Space can be viewed as networksof loops called spin networks5. A spin network, as formulated byPenrose 6 is a kind of graph in which each line segment representsthe world line of a system. The junction wherethree line segments join is called a vertex. A vertex can be thought of as anevent in which either a single system splits into two or the time reversal ofthe same, two systems colliding and joining into a single system.
Penrose’sbasic idea was to reformulate spacetime and quantum mechanics fromcombinatorial principle alone.More technically, aspin network is “a directed graph whose edges areassociated with irreducible representations ofa compact Lie group andwhose vertices are associatedwith intertwiners of the edge representationsadjacent to it”. A spin network, embedded into a manifold, is used to definea functional on the spaceof connections on thismanifold. In fact a loop is a closed spin network (For example, certainlinear combinations of Wilson loops are called spin network states).Spin foamis the evolution of a spin network over time and has the size of the Plancklength. Spin foam is a topological structure made out of two-dimensional facesthat represents one of the configurations that must be summed to obtain aFeynman’s path integral description of quantum gravity. A spin networkrepresents a “quantum state” of the gravitational field on a3-dimensional hypersurface.
The set of all possible spin networks is countable;it constitutes a basis of LQG Hilbert space. In LQG space and timeare quantized i.e. they arephysically “granular”, analogous to photons in electromagnetic fieldor discrete values of angular momentum and energy in quantum mechanics. In quantization of areas theoperator of the area A of a two-dimensional surface ? should have adiscrete spectrum. Every spinnetwork is an eigenstate ofeach such operator, and the area eigenvalue equals Here summation is over all intersections i of? with the spin network and is the Planck length isthe Immirzi parameter and = 0, 1/2, 1, 3/2,…
is the spin associatedwith the link i of the spin network. The lowest possible non-zeroeigenvalue of the area operator corresponds, assuming to beon the order of 1, to the smallest possible measurable area of ~10?66 cm2.