spinnetwork

Spin network (also spelled spinnetwork) is a combinatorial and algebraic structure used in loop quantum gravity to describe quantum states of the gravitational field. A spin network consists of a graph whose edges are labeled by irreducible representations of SU(2) (spins j = 0, 1/2, 1, 3/2, ...), and whose vertices are equipped with intertwiners that couple the incident representations in a gauge-invariant way. It provides a basis for the kinematical Hilbert space of quantum geometry.

Mathematically, one assigns to each edge a representation and to each vertex an intertwiner that maps the tensor product of the incident edge representations to the trivial representation. When embedded in a three-dimensional manifold, the network defines a gauge-invariant function of a SU(2) connection via holonomies along edges contracted with intertwiners. Abstract or non-embedded spin networks also play a role, emphasizing combinatorial and topological features independent of a embedding.

Physically, spin networks encode quantum geometry. Operators corresponding to geometric quantities have discrete spectra on spin network states: the area of a surface is quantized in terms of the spins of edges crossing the surface, and the volume is determined by the intertwiners at the vertices. The spectra depend on the Barbero-Immirzi parameter, a dimensionless constant in the theory.

Dynamics of spin networks are captured by spin foams, two-dimensional histories that describe how networks evolve over time and are weighted by amplitudes in a path-integral-like formulation. The concept originated with Roger Penrose as a combinatorial model, and was incorporated into loop quantum gravity by Ashtekar, Lewandowski, Smolin, and Thiemann.