin networks in 1971.Spin networks have since been applied to the theory ofquantum gravitybyCarlo Rovelli,Lee Smolin,Jorge Pullin,Rodolfo Gambiniand others.

Spin networks can also be used to construct a particularfunctionalon the space ofconnectionswhich is invariant under localgauge transformations.

Definition

Penrose's definition

A spin network, as described in Penrose (1971),is a kind of diagram in which each line segment represents theworld lineof a "unit" (either anelementary particleor 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 numbernis called ann-unit and hasangular momentumnħ/2,whereħis the reducedPlanck constant.Forbosons,such asphotonsandgluons,nis an even number. Forfermions,such aselectronsandquarks,nis odd.

Given any closed spin network, a non-negative integer can be calculated which is called thenormof the spin network. Norms can be used to calculate theprobabilitiesof 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:

• Triangle inequality:a≤b+candb≤a+candc≤a+b.
• Fermion conservation:a+b+cmust be an even number.

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+cmust be a whole number.

Formal approach to definition

Formally, a spin network may be defined as a (directed)graphwhoseedgesare associated withirreduciblerepresentationsof acompactLie groupand whoseverticesare associated withintertwinersof the edge representations adjacent to it.

Properties

A spin network, immersed into a manifold, can be used to define afunctionalon the space ofconnectionson this manifold. One computesholonomiesof 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.

Usage in physics

In the context of loop quantum gravity

Inloop quantum gravity(LQG), a spin network represents a "quantum state" of thegravitational fieldon 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 abasisof LQGHilbert space.

One of the key results of loop quantum gravity isquantizationof areas: the operator of the areaAof a two-dimensional surface Σ should have a discretespectrum.Everyspin networkis aneigenstateof each such operator, and the area eigenvalue equals

$A_{\Sigma }=8\pi \ell _{\text{PL}}^{2}\gamma \sum _{i}{\sqrt {j_{i}(j_{i}+1)}}$

where the sum goes over all intersectionsiof Σ with the spin network. In this formula,

• ℓ_PLis thePlanck length,
• $\gamma$is theImmirzi parameterand
• j_i= 0, 1/2, 1, 3/2,... is thespinassociated with the linkiof 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 anImmirzi parameteron the order of 1, this gives the smallest possible measurable area of ~10⁻⁶⁶cm².

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 operatorAare 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.

More general gauge theories

Similar constructions can be made for general gauge theories with a compact Lie group G and aconnection form.This is actually an exactdualityover a lattice. Over amanifoldhowever, assumptions likediffeomorphism invarianceare needed to make the duality exact (smearingWilson loopsis tricky). Later, it was generalized byRobert Oecklto representations ofquantum groupsin 2 and 3 dimensions using theTannaka–Krein duality.

Michael A. LevinandXiao-Gang Wenhave also definedstring-netsusingtensor categoriesthat are objects very similar to spin networks. However the exact connection with spin networks is not clear yet.String-net condensationproducestopologically orderedstates in condensed matter.

Usage in mathematics

In mathematics, spin networks have been used to studyskein modulesandcharacter varieties,which correspond to spaces ofconnections.

See also

• Spin connection
• Spin structure
• Character variety
• Penrose graphical notation
• Spin foam
• String-net
• Trace diagram
• Tensor network

Further reading

Early papers

• I. B. Levinson, "Sum of Wigner coefficients and their graphical representation,"Proceed. Phys-Tech Inst. Acad Sci. Lithuanian SSR2, 17-30 (1956)
• Kogut, John; Susskind, Leonard (1975). "Hamiltonian formulation of Wilson's lattice gauge theories".Physical Review D.11(2): 395–408.Bibcode:1975PhRvD..11..395K.doi:10.1103/PhysRevD.11.395.
• Kogut, John B. (1983). "The lattice gauge theory approach to quantum chromodynamics".Reviews of Modern Physics.55(3): 775–836.Bibcode:1983RvMP...55..775K.doi:10.1103/RevModPhys.55.775.(see the Euclidean high temperature (strong coupling) section)
• Savit, Robert (1980). "Duality in field theory and statistical systems".Reviews of Modern Physics.52(2): 453–487.Bibcode:1980RvMP...52..453S.doi:10.1103/RevModPhys.52.453.(see the sections on Abelian gauge theories)

Modern papers

• Rovelli, Carlo; Smolin, Lee (1995). "Spin networks and quantum gravity".Phys. Rev. D.52(10): 5743–5759.arXiv:gr-qc/9505006.Bibcode:1995PhRvD..52.5743R.doi:10.1103/PhysRevD.52.5743.PMID10019107.S2CID16116269.
• Pfeiffer, Hendryk; Oeckl, Robert (2002). "The dual of non-Abelian Lattice Gauge Theory".Nuclear Physics B - Proceedings Supplements.106–107: 1010–1012.arXiv:hep-lat/0110034.Bibcode:2002NuPhS.106.1010P.doi:10.1016/S0920-5632(01)01913-2.S2CID14925121.
• Pfeiffer, Hendryk (2003). "Exact duality transformations for sigma models and gauge theories".Journal of Mathematical Physics.44(7): 2891–2938.arXiv:hep-lat/0205013.Bibcode:2003JMP....44.2891P.doi:10.1063/1.1580071.S2CID15580641.
• Oeckl, Robert (2003). "Generalized lattice gauge theory, spin foams and state sum invariants".Journal of Geometry and Physics.46(3–4): 308–354.arXiv:hep-th/0110259.Bibcode:2003JGP....46..308O.doi:10.1016/S0393-0440(02)00148-1.S2CID13226932.
• Baez, John C. (1996)."Spin Networks in Gauge Theory".Advances in Mathematics.117(2): 253–272.arXiv:gr-qc/9411007.doi:10.1006/aima.1996.0012.S2CID17050932.
• Xiao-Gang Wen, "Quantum Field Theory of Many-body Systems – from the Origin of Sound to an Origin of Light and Fermions,"[1].(Dubbedstring-netshere.)
• Major, Seth A. (1999). "A spin network primer".American Journal of Physics.67(11): 972–980.arXiv:gr-qc/9905020.Bibcode:1999AmJPh..67..972M.doi:10.1119/1.19175.S2CID9188101.

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.