Spin network

In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of matrix groups. The diagrammatic notation often simplifies calculation because simple diagrams may be used to represent complicated functions. A spin network, as described in Penrose (1971), is a kind of diagram in which each line segment represents the world line of a 'unit' (either an elementary particle or a compound system of particles). Three line segments join at each vertex. A vertex may be interpreted as an event in which either a single unit splits into two or two units collide and join into a single unit. Diagrams whose line segments are all joined at vertices are called closed spin networks. Time may be viewed as going in one direction, such as from the bottom to the top of the diagram, but for closed spin networks the direction of time is irrelevant to calculations.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-knots' – that is, equivalence classes of spin networks under diffeomorphisms) is countable; it constitutes a basis of LQG Hilbert space.In mathematics, spin networks have been used to study skein modules and character varieties, which correspond to spaces of connections.