Spin Network: Quantized Space in Loop Quantum Gravity

Spin Network

Selected link e0

Spin j (half-integer) 0.5

Quantum Geometry

Link spin j

0.5

Link area (l_P²)

6.66

Total network area

26.6

Nodes / links

5 / 4

How it works

In Loop Quantum Gravity, space itself is not continuous — geometry is built from a spin network: a graph whose nodes represent quanta of volume and whose links represent quanta of area between adjacent chunks of space. Each link carries a spin label j ∈ {½, 1, 3/2, 2, …}, an SU(2) representation, exactly like the spin of an electron.

The area operator's eigenvalues (Rovelli–Smolin / Ashtekar–Lewandowski) give the area punctured by a link of spin j:

A_j = 8π γ l_P² √(j(j+1))

γ ≈ 0.2375   Barbero–Immirzi parameter
l_P = √(ħG/c³)   Planck length (set to 1 here)

The seed network here is a quantum tetrahedron: one central node (a chunk of volume) with 4 links to boundary nodes — the tetrahedron's 4 faces, each an independent area quantum. Pick a link and drag its spin to see its area jump between discrete allowed values — area in LQG is not continuous, it comes in a quantized spectrum.

Evolve performs a 1→4 Pachner move on a random node: it splits one chunk of volume into four smaller ones, wiring the new sub-nodes into a small tetrahedral cluster and handing each of the split node's old links to one of the four children. This is the combinatorial move that generates the sum-over-triangulations in spin foam models — the path-integral formulation of LQG's dynamics — refining the discrete geometry the way subdividing a mesh refines a surface.

Link selector / spin slider — pick any link, set its spin, watch the quantized area and the network total update live.
Evolve — one Pachner refinement step (capped so the graph stays readable).
Reset — back to the single quantum tetrahedron.

Node · quantum of volume Selected link

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