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We know from quantum mechanics that if we replace the wave-function, ${\displaystyle \psi (x)}$, describing the electron field by

${\displaystyle \psi ‘(x)=\exp(i\theta )\psi (x)}$

that it leaves physical predictions unchanged. We consider the imposition of local invariance on the phase of the electron field,

${\displaystyle \psi ‘(x)=\Omega \psi (x)=\exp(i\theta (x))\psi (x)}$

The problem is that derivatives of ${\displaystyle \psi (x)}$ are not covariant under this transformation:

${\displaystyle \partial _{\mu }(\exp(i\theta (x))\psi (x))=\Omega \partial _{\mu }\psi (x)+\partial _{\mu }\Omega \psi (x)}$.

In order to cancel out the second unwanted term, one introduces a new derivative operator ${\displaystyle {\mathcal {D}}_{\mu }}$ that is covariant. To construct ${\displaystyle {\mathcal {D}}_{\mu }}$, one introduces a new field, the connection ${\displaystyle A_{\mu }}$:

${\displaystyle {\mathcal {D}}_{\mu }=\partial _{\mu }+igA_{\mu }(x)}$.

Then

${\displaystyle ({\mathcal {D}}_{\mu }\psi )’=\partial _{\mu }\psi ‘+igA_{\mu }’\psi ‘=\Omega \partial _{\mu }\psi +(\partial \Omega )\psi +igA_{\mu }’\Omega \psi }$

The term ${\displaystyle \partial _{\mu }\Omega }$ is precisely cancelled out by requiring the connection field transforms as

${\displaystyle A_{\mu }'(x)=A_{\mu }(x)+{i \over g}[\partial _{\mu }\Omega (x)]\Omega ^{-1}(x)\quad Eq1.}$.

We then have that

${\displaystyle ({\mathcal {D}}_{\mu }\psi )’=\Omega {\mathcal {D}}_{\mu }\psi }$.

Note that ${\displaystyle Eq1}$ is equivalent to

${\displaystyle A_{\mu }'(x)=A_{\mu }(x)+{1 \over g}\partial _{\mu }\theta (x)}$

which looks the same as a gauge transformation of the gauge potential of Maxwell’s theory. It is possible to construct an invariant action for the connection field itself. We want an action that only has two derivatives (since actions with higher derivatives are not unitary). Define the quantity:

${\displaystyle F_{\mu \nu }={-i \over g}[{\mathcal {D}}_{\mu },{\mathcal {D}}_{\mu }]={-i \over g}[\partial _{\mu }+igA_{\mu }(x),\partial _{\nu }+igA_{\nu }(x)]}$

${\displaystyle ={-i \over g}{\Big (}[\partial _{\mu },\partial _{\nu }]+ig(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })-g^{2}[A_{\mu },A_{\nu }]{\Big )}}$

${\displaystyle =\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\nu }}$.

The unique action with only two derivatives is given by:

${\displaystyle S=-{\frac {1}{2}}\int d^{4}x{\Big (}F_{\mu \nu }F^{\mu \nu }{\Big )}}$.

Therefore, one can derive electromagnetic theory from arguments based solely on symmetry.

We now generalize the above reasoning to general gauge groups. One begins with the generators of some Lie algebra:

${\displaystyle [T_{i},T_{j}]=if^{ijk}T^{k}}$

Let there be a fermion field that transforms as

${\displaystyle \mathbf {\Psi } ‘\mapsto {\hat {\Omega }}(x)\mathbf {\Psi } (x)=\exp(i\theta ^{i}(x)T^{i})\mathbf {\Psi } (x)}$

Again the derivatives of ${\displaystyle \mathbf {\Psi } (x)}$ are not covariant under this transformation. We introduce a covariant derivative

${\displaystyle \mathbf {\mathcal {D}} _{\mu }=\mathbf {I} \partial _{\mu }+ig\mathbf {A} _{\mu }(x)}$

with connection field given by

${\displaystyle \mathbf {A} _{\mu }(x)=A_{\mu }^{i}(x)T^{i}}$

We require that ${\displaystyle \mathbf {A} _{\mu }(x)}$ transforms as:

${\displaystyle \mathbf {A} _{\mu }'(x)={\hat {\Omega }}\mathbf {A} _{\mu }(x){\hat {\Omega }}^{-1}+{i \over g}{\hat {\Omega }}(\partial _{\mu }{\hat {\Omega }}^{-1})}$.

