Spin networks are states of quantum geometry in a theory of quantum gravity, discovered by Lee Smolin and Carlo Rovelli, which is the conceptual ancestor of the imaginary physics of Schild's Ladder.
One way to characterise the geometry of space is to describe how vectors are carried along any path, a process known as parallel transport. In curved space, parallel transport around a loop will generally produce a vector that is rotated compared to its original orientation; this leads to such well-known phenomena as the sum of angles of a triangle being different from 180°.
If a quantum-mechanical particle starting out in a certain spin state (characterised by a total spin, j, and a component in the direction of the z-axis, m) is carried along a path through space, parallel transport will generally change the particle's spin state; this is the quantum-mechanical equivalent of the rotation of a classical vector. For example, an electron that starts out with its spin pointing up might be converted into a superposition of spin-up and spin-down components, or it might undergo a change of phase, depending on the particular rotation that it experiences — which in turn depends on the curvature of the region of space through which it passes. So, a simple way to probe the geometry of space would be to carry an electron around a loop, and see how much its spin state changed.
Spin networks generalise this idea, by allowing for a more elaborate process of comparison. Each edge of a spin network is labelled by a spin value, j, and you can imagine parallel-transporting particles with that amount of total spin along each edge. At each node, an amplitude can be computed which describes how similar the incoming and outgoing spin states are, and the product of these amplitudes for all nodes gives a value for the network as a whole, which will depend on the geometry of space along all the edges of the network.
The total spins on the edges aren't enough to fully describe a particle's spin state, though; there is still freedom to choose different values for m, the components of spin along the z-axis. The trouble is, if any particular choice was adopted — such as making m equal to j on all edges — then the amplitude assigned to each geometry would depend on the orientation of the z-axis. However, there's an easy way to overcome this problem: summing the network's amplitude for all possible combinations of m values, where m ranges from –j to +j for each edge, gives a value that is completely independent of any choices of orientation. By using this sum, a spin network defines a state of quantum geometry with the crucial property of gauge invariance: the amplitude does not depend on the way things are measured, only on the geometry itself.
This applet gives examples of spin network states evaluated over a range of geometries.