# Spin network

A spin network is a graph whose edges are associated with representations of a 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 and gauge theory.

One of the key results of loop quantum gravity is quantization of areas: according to several related derivations based on loop quantum gravity, the operator of the area [itex]A[itex] of a two-dimensional surface [itex]\Sigma[itex] should have discrete spectrum. Every spin network is an eigenstate of each such operator, and the area eigenvalue equals

[itex]A_{\Sigma} = 8\pi G_{\mathrm{Newton}} \gamma

\sum_i \sqrt{j_i(j_i+1)}[itex]

where the sum goes over all intersections [itex]i[itex] of [itex]\Sigma[itex] with the spin network. In this formula,

• [itex]G_{\mathrm{Newton}}[itex] is the gravitational constant,
• [itex]\gamma[itex] is the Immirzi parameter and
• [itex]j_i=0,0.5,1,1.5,\dots[itex] is the spin associated with the link [itex]i[itex] of the spin network. The two-dimensional area is therefore "concentrated" in the intersections with the spin network.

Similar quantization applies to the volume operators but mathematics behind these derivations is less convincing.

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