## 6.4 Relation between spin network states and loop states and diagrammatic representation of the states

A diagrammatic representation for the states in is very useful in concrete calculations. First, associate to a loop state a diagram in M, formed by the loop itself. Next, notice that we can multiply two loop states, obtaining a normalizable state. We represent the product of n loop states by the diagram formed by the set of the n (possibly overlapping) corresponding loops (we denote this set ``multiloop''). Thus, linear combinations of multiloops diagrams represent states in . Representing states as linear combinations of multiloops diagrams makes computation in easy.

Now, the spin network state defined by the graph with no nodes , with color 1, is clearly, by definition, the loop state , and we represent it by the diagram . The spin network state determined by the graph without nodes , with color n can be obtained as follows. Draw n parallel lines along the loop ; cut all lines at an arbitrary point of , and consider the n ! diagrams obtained by joining the legs after a permutation. The linear combination of these n ! diagrams, taken with alternate signs (namely with the sign determined by the parity of the permutation) is precisely the state . The reason for this key result can be found in the fact that an irreducible representation of SU (2) can be obtained as the totally symmetric tensor product of the fundamental representation with itself. For details, see [77].

Next, consider a graph with nodes. Draw parallel lines along each link . Join pairwise the end points of these lines at each node (in an arbitrary manner), in such a way that each line is joined with a line from a different link (see Figure 1). In this manner, one obtains a multiloop diagram. Now antisymmetrize the parallel lines along each link, obtaining a linear combination of diagrams representing a state in . One can show that this state is a spin network state, where is the color of the links, and the color of the nodes is determined by the pairwise joining of the legs chosen [77]. Again, simple SU (2) representation theory is behind this result.

More in detail, if a node is trivalent (has 3 adjacent links), then we can join legs pairwise only if the total number of the legs is even, and if the number of the legs in each link is smaller or equal than the sum of the number of the other two. This can be recognized immediately as the Clebsch-Gordan condition. If these conditions are satisfied, there is only a single way of joining legs. This corresponds to the fact that there is only one invariant tensor in the product of three irreducible of SU (2). Higher valence nodes can be decomposed in trivalent ``virtual'' nodes, joined by ``virtual'' links. Orthogonal independent invariant tensors are obtained by varying over all allowed colorings of these virtual links (compatible with the Clebsch-Gordan conditions at the virtual nodes). Different decompositions of the node give different orthogonal bases. Thus the total (links and nodes) coloring of a spin network can be represented by means of the coloring of the real and the virtual links. See Figure 1 .

Figure 1: Construction of ``virtual'' nodes and ``virtual'' links over an n -valent node.

Vice versa, multiloop states can be decomposed in spin network states by simply symmetrizing along (real and virtual) nodes. This can be done particularly easily diagrammatically, as illustrated by the graphical formulae in [187, 77]. These are standard formulae. In fact, it is well known that the tensor algebra of the SU (2) irreducible representations admits a completely graphical notation. This graphical notation has been widely used for instance in nuclear and atomic physics. One can find it presented in detail in books such as [214, 52, 66]. The application of this diagrammatic calculus to quantum gravity is described in detail in [77], which I recommend to anybody who intends to perform concrete calculations in loop gravity.

It is interesting to notice that loop quantum gravity was first constructed in a pure diagrammatic notation, in [184]. The graphical nature of this calculus puzzled some, and the theory was accused of being vague and strange. Only later did researchers notice that the diagrammatic notation is actually very conventional SU (2) diagrammatic calculus, well known in atomic and nuclear physics.

 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