6.4 Relation between spin network 6 The Formalism6.2 Loop quantum gravity

6.3 Loop states and spin network states 

A subspace tex2html_wrap_inline2726 of tex2html_wrap_inline2484 is formed by states invariant under SU (2) gauge transformations. We now define an orthonormal basis in tex2html_wrap_inline2726 . This basis represents a very important tool for using the theory. It was introduced in [187Jump To The Next Citation Point In The Article] and developed in [31, 32]; it is denoted spin network basis.

First, given a loop tex2html_wrap_inline2588 in M, there is a normalized state tex2html_wrap_inline2738 in tex2html_wrap_inline2484, which is obtained by taking tex2html_wrap_inline2742 and f (g)=- Tr (g). Namely


We introduce a Dirac notation for the abstract states, and denote this state as tex2html_wrap_inline2746 . 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 tex2html_wrap_inline2588 also a multiloop, namely a collection of (possibly overlapping) loops tex2html_wrap_inline2750, 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 tex2html_wrap_inline2484, and therefore a generic state tex2html_wrap_inline2754 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 tex2html_wrap_inline2756 over loop space defined in(19Popup Equation). Equation (19Popup Equation) can be explicitly written as an integral transform, as we will see in section 6.7 .

Next, consider a graph tex2html_wrap_inline2650 . A ``coloring'' of tex2html_wrap_inline2650 is given by the following.

  1. Associate an irreducible representation of SU (2) to each link of tex2html_wrap_inline2650 . Equivalently, we may associate to each link tex2html_wrap_inline2662 a half integer number tex2html_wrap_inline2768, the spin of the irreducible, or, equivalently, an integer number tex2html_wrap_inline2770, the ``color'' tex2html_wrap_inline2772 .
  2. Associate an invariant tensor v in the tensor product of the representations tex2html_wrap_inline2776, to each node of tex2html_wrap_inline2650 in which links with spins tex2html_wrap_inline2776 meet. An invariant tensor is an object with n indices in the representations tex2html_wrap_inline2776 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 tex2html_wrap_inline2776, the invariant tensors form a finite dimensional linear space. Pick a basis tex2html_wrap_inline2790 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 tex2html_wrap_inline2776 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 tex2html_wrap_inline2796, or simply tex2html_wrap_inline2798, 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 tex2html_wrap_inline2802 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 tex2html_wrap_inline2802, which we also write as


One can then show the following.

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 [17Jump To The Next Citation Point In The Article, 51Jump To The Next Citation Point In The Article], and references therein).

6.4 Relation between spin network 6 The Formalism6.2 Loop quantum gravity

image Loop Quantum Gravity
Carlo Rovelli
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