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).

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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.

x

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