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.

 

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