We define the field strength operator

${\displaystyle \mathbf {F} _{\mu \nu }=-{i \over g}[\mathbf {\mathcal {D}} _{\mu },\mathbf {\mathcal {D}} _{\nu }]=\partial _{\mu }\mathbf {A} _{\nu }-\partial _{\nu }\mathbf {A} _{\mu }+ig[\mathbf {A} _{\mu },\mathbf {A} _{\nu }]=(\partial _{\mu }A_{\nu }^{i}-\partial _{\nu }A_{\mu }^{i}+gf^{ijk}A_{\mu }^{j}A_{\nu }^{k})T^{i}}$.

As ${\displaystyle \mathbf {\mathcal {D}} _{\mu }}$ is covariant, this means that the ${\displaystyle F_{\mu \nu }^{i}}$ tensor is also covariant:

${\displaystyle \mathbf {F} _{\mu \nu }\mapsto \mathbf {F} _{\mu \nu }’={\hat {\Omega }}\mathbf {F} _{\mu \nu }{\hat {\Omega }}^{-1}}$

Note that ${\displaystyle \mathbf {F} _{\mu \nu }}$ is only invariant under gauge transformations if ${\displaystyle {\hat {\Omega }}}$ is a scalar, that is, only in the case of electromagnetism.

We can now construct an invariant action out of this tensor. Again we want an action that only has two derivatives. The simplest choice is the trace of the commutator:

${\displaystyle \operatorname {Tr} ({\hat {\Omega }}\mathbf {F} _{\mu \nu }{\hat {\Omega }}^{-1}{\hat {\Omega }}\mathbf {F} ^{\mu \nu }{\hat {\Omega }}^{-1})=\operatorname {Tr} (\mathbf {F} _{\mu \nu }\mathbf {F} ^{\mu \nu })}$

The unique action with only two derivatives is given by:

${\displaystyle S=-{1 \over 2}\int d^{4}xTr(\mathbf {F} _{\mu \nu }\mathbf {F} ^{\mu \nu })=-{1 \over 2}\int d^{4}x\operatorname {Tr} {\Big (}F_{\mu \nu }^{i}T^{j}F_{j}^{\mu \nu }T^{j}{\Big )}}$

This is the action for Yang-mills theory.

In the connection representation the configuration variable is ${\displaystyle A_{a}^{i}}$ and its conjugate momentum is the (densitized) triad ${\displaystyle {\tilde {E}}_{i}^{a}}$. It is most natural to consider wavefunctions ${\displaystyle \Psi (A_{a}^{i})}$. This is known as the connection representation. The canonical variables get promoted to quantum operators:

${\displaystyle {\hat {A}}_{a}^{i}\Psi [A]=A_{a}^{i}\Psi [A]}$

(analogous to the position representation ${\displaystyle {\hat {q}}\psi (q)=q\psi (q)}$) and the triads are functional derivatives,

${\displaystyle {\hat {\tilde {E}}}_{a}^{i}\Psi [A]=-i{\delta \Psi [A] \over \delta A_{a}^{i}}}$

(analogous to ${\displaystyle {\hat {p}}\psi (q)=-i{d\psi (q) \over dq}}$)

Let us return to the classical Yang–Mills theory. It is possible to encode the gauge invariant information of the theory in terms of `loop-like’ variables.

We need the notion 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 ${\displaystyle \gamma }$  ; it is denoted

${\displaystyle h_{\gamma }[A]}$

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

${\displaystyle (h’_{e})_{\alpha \beta }=U_{\alpha \gamma }^{-1}(x)(h_{e})_{\gamma \sigma }U_{\sigma \beta }(y).}$

For a closed loop ${\displaystyle x=y}$ if we take the trace of this, that is, putting ${\displaystyle \alpha =\beta }$ and summing we obtain

${\displaystyle (h’_{e})_{\alpha \alpha }=U_{\alpha \gamma }^{-1}(x)(h_{e})_{\gamma \sigma }U_{\sigma \alpha }(x)=[U_{\sigma \alpha }(x)U_{\alpha \gamma }^{-1}(x)](h_{e})_{\gamma \sigma }=\delta _{\sigma \gamma }(h_{e})_{\gamma \sigma }=(h_{e})_{\gamma \gamma }}$

or

${\displaystyle \operatorname {Tr} h’_{\gamma }=\operatorname {Tr} h_{\gamma }.}$

