Editing Loop quantum gravity (section)

=== Geometric operators, the need for intersecting Wilson loops and spin network states ===
The easiest geometric quantity is the area. Let us choose coordinates so that the surface <math>\Sigma</math> is characterized by <math>x^3 = 0</math>. The area of small parallelogram of the surface <math>\Sigma</math> is the product of length of each side times <math>\sin \theta</math> where <math>\theta</math> is the angle between the sides. Say one edge is given by the vector <math>\vec{u}</math> and the other by <math>\vec{v}</math> then,
<math display="block">A = \| \vec{u} \| \| \vec{v} \| \sin \theta = \sqrt{\| \vec{u} \|^2 \| \vec{v} \|^2 (1 - \cos^2 \theta)} = \sqrt{\| \vec{u} \|^2 \| \vec{v} \|^2 - (\vec{u} \cdot \vec{v})^2}</math>

In the space spanned by <math>x^1</math> and <math>x^2</math> there is an infinitesimal parallelogram described by <math>\vec{u} = \vec{e}_1 dx^1</math> and <math>\vec{v} = \vec{e}_2 dx^2</math>. Using <math>q_{AB}^{(2)} = \vec{e}_A \cdot \vec{e}_B</math> (where the indices <math>A</math> and <math>B</math> run from 1 to 2), yields the area of the surface <math>\Sigma</math> given by
<math display="block">A_\Sigma = \int_\Sigma dx^1 dx^2 \sqrt{\det \left(q^{(2)}\right)}</math>
where <math>\det (q^{(2)}) = q_{11} q_{22} - q_{12}^2</math> and is the determinant of the metric induced on <math>\Sigma</math>. The latter can be rewritten <math>\det (q^{(2)}) = \epsilon^{AB} \epsilon^{CD} q_{AC} q_{BD} / 2</math> where the indices <math>A \dots D</math> go from 1 to 2. This can be further rewritten as
<math display="block">\det (q^{(2)}) = {\epsilon^{3ab} \epsilon^{3cd} q_{ac} q_{bc} \over 2}.</math>

The standard formula for an inverse matrix is
<math display="block">q^{ab} = {\epsilon^{bcd} \epsilon^{aef} q_{ce} q_{df} \over 2!\det (q)}.</math>

There is a similarity between this and the expression for <math>\det(q^{(2)})</math>. But in Ashtekar variables, <math>\tilde{E}^a_i\tilde{E}^{bi} = \det (q) q^{ab}</math>. Therefore,
<math display="block">A_\Sigma = \int_\Sigma dx^1 dx^2 \sqrt{\tilde{E}^3_i \tilde{E}^{3i}}.</math>

According to the rules of canonical quantization the triads <math>\tilde{E}^3_i</math> should be promoted to quantum operators,
<math display="block">\hat{\tilde{E}}^3_i \sim {\delta \over \delta A_3^i}.</math>

The area <math>A_\Sigma</math> can be promoted to a well defined quantum operator despite the fact that it contains a product of two functional derivatives and a square-root.{{sfn|Gambini|Pullin|2011|loc=Section 8.2}} Putting <math>N = 2J</math> (<math>J</math>-th representation),
<math display="block">\sum_i T^i T^i = J (J+1) 1.</math>

This quantity is important in the final formula for the area spectrum. The result is
<math display="block">\hat{A}_\Sigma W_\gamma [A] = 8 \pi \ell_{\text{Planck}}^2 \beta \sum_I \sqrt{j_I (j_I + 1)} W_\gamma [A]</math>
where the sum is over all edges <math>I</math> of the Wilson loop that pierce the surface <math>\Sigma</math>.

The formula for the volume of a region <math>R</math> is given by
<math display="block">V = \int_R d^3 x \sqrt{\det (q)} = \int_R dx^3 \sqrt{\frac{1}{3!} \epsilon_{abc} \epsilon^{ijk} \tilde{E}^a_i \tilde{E}^b_j \tilde{E}^c_k}.</math>

The quantization of the volume proceeds the same way as with the area. Each time the derivative is taken, it brings down the tangent vector <math>\dot{\gamma}^a</math>, and when the volume operator acts on non-intersecting Wilson loops the result vanishes. Quantum states with non-zero volume must therefore involve intersections. Given that the anti-symmetric summation is taken over in the formula for the volume, it needs intersections with at least three non-[[coplanar]] lines. At least four-valent vertices are needed for the volume operator to be non-vanishing.

Assuming the real representation where the gauge group is <math>\operatorname{SU}(2)</math>, Wilson loops are an over complete basis as there are identities relating different Wilson loops. These occur because Wilson loops are based on matrices (the holonomy) and these matrices satisfy identities. Given any two <math>\operatorname{SU}(2)</math> matrices <math>\mathbb{A}</math> and <math>\mathbb{B}</math>,
<math display="block">\operatorname{Tr}(\mathbb{A}) \operatorname{Tr}(\mathbb{B}) = \operatorname{Tr}(\mathbb{A}\mathbb{B}) + \operatorname{Tr}(\mathbb{A}\mathbb{B}^{-1}).</math>

This implies that given two loops <math>\gamma</math> and <math>\eta</math> that intersect,
<math display="block">W_\gamma [A] W_\eta [A] = W_{\gamma \circ \eta} [A] + W_{\gamma \circ \eta^{-1}} [A]</math>
where by <math>\eta^{-1}</math> we mean the loop <math>\eta</math> traversed in the opposite direction and <math>\gamma \circ \eta</math> means the loop obtained by going around the loop <math>\gamma</math> and then along <math>\eta</math>. See figure below. Given that the matrices are unitary one has that <math>W_\gamma [A] = W_{\gamma^{-1}} [A]</math>. Also given the cyclic property of the matrix traces (i.e. <math>\operatorname{Tr} (\mathbb{A} \mathbb{B}) = \operatorname{Tr}(\mathbb{B} \mathbb{A})</math>) one has that <math>W_{\gamma \circ \eta} [A] = W_{\eta \circ \gamma} [A]</math>. These identities can be combined with each other into further identities of increasing complexity adding more loops. These identities are the so-called Mandelstam identities. Spin networks certain are linear combinations of intersecting Wilson loops designed to address the over-completeness introduced by the Mandelstam identities (for trivalent intersections they eliminate the over-completeness entirely) and actually constitute a basis for all gauge invariant functions.

[[File:The Mandelstam identity.jpg|right|thumb|upright=2.2|Graphical representation of the simplest non-trivial Mandelstam identity relating different [[Wilson loops]]]]

As mentioned above the holonomy tells one how to propagate test spin half particles. A spin network state assigns an amplitude to a set of spin half particles tracing out a path in space, merging and splitting. These are described by spin networks <math>\gamma</math>: the edges are labelled by spins together with 'intertwiners' at the vertices which are prescription for how to sum over different ways the spins are rerouted. The sum over rerouting are chosen as such to make the form of the intertwiner invariant under Gauss gauge transformations.