## 6.3 Loop states and spin network states

A subspace of is formed by states invariant under SU (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 [187] and developed in [31, 32]; it is denoted spin network basis.

First, given a loop in M, there is a normalized state in , which is obtained by taking and f (g)=- Tr (g). Namely

We introduce a Dirac notation for the abstract states, and denote this state as . These sates are called loop states. Using Dirac notation, we can write

It is easy to show that loop states are normalizable. Products of loop states are normalizable as well. Following tradition, we denote with also a multiloop, namely a collection of (possibly overlapping) loops , 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 (over-)span , 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 6.7 .

Next, consider a graph . A ``coloring'' of is given by the following.

1. Associate an irreducible representation of SU (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 number , the ``color'' .
2. Associate an invariant tensor v in the tensor product of the representations , to each node of in which links with spins meet. An invariant tensor is an object with n indices in the representations that transform covariantly. If n =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 intertwining tensor . 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 n =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.
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, 153].)

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 are normalizable. The normalization factor is computed in [77].
• They are SU (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 SU (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 [77] for details.
The spin network states provide a very convenient basis for the quantum theory.

The spin network states defined above are SU (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 Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de