Thus the trace of an holonomy around a closed loop is gauge invariant. It is denoted

${\displaystyle W_{\gamma }[A]}$

and is called a Wilson loop. The explicit form of the holonomy is

${\displaystyle h_{\gamma }[A]={\mathcal {P}}\exp {\Big \{}-\int _{\gamma _{0}}^{\gamma _{1}}\,ds{\dot {\gamma }}^{a}A_{a}^{i}(\gamma (s))T_{i}{\Big \}}}$

where ${\displaystyle \gamma }$ is the curve along which the holonomy is evaluated, and ${\displaystyle s}$ is a parameter along the curve, ${\displaystyle {\mathcal {P}}}$ denotes path ordering meaning factors for smaller values of ${\displaystyle s}$ appear to the left, and ${\displaystyle T_{i}}$ are matrices that satisfy the ${\displaystyle su(2)}$ algebra

${\displaystyle [T^{i},T^{j}]=2i\epsilon ^{ijk}T^{k}.\,}$

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 ${\displaystyle (N+1)\times (N+1)}$ matrices with ${\displaystyle N=1,2,3,\dots }$, 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 ${\displaystyle su(2)}$ algebra. The most fundamental representation being the Pauli matrices. The holonomy is labelled by a half integer ${\displaystyle N/2}$ according to the irreducible representation used.

An important theorem about Yang–Mills gauge theories is Giles’ theorem, according to which if one gives the trace of the holonomy of a connection for all possible loops on a manifold one can, in principle, reconstruct all the gauge invariant information of the connection.^[3] That is, Wilson loops constitute a basis of gauge invariant functions of the connection. This key result is the basis for the loop representation for gauge theories and gravity.

The use of Wilson loops explicitly solves the Gauss gauge constraint. As Wilson loops form a basis we can formally expand any Gauss gauge invariant function as,

${\displaystyle \Psi [A]=\sum _{\gamma }\Psi [\gamma ]W_{\gamma }[A]}$.

This is called the loop transform. We can see the analogy with going to the momentum representation in quantum mechanics. There one has a basis of states ${\displaystyle \exp(ikx)}$ labelled by a number ${\displaystyle k}$ and one expands

${\displaystyle \psi [x]=\int dk\psi (k)\exp(ikx).}$

and works with the coefficients of the expansion ${\displaystyle \psi (k)}$.

The inverse loop transform is defined by

${\displaystyle \Psi [\gamma ]=\int [dA]\Psi [A]W_{\gamma }[A].}$

This defines the loop representation. Given an operator ${\displaystyle {\hat {O}}}$ in the connection representation,

${\displaystyle \Phi [A]={\hat {O}}\Psi [A],\qquad {\text{Eq 1}}}$

one should define the corresponding operator ${\displaystyle {\hat {O}}’}$ on ${\displaystyle \Psi [\gamma ]}$ in the loop representation via,

${\displaystyle \Phi [\gamma ]={\hat {O}}’\Psi [\gamma ],\qquad {\text{Eq 2}}}$

where ${\displaystyle \Phi [\gamma ]}$ is defined by the usual inverse loop transform,

${\displaystyle \Phi [\gamma ]=\int [dA]\Phi [A]W_{\gamma }[A].\qquad {\text{Eq 3}}}$

A transformation formula giving the action of the operator ${\displaystyle {\hat {O}}’}$ on ${\displaystyle \Psi [\gamma ]}$ in terms of the action of the operator ${\displaystyle {\hat {O}}}$ on ${\displaystyle \Psi [A]}$ is then obtained by equating the R.H.S. of ${\displaystyle Eq\;2}$ with the R.H.S. of ${\displaystyle Eq\;3}$ with ${\displaystyle Eq\;1}$ substituted into ${\displaystyle Eq\;3}$, namely

${\displaystyle {\hat {O}}’\Psi [\gamma ]=\int [dA]W_{\gamma }[A]{\hat {O}}\Psi [A],}$

or

${\displaystyle {\hat {O}}’\Psi [\gamma ]=\int [dA]({\hat {O}}^{\dagger }W_{\gamma }[A])\Psi [A],}$

where by ${\displaystyle {\hat {O}}^{\dagger }}$ we mean the operator ${\displaystyle {\hat {O}}}$ but with the reverse factor ordering (remember from simple quantum mechanics where the product of operators is reversed under conjugation). We evaluate the action of this operator on the Wilson loop as a calculation in the connection representation and rearranging the result as a manipulation purely in terms of loops (one should remember that when considering the action on the Wilson loop one should choose the operator one wishes to transform with the opposite factor ordering to the one chosen for its action on wavefunctions ${\displaystyle \Psi [A]}$).

