Editing Loop quantum gravity (section)

=== Introduction of the loop representation ===
{{main| Holonomy| Wilson loop| Knot invariant}}
It was in particular the inability to have good control over the space of solutions to Gauss's law and spatial diffeomorphism constraints that led Rovelli and Smolin to consider the [[loop representation in gauge theories and quantum gravity]].{{sfn|Rovelli|Smolin|1988|pp=1155–1958}}

LQG includes the concept of a [[holonomy]]. A holonomy is a measure of how much the initial and final values of a spinor or vector differ after [[parallel transport]] around a closed loop; it is denoted
<math display="block">h_\gamma [A] .</math>

Knowledge of the holonomies is equivalent to knowledge of the connection, up to gauge equivalence. Holonomies can also be associated with an edge; under a Gauss Law these transform as
<math display="block">(h'_e)_{\alpha \beta} = U_{\alpha \gamma}^{-1} (x) (h_e)_{\gamma \sigma} U_{\sigma \beta} (y).</math>

For a closed loop <math>x = y</math> and assuming <math>\alpha = \beta</math>, yields
<math display="block">(h'_e)_{\alpha \alpha} = U_{\alpha \gamma}^{-1} (x) (h_e)_{\gamma \sigma} U_{\sigma \alpha} (x) = [U_{\sigma \alpha} (x) U_{\alpha \gamma}^{-1} (x)] (h_e)_{\gamma \sigma} = \delta_{\sigma \gamma} (h_e)_{\gamma \sigma} = (h_e)_{\gamma \gamma}</math>
or
<math display="block">\operatorname{Tr} h'_\gamma = \operatorname{Tr} h_\gamma.</math>

The trace of an holonomy around a closed loop is written
<math display="block">W_\gamma [A]</math>
and is called a Wilson loop. Thus Wilson loops are gauge invariant. The explicit form of the Holonomy is
<math display="block">h_\gamma [A] = \mathcal{P} \exp \left \{-\int_{\gamma_0}^{\gamma_1} ds \dot{\gamma}^a A_a^i (\gamma (s)) T_i \right \}</math>
where <math>\gamma</math> is the curve along which the holonomy is evaluated, and <math>s</math> is a parameter along the curve, <math>\mathcal{P}</math> denotes path ordering meaning factors for smaller values of <math>s</math> appear to the left, and <math>T_i</math> are matrices that satisfy the <math>\operatorname{SU}(2)</math> algebra
<math display="block">[T^i ,T^j] = 2i \epsilon^{ijk} T_k.</math>

The [[Pauli matrices]] satisfy the above relation. It turns out that there are infinitely many more examples of sets of matrices that satisfy these relations, where each set comprises <math>(N+1) \times (N+1)</math> matrices with <math>N = 1,2,3,\dots</math>, and where none of these can be thought to 'decompose' into two or more examples of lower dimension. They are called different [[irreducible representations]] of the <math>\operatorname{SU}(2)</math> algebra. The most fundamental representation being the Pauli matrices. The holonomy is labelled by a half integer <math>N/2</math> according to the irreducible representation used.

The use of [[Wilson loop]]s explicitly solves the Gauss gauge constraint. [[Loop representation]] is required to handle the spatial diffeomorphism constraint. With Wilson loops as a basis, any Gauss gauge invariant function expands as,
<math display="block">\Psi [A] = \sum_\gamma \Psi [\gamma] W_\gamma [A].</math>

This is called the loop transform and is analogous to the momentum representation in quantum mechanics (see [[Position and momentum space]]). The QM representation has a basis of states <math>\exp (ikx)</math> labelled by a number <math>k</math> and expands as
<math display="block">\psi [x] = \int dk \psi (k) \exp (ikx). </math>
and works with the coefficients of the expansion <math>\psi (k).</math>

The inverse loop transform is defined by
<math display="block">\Psi [\gamma] = \int [dA] \Psi [A] W_\gamma [A].</math>

