is formed by states invariant under
(2) gauge transformations. We now define an orthonormal basis in
. This basis represents a very important tool for using the
theory. It was introduced in  and developed in [31,
32]; it is denoted spin network basis.
First, given a loop
M, there is a normalized state
, which is obtained by taking
We introduce a Dirac notation for the abstract states, and
denote this state as
. These sates are called loop states. Using Dirac notation, we
It is easy to show that loop states are normalizable. Products
of loop states are normalizable as well. Following tradition, we
also a multiloop, namely a collection of (possibly overlapping)
, and we call
a multiloop state. (Multi-)loop states represented the main
tool for loop quantum gravity before the discovery of the spin
network basis. Linear combinations of multiloop states
, and therefore a generic state
is fully characterized by its projections on the multiloop
states, namely by
The ``old'' loop representation was based on representing
quantum states in this manner, namely by means of the functionals
over loop space defined in(19). Equation (19) can be explicitly written as an integral transform, as we will
see in section
Next, consider a graph
. A ``coloring'' of
is given by the following.
We indicate a colored graph by
, or simply
, and denote it a ``spin network''. (It was Penrose who first
had the intuition that this mathematics could be relevant for
describing the quantum properties of the geometry, and who gave
the first version of spin network theory [152,
- Associate an irreducible representation of
(2) to each link of
. Equivalently, we may associate to each link
a half integer number
, the spin of the irreducible, or, equivalently, an integer
, the ``color''
- Associate an invariant tensor
in the tensor product of the representations
, to each node of
in which links with spins
meet. An invariant tensor is an object with
indices in the representations
that transform covariantly. If
=3, there is only one invariant tensor (up to a multiplicative
factor), given by the Clebsh-Gordon coefficient. An invariant
tensor is also called an
. All invariant tensors are given by the standard
Clebsch-Gordon theory. More precisely, for fixed
, the invariant tensors form a finite dimensional linear
space. Pick a basis
is this space, and associate one of these basis elements to
the node. Notice that invariant tensors exist only if the
tensor product of the representations
contains the trivial representation. This yields a condition
on the coloring of the links. For
=3, this is given by the well known Clebsh-Gordan condition:
Each color is not larger than the sum of the other two, and the
sum of the three colors is even.
Given a spin network
S, we can construct a state
as follows. We take the propagator of the connection along each
link of the graph, in the representation associated to that link,
and then, at each node, we contract the matrices of the
representation with the invariant tensor. We obtain a state
, which we also write as
One can then show the following.
The spin network states provide a very convenient basis for the
- The spin network states are normalizable. The normalization
factor is computed in .
- They are
(2) gauge invariant.
- Each spin network state can be decomposed into a finite
linear combination of products of loop states.
- The (normalized) spin network states form an orthonormal
basis for the gauge
(2) invariant states in
(choosing the basis of invariant tensors appropriately).
- The scalar product between two spin network states can be
easily computed graphically and algebraically. See  for details.
The spin network states defined above are
(2) gauge invariant. There exists also an extension of the spin
network basis to the full Hilbert space (see for instance [17,
51], and references therein).
||Loop Quantum Gravity
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