Space can be viewed as networks

of loops called spin networks5. A spin network, as formulated by

Penrose 6 is a kind of graph in which each line segment represents

the world line of a system. The junction where

three line segments join is called a vertex. A vertex can be thought of as an

event in which either a single system splits into two or the time reversal of

the same, two systems colliding and joining into a single system. Penrose’s

basic idea was to reformulate spacetime and quantum mechanics from

combinatorial principle alone.

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, is 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).

Spin foam

is the evolution of a spin network over time and has 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 i.e. they are

physically “granular”, analogous to photons in electromagnetic field

or discrete values of angular momentum and energy in quantum mechanics. In 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

Here summation is 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 lowest possible non-zero

eigenvalue of the area operator corresponds, assuming

to be

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