This defines the loop representation. Given an operator <math>\hat{O}</math> in the connection representation,
<math display="block">\Phi [A] = \hat{O} \Psi [A] \qquad Eq \; 1,</math>
one should define the corresponding operator <math>\hat{O}'</math> on <math>\Psi [\gamma]</math> in the loop representation via,
<math display="block">\Phi [\gamma] = \hat{O}' \Psi [\gamma] \qquad Eq \; 2,</math>
where <math>\Phi [\gamma]</math> is defined by the usual inverse loop transform,
<math display="block">\Phi [\gamma] = \int [dA] \Phi [A] W_\gamma [A] \qquad Eq \; 3.</math>

A transformation formula giving the action of the operator <math>\hat{O}'</math> on <math>\Psi [\gamma]</math> in terms of the action of the operator <math>\hat{O}</math> on <math>\Psi [A]</math> is then obtained by equating the R.H.S. of <math>Eq \; 2</math> with the R.H.S. of <math>Eq \; 3</math> with <math>Eq \; 1</math> substituted into <math>Eq \; 3</math>, namely
<math display="block">\hat{O}' \Psi [\gamma] = \int [dA] W_\gamma [A] \hat{O} \Psi [A],</math>
or
<math display="block">\hat{O}' \Psi [\gamma] = \int [dA] (\hat{O}^\dagger W_\gamma [A]) \Psi [A],</math>
where <math>\hat{O}^\dagger</math> means the operator <math>\hat{O}</math> but with the reverse factor ordering (remember from simple quantum mechanics where the product of operators is reversed under conjugation). The action of this operator on the Wilson loop is evaluated as a calculation in the connection representation and the result is rearranged purely as a manipulation in terms of loops (with regard to the action on the Wilson loop, the chosen transformed operator is the one with the opposite factor ordering compared to the one used for its action on wavefunctions <math>\Psi [A]</math>). This gives the physical meaning of the operator <math>\hat{O}'</math>. For example, if <math> \hat{O}^\dagger</math> corresponded to a spatial diffeomorphism, then this can be thought of as keeping the connection field <math>A</math> of <math>W_\gamma [A]</math> where it is while performing a spatial diffeomorphism on <math>\gamma</math> instead. Therefore, the meaning of <math>\hat{O}'</math> is a spatial diffeomorphism on <math>\gamma</math>, the argument of <math>\Psi [\gamma]</math>.

In the loop representation, the spatial diffeomorphism constraint is solved by considering functions of loops <math>\Psi [\gamma]</math> that are invariant under spatial diffeomorphisms of the loop <math>\gamma</math>. That is, [[knot invariant]]s are used. This opens up an unexpected connection between [[knot theory]] and quantum gravity.

Any collection of non-intersecting Wilson loops satisfy Ashtekar's quantum Hamiltonian constraint. Using a particular ordering of terms and replacing <math>\tilde{E}^a_i</math> by a derivative, the action of the quantum Hamiltonian constraint on a Wilson loop is
<math display="block">\hat{\tilde{H}}^\dagger W_\gamma [A] = - \epsilon_{ijk} \hat{F}^k_{ab} \frac{\delta}{\delta A_a^i} \frac{\delta}{\delta A_b^j} W_\gamma [A].</math>

When a derivative is taken it brings down the tangent vector, <math>\dot{\gamma}^a</math>, of the loop, <math>\gamma</math>. So,
<math display="block">\hat{F}^i_{ab} \dot{\gamma}^a \dot{\gamma}^b.</math>

However, as <math>F^i_{ab}</math> is anti-symmetric in the indices <math>a</math> and <math>b</math> this vanishes (this assumes that <math>\gamma</math> is not discontinuous anywhere and so the tangent vector is unique).

With regard to loop representation, the wavefunctions <math>\Psi [\gamma]</math> vanish when the loop has discontinuities and are knot invariants. Such functions solve the Gauss law, the spatial diffeomorphism constraint and (formally) the Hamiltonian constraint. This yields an infinite set of exact (if only formal) solutions to all the equations of quantum general relativity!{{sfn|Rovelli|Smolin|1988|pp=1155–1958}} This generated a lot of interest in the approach and eventually led to LQG.