Math 2.0

Relativity and gravitational theory Definition

Spin network

Builds on GraphRepresentation theory of SU(2)Holonomy

Definition

For j∈{0,12,1,32,… }j \in \set{0, \frac12, 1, \frac32, \dots}, let VjV_j be the irreducible representation of SU(2)SU(2) of dimension 2j+12j + 1 (see representation theory of SU(2)). An (SU(2)SU(2)) spin network is a triple (Γ,j,ι)(\Gamma, j, \iota) where:

  1. Γ\Gamma is a finite oriented graph, possibly with loops and multiple edges;
  2. jj assigns to each edge ee a spin je≠0j_e \ne 0;
  3. ι\iota assigns to each vertex vv an intertwiner ιv\iota_v, that is, an SU(2)SU(2)-invariant vector of ⨂e incoming at vVje⊗⨂e outgoing at vVje∗\bigotimes_{e \text{ incoming at } v} V_{j_e} \otimes \bigotimes_{e \text{ outgoing at } v} V_{j_e}^*.

In loop quantum gravity, Γ\Gamma is embedded in a three-manifold Σ\Sigma. The spin network then defines the function Ψ(A)=(⨂vιv)⋅(⨂eρje(he(A)))\Psi(A) = \left( \bigotimes_v \iota_v \right) \cdot \left( \bigotimes_e \rho_{j_e}(h_e(A)) \right) of a connection AA. Here he(A)h_e(A) is the holonomy of AA along ee, ρj\rho_j is the representation on VjV_j, and ⋅\cdot means contracting all indices.

Remarks

Example: let Γ\Gamma be one edge ee that is a closed loop, with a two-valent vertex vv and ιv\iota_v the identity of VjV_j, an invariant element of Vj⊗Vj∗V_j \otimes V_j^*. Then Ψ(A)=tr⁡ρj(he(A))\Psi(A) = \tr \rho_j(h_e(A)), the Wilson loop of AA in spin jj. Non-example: a trivalent vertex with three edges of spin 12\frac12 admits no nonzero intertwiner. Indeed V1/2⊗V1/2⊗V1/2≅V1/2⊕V1/2⊕V3/2V_{1/2} \otimes V_{1/2} \otimes V_{1/2} \cong V_{1/2} \oplus V_{1/2} \oplus V_{3/2} contains no copy of V0V_0. In general, three spins at a trivalent vertex must satisfy the triangle inequalities and have an integer sum. Spin network states on embedded graphs, with orthonormal intertwiners at the vertices, form an orthonormal basis of the gauge-invariant Hilbert space of loop quantum gravity.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

Show the path as a graph
Cross product, concept, no evidence levelCross productconceptDeterminant, concept, no evidence levelDeterminantconceptGraph, concept, no evidence levelGraphconceptGroup, concept, no evidence levelGroupconceptHolonomy, concept, no evidence levelHolonomyconceptHomogeneous polynomial, concept, no evidence levelHomogeneous polynomialconceptIdentity matrix, object, no evidence levelIdentity matrixobjectIrreducible representation, concept, no evidence levelIrreducible represent…conceptLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionReal projective space, object, no evidence levelReal projective spaceobjectRepresentation of a Lie group, concept, no evidence levelRepresentation of a L…conceptVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleDeterminant, definition, no evidence levelDeterminantdefinitionUnitary group, object, no evidence levelUnitary groupobjectSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleRotation group SO(3), example, no evidence levelRotation group SO(3)exampleSU(2), example, no evidence levelSU(2)exampleRepresentation theory of SU(2), example, no evidence levelRepresentation theory…exampleSpin network, definition, no evidence levelSpin networkdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cross product concept
  2. Determinant concept
  3. Graph concept
  4. Group concept
  5. Holonomy concept
  6. Homogeneous polynomial concept
  7. Identity matrix object
  8. Irreducible representation concept
  9. Lie group concept
  10. Matrix definition
  11. Real projective space object
  12. Representation of a Lie group concept
  13. Vector space definition
  14. Ring definition
  15. Square matrix definition
  16. Transpose definition
  17. Linear map definition
  18. Unit of a ring definition
  19. Conjugate transpose definition
  20. Multilinear map definition
  21. Orthogonal matrix definition
  22. Division ring definition
  23. Alternating multilinear map definition
  24. Orthogonal group object
  25. Unitary matrix definition
  26. The quaternions example
  27. Determinant definition
  28. Unitary group object
  29. Special orthogonal group object
  30. Special unitary group SU(n) example
  31. Rotation group SO(3) example
  32. SU(2) example
  33. Representation theory of SU(2) example
  34. Spin network definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Spin_network
About this page
TagH77Y8DTXQC
URIm2://H77Y8DTXQC
Kinddefinition
WritingW0 stub
MSC83C45, 22E70
DefinesSpin network
UsesGraph, Representation theory of SU(2), Holonomy
Changed3 November 2026

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