The introduction of Ashtekar variables cast general relativity in the same language as gauge theories. It was in particular the inability to have good control over the space of solutions to the Gauss’ law and spatial diffeomorphism constraints that led Rovelli and Smolin to consider a new representation – the loop representation.^[4]

To handle the spatial diffeomorphism constraint we need to go over to the loop representation. The above reasoning gives the physical meaning of the operator ${\displaystyle {\hat {O}}’}$. For example, if ${\displaystyle {\hat {O}}^{\dagger }}$ corresponded to a spatial diffeomorphism, then this can be thought of as keeping the connection field ${\displaystyle A}$ of ${\displaystyle W_{\gamma }[A]}$ where it is while performing a spatial diffeomorphism on ${\displaystyle \gamma }$ instead. Therefore, the meaning of ${\displaystyle {\hat {O}}’}$ is a spatial diffeomorphism on ${\displaystyle \gamma }$, the argument of ${\displaystyle \Psi [\gamma ]}$.

In the loop representation we can then solve the spatial diffeomorphism constraint by considering functions of loops ${\displaystyle \Psi [\gamma ]}$ that are invariant under spatial diffeomorphisms of the loop ${\displaystyle \gamma }$. That is, we construct what mathematicians call knot invariants. This opened up an unexpected connection between knot theory and quantum gravity.

The easiest geometric quantity is the area. Let us choose coordinates so that the surface ${\displaystyle \Sigma }$ is characterized by ${\displaystyle x^{3}=0}$. The area of small parallelogram of the surface ${\displaystyle \Sigma }$ is the product of length of each side times ${\displaystyle \sin \theta }$ where ${\displaystyle \theta }$ is the angle between the sides. Say one edge is given by the vector ${\displaystyle {\vec {u}}}$ and the other by ${\displaystyle {\vec {v}}}$ then,

${\displaystyle {\begin{aligned}A&=\|{\vec {u}}\|\|{\vec {v}}\|\sin \theta ={\sqrt {\|{\vec {u}}\|^{2}\|{\vec {v}}\|^{2}(1-\cos ^{2}\theta )}}\\[6pt]&={\sqrt {\|{\vec {u}}\|^{2}\|{\vec {v}}\|^{2}-({\vec {u}}\cdot {\vec {v}})^{2}}}\end{aligned}}}$

From this we get the area of the surface ${\displaystyle \Sigma }$ to be given by

${\displaystyle A_{\Sigma }=\int _{\Sigma }\,dx^{1}\,dx^{2}{\sqrt {\det \;q^{(2)}}}}$

where ${\displaystyle \det q^{(2)}=q_{11}q_{22}-q_{12}^{2}}$ and is the determinant of the metric induced on ${\displaystyle \Sigma }$. This can be rewritten as

${\displaystyle \det \;q^{(2)}={\epsilon ^{3ab}\epsilon ^{3cd}q_{ac}q_{bc} \over 2}.}$

The standard formula for an inverse matrix is

${\displaystyle q^{ab}={\epsilon ^{bcd}\epsilon ^{aef}q_{ce}q_{df} \over 2!\det(q)}}$

Note the similarity between this and the expression for ${\displaystyle \det q^{(2)}}$. But in Ashtekar variables we have ${\displaystyle {\tilde {E}}_{i}^{a}{\tilde {E}}^{bi}=\det(q)q^{ab}}$. Therefore,

${\displaystyle A_{\Sigma }=\int _{\Sigma }\,dx^{1}\,dx^{2}{\sqrt {{\tilde {E}}_{i}^{3}{\tilde {E}}^{3i}}}.}$

