Editing Spin network (section)

== Usage in physics ==

=== In the context of loop quantum gravity ===
In [[loop quantum gravity]] (LQG), a spin network represents a "quantum state" of the [[gravitational field]] on a 3-dimensional [[hypersurface]]. The set of all possible spin networks (or, more accurately, "[[s-knot]]s"{{snd}}that is, equivalence classes of spin networks under [[diffeomorphisms]]) is [[countable]]<!-- inaccurate - diffeomorphism class? -->; it constitutes a [[basis (linear algebra)|basis]] of LQG [[Hilbert space]].

One of the key results of loop quantum gravity is [[quantization (physics)|quantization]] of areas: the operator of the area ''A'' of a two-dimensional surface Σ should have a discrete [[Spectrum of a matrix|spectrum]]. Every '''spin network''' is an [[eigenstate]] of each such operator, and the area eigenvalue equals

:<math>A_{\Sigma} = 8\pi \ell_\text{PL}^2\gamma
\sum_i  \sqrt{j_i(j_i+1)}</math>

where the sum goes over all intersections ''i'' of Σ with the spin network. In this formula,
*{{ell}}<sub>PL</sub> is the [[Planck length]],
*<math>\gamma</math> is the [[Immirzi parameter]] and
*''j<sub>i</sub>'' = 0, 1/2, 1, 3/2, ... is the [[Spin (physics)|spin]] associated with the link ''i'' of 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 an [[Immirzi parameter]] on the order of 1, this gives the smallest possible measurable area of ~10<sup>−66</sup> cm<sup>2</sup>.

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 operator ''A'' are constrained by [[ladder 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 a [[connection form]]. This is actually an exact [[duality (mathematics)|duality]] over a lattice. Over a [[manifold]] however, assumptions like [[diffeomorphism invariance]] are needed to make the duality exact (smearing [[Wilson loop]]s is tricky). Later, it was generalized by [[Robert Oeckl]] to representations of [[quantum group]]s in 2 and 3 dimensions using the [[Tannaka–Krein duality]].

[[Michael A. Levin]] and [[Xiao-Gang Wen]] have also defined [[string-net]]s using [[monoidal category|tensor categories]] that are objects very similar to spin networks. However the exact connection with spin networks is not clear yet. [[String-net condensation]] produces [[topological order|topologically ordered]] states in condensed matter.