More technically, a

spin network is a directed graph whose edges are

associated with irreducible representations of

a compact Lie group and

whose vertices are associated

with intertwiners of the edge representations

adjacent to it. A spin network, embedded into a manifold, can be used to define

a functional on the space

of connections on this

manifold. In fact a loop is a closed spin network (For example, certain

linear combinations of Wilson loops are called spin network states).

The evolution of a spin network over

time is called a spin foam which is about the size of the Planck

length. Spin foam is a topological structure made out of two-dimensional faces

that represents one of the configurations that must be summed to obtain a

Feynman’s path integral description of quantum gravity. A spin network

represents a “quantum state” of the gravitational field on a

3-dimensional hypersurface. The set of all possible spin networks is countable;

it constitutes a basis of LQG Hilbert space.

In LQG space and time

are quantized. It gives a

physical picture of spacetime where space and time are “granular”,

analogous to photons in quantum electrodynamics or discrete values of angular

momentum and energy in quantum mechanics. For example, quantization of areas: the operator

of the area A of a two-dimensional surface ? should have a

discrete spectrum. Every spin network is

an eigenstate of

each such operator, and the area eigenvalue equals

Where the sum goes over all intersections i of ? with

the spin network and

is the Planck length

is

the Immirzi parameter and

= 0, 1/2, 1, 3/2,… is the spin associated

with the link i of the spin network. The two-dimensional area is

therefore “concentrated” in the intersections with the spin network. The

lowest possible non-zero eigenvalue of the area operator corresponds, assuming

to be

on the order of 1, gives the smallest possible measurable area of ~10?66 cm2.