According to the rules of canonical quantization we should promote the triads ${\displaystyle {\tilde {E}}_{i}^{3}}$ to quantum operators,

${\displaystyle {\hat {\tilde {E}}}_{i}^{3}\sim {\delta \over \delta A_{3}^{i}}.}$

It turns out that the area ${\displaystyle A_{\Sigma }}$ can be promoted to a well defined quantum operator despite the fact that we are dealing with product of two functional derivatives and worse we have a square-root to contend with as well.^[5] Putting ${\displaystyle N=2J}$, we talk of being in the J-th representation. We note that ${\displaystyle \sum _{i}T^{i}T^{i}=J(J+1)1}$. This quantity is important in the final formula for the area spectrum. We simply state the result below,

${\displaystyle {\hat {A}}_{\Sigma }W_{\gamma }[A]=8\pi \ell _{Planck}^{2}\beta \sum _{I}{\sqrt {j_{I}(j_{I}+1)}}W_{\gamma }[A]}$

where the sum is over all edges ${\displaystyle I}$ of the Wilson loop that pierce the surface ${\displaystyle \Sigma }$.

The formula for the volume of a region ${\displaystyle R}$ is given by

${\displaystyle V=\int _{R}d^{3}x{\sqrt {\det(q)}}={1 \over 6}\int _{R}dx^{3}{\sqrt {\epsilon _{abc}\epsilon ^{ijk}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b}{\tilde {E}}_{k}^{c}}}.}$

The quantization of the volume proceeds the same way as with the area. As we take the derivative, and each time we do so we bring down the tangent vector ${\displaystyle {\dot {\gamma }}^{a}}$, 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 we would need at least intersections with three non-coplanar lines. Actually it turns out that one needs at least four-valent vertices for the volume operator to be non-vanishing.

We now consider Wilson loops with intersections. We assume the real representation where the gauge group is ${\displaystyle SU(2)}$. Wilson loops are an over complete basis as there are identities relating different Wilson loops. These come about from the fact that Wilson loops are based on matrices (the holonomy) and these matrices satisfy identities, the so-called Mandelstam identities. Given any two ${\displaystyle SU(2)}$ matrices ${\displaystyle \mathbb {A} }$ and ${\displaystyle \mathbb {B} }$ it is easy to check that,

${\displaystyle \operatorname {Tr} (\mathbb {A} )\operatorname {Tr} (\mathbb {B} )=\operatorname {Tr} (\mathbb {A} \mathbb {B} )+\operatorname {Tr} (\mathbb {A} \mathbb {B} ^{-1}).}$

This implies that given two loops ${\displaystyle \gamma }$ and ${\displaystyle \eta }$ that intersect, we will have,

${\displaystyle W_{\gamma }[A]W_{\eta }[A]=W_{\gamma \circ \eta }[A]+W_{\gamma \circ \eta ^{-1}}[A]}$

where by ${\displaystyle \eta ^{-1}}$ we mean the loop ${\displaystyle \eta }$ traversed in the opposite direction and ${\displaystyle \gamma \circ \eta }$ means the loop obtained by going around the loop ${\displaystyle \gamma }$ and then along ${\displaystyle \eta }$. See figure below. This is called a Mandelstam identity of the second kind. There is the Mandelstam identity of the first kind ${\displaystyle W(\gamma _{1}\circ \gamma _{2})=W(\gamma _{2}\circ \gamma _{1})}$. Spin networks are certain linear combinations of intersecting Wilson loops designed to address the over-completeness introduced by the Mandelstam identities.

In fact spin networks constitute a basis for all gauge invariant functions which minimize the degree of over-completeness of the loop basis, and for trivalent intersections eliminate it entirely.

As mentioned above the holonomy tells you 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 ${\displaystyle \gamma }$: 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.

Theorems establishing the uniqueness of the loop representation as defined by Ashtekar et al. (i.e. a certain concrete realization of a Hilbert space and associated operators reproducing the correct loop algebra – the realization that everybody was using) have been given by two groups (Lewandowski, Okolow, Sahlmann and Thiemann)^[6] and (Christian Fleischhack).^[7] Before this result was established it was not known whether there could be other examples of Hilbert spaces with operators invoking the same loop algebra, other realizations, not equivalent to the one that had been used so far.

• Conway knot 11n34
• Kinoshita–Terasaka knot 11n42