Lattice gauge and spin network

Question

I see the similarity between the Lattice Gauge and Spin Network. (For example, both theories depict the node part as quantum (the latter is explained as spin).) Are there any other mathematical, physical similarity between the two theories?

Answer 1

This question is already quite old, but since it was modified recently, let my provide a further answer for other latecomers to this question.

There is a relation between lattige gauge theory and spin-network states, namely the latter one form a basis of the Hilbert space of lattice gauge theory. Let me be a little bit more explicit.

First of all, a spin-network corresponding to a (compact) Lie group $G$ can abstractly be defined to be a triple $(\gamma,\rho,i)$ consisting of the following data:

1. A finite directed graph $\gamma=(\mathcal{V},\mathcal{E})$ with target and source map $t,s:\mathcal{E}\to\mathcal{V}$.
2. A map $\rho$ assigning to each edge $e\in\mathcal{E}$ an irreducible representation of $G$.
3. A map $i$ assigning to each vertex $v\in\mathcal{V}$ an intertwiner of the type \begin{align*}i_{v}:\bigotimes_{e \in\mathcal{T}(v)}\mathcal{H}_{e}\to\bigotimes_{e^{\prime}\in\mathcal{S}(v)}\mathcal{H}_{e^{\prime}},\end{align*} where $\mathcal{H}_{e}$ and $\mathcal{H}_{e^{\prime}}$ are the vector spaces corresponding to the representations associated to the edges $e$ and $e^{\prime}$.

Now, spin-networks appear naturally in the discussion of gauge theory on a graph. For this, consider a (compact, connected, finite-dimensional) Lie group $G$ as well as a principal $G$-bundle $P$ over a $d$-dimensional manifold $\mathcal{M}\cong\mathbb{R}\times \Sigma$, where $\Sigma$ is some space-like hypersurface representing space. Furthermore, let $\gamma=(\mathcal{V},\mathcal{E})$ be a finite directed graph embedded in $\Sigma$. Now as usual in lattice gauge theory, we discretize a connection $1$-form of our principal bundle (i.e. a gauge field in physics terminology) by assigning a group element $g\in G$ to each edge. Hence, the space of discrete connections is given by $$\mathcal{A}_{\gamma}:= G^{\vert\mathcal{E}\vert}$$ A gauge transformation can be described via a group element $h_{v}$ living on the vertices of $\gamma$ and acts as $$g_{e}\mapsto h^{-1}_{t(e)}g_{e}h_{s(e)}$$ on some group element $g_{e}$. Hence, the set of gauge transformations is given by $$\mathcal{G}_{\gamma}:= G^{\vert\mathcal{V}\vert}.$$ Using the (normalized) Haar measure on each copy of $G$ in $\mathcal{A}_{\gamma}$, we get a well-defined Hilbert space $L^{2}(\mathcal{A}_{\gamma}/\mathcal{G}_{\gamma})$, which can be identified with the subspace of $L^{2}(\mathcal{A}_{\gamma})$ consisting of gauge-invariant functions.

Now, the main point is the following. As discussed at length in arXiv:gr-qc/9411007, one can show, using the theorem of Peter-Weyl and some tricks, that there is the following isormophism:

$$L^{2}(\mathcal{A}_{\gamma}/\mathcal{G}_{\gamma})\cong\bigoplus_{\rho\in\Lambda^{\mathcal{E}}}\bigotimes_{v\in\mathcal{V}}\mathrm{Int}\bigg(\bigotimes_{e\in\mathcal{S}(v)}\mathcal{H}_{\rho_{e}},\bigotimes_{e\in\mathcal{T}(v)}\mathcal{H}_{\rho_{e}}\bigg ).$$

where $\Lambda$ is the set of all (equivalence classes of) irreducible and unitary representations of $G$ and where $\mathcal{H}_{\rho}$ denotes the representation space corresponding to some $\rho\in\Lambda$. Using this as well as the above definition of spin-networks, it is clear that spin-network states actually span the Hilbert space of gauge theory on a graph.

Answer 2

You ask if there are mathematical similarities between the two. In fact there is a mathematical structure called a gauge network that accommodates both lattice gauge theory and spin networks. It was introduced by Marcolli and van Suijlekom in

https://arxiv.org/abs/1301.3480

(it was published in the same year you posted this question).

Here is a rough description: a spin network is in some sense a discrete version of a spin manifold ("spin" manifold being one whose global structure allows for spinors). A lattice gauge theory is a discrete version of a manifold with a gauge theory defined on it (a 'principal G bundle over a manifold' in jargon).

A gauge network attaches more general algebraic data than either of these two theories to the nodes and edges of a graph. Both spin networks and lattice gauge theories are special cases of gauge networks. So if you work with the gauge network structure, you are in the right discrete setting to start talking simultaneously about gravity and gauge theories.