General relativity can be formulated either using the metric tensor $g_{\mu\nu}$ (and implicitly the affine Levi-Civita connection defined in terms of Christoffel symbols $\Gamma_{\mu\nu}^{\rho}$) or in terms of the vielbein field or tetrad $e_{\mu}^m$ and the spin connection $\omega_{\mu}^{mn}$. The two formulations are equivalent but contrary to widespread belief the formulation based on the spin connection is more fundamental for two reasons. First, fundamental matter fields in Nature are described by chiral fermions (leptons and quarks) which when propagating in a curved spacetime feels the metric only through the vielbein field. The second reason is the fact that the most successful canonical quantization of general relativity today (loop quantum gravity) uses as canonical conjugate variables the (densitized) vielbein field from one hand and the (self-dual part of the) spin connection from the other hand as the canonical momentum and the configuration variable respectively (and not the $3-$dimensional metric $h_{ij}$ or euivalently the vielbein field or triads $e_{i}^m$ and the extrinsic curvature $K_{ij}$ used in the classic ADM formulation). It will be seen that general relativity can then be reformulated as a complex $SU(2)$ gauge theory of the self-dual spin connections and as a consequence general relativity becomes a dynamical theory for three-dimensional connections and not for three-dimensional geometries and this makes it embeddable into Yang-Mills gauge theory.
The Hilbert-Einstein action
The vielbein formalism
ADM formulation and geometrodynamics
Three exercises in general relativity
The Palatini action and Ashtekar variables
The Hilbert-Einstein action of general relativity expressed in terms of the $4-$dimensional metric $g$ is equivalent to the Palatini action expressed in terms of the vielbein field $e$ and the spin connection $\omega$. In the first formulation the affine connection is used implicitly since it is determined by the metric tensor whereas in the second formulation the spin connection is used explicitly since it is an independent dynamical variable. The Palatini action is of the form ($G$ being Newton's constant)
\begin{eqnarray}
S=\frac{1}{16\pi G}\int d^4x\epsilon_{mnkl}\tilde{\eta}^{\mu\nu\alpha\beta}e_{\mu}^me_{\nu}^nR_{\alpha\beta}^{kl}.\label{pal1}
\end{eqnarray}
The indices $m,n,...$ are internal indices associated with the local $SO(1,3)$ Lorentz group whereas the indices $\mu,\nu,...$ are external indices associated with spacetime (and consequently with the local diffeomorphism group of general coordinate transformations). The tensor $\tilde{\eta}$ is the Levi-Civita tensor density corresponding to the curved indices $\mu,\nu,...$ whereas $\epsilon$ is the flat Levi-Civita symbol. The curvature (from the previous results) is defined by the relation
\begin{eqnarray}
R_{\rho\sigma\mu\nu}=R_{\rho\sigma n}^me_{\mu}^ne_m^{\alpha}g_{\alpha\nu}.
\end{eqnarray}
Now we use the result (with $g={\rm det}(g_{\mu\nu})$)
\begin{eqnarray}
\tilde{\eta}^{\alpha\beta\mu\nu}\epsilon_{klmn}e_{\mu}^me_{\nu}^n=\frac{\sqrt{-g}}{2}(e_k^{\alpha}e_l^{\beta}-e_k^{\beta}e_l^{\alpha}).
\end{eqnarray}
The Palatini action becomes then (recall that the Ricci curvature tensor and the Ricci scalar are defined by $R_{\mu\nu}=R^{\rho}_{\mu\rho\nu}$ and $R=g^{\mu\nu}R_{\mu\nu}$)
\begin{eqnarray}
S&=&\frac{1}{16\pi G}\int \sqrt{-g} d^4x e_k^{\alpha}e_l^{\beta}R_{\alpha\beta}^{kl}\nonumber\\
&=&\frac{1}{16\pi G}\int \sqrt{-g} d^4x R.
\end{eqnarray}
This is the Hilbert-Einstein action.
Alternatively, quantization based on the Palatini action gives immediately geometrodynamics of Wheeler, DeWitt and others. Indeed, the conjugate momentum associated with the spin connection $\omega_{\mu}^{mn}$ is found to be given by $\Pi^{\mu}_{mn}=\tilde{\eta}^{\mu\nu\alpha}\epsilon_{mnkl}e_{\nu}^ke_{\alpha}^l$. The theory has thus an additional (second class) constraint consisting in the fact that the momentum is decomposable as a product of two vielbein fields. By solving the second class constraint (using Dirac's formalism) we obtain new canonical variables in which the spin connection is lost as a dynamical variable and we end up again with geometrodynamics \cite{Ashtekar:1988sw}.
The revolutionary solution provided by Ashtekar (see \cite{Ashtekar} for a modern review and for the original references) consists in insisting that the Palatini action is the correct starting point but with the additional twist that the spin connection must be self-dual. In other words, we must replace in the Palatini action the real $SO(1,3)$ spin connection $\omega_{\mu}^{mn}$ by the complex self-dual connection $A_{\mu}^{mn}$defined by
\begin{eqnarray}
A_{\mu}^{mn}=\frac{1}{2G}(\omega_{\mu}^{mn}-\frac{i}{2}\epsilon^{mn}~_{kl}\omega_{\mu}^{kl}).
\end{eqnarray}
The complex connection $A_{\mu}^{mn}$ is self-dual because it satisfies the self-dual condition
\begin{eqnarray}
iA_{\mu}^{mn}=\frac{1}{2}\epsilon^{mn}~_{kl}A_{\mu}^{kl}.\label{sdc}
\end{eqnarray}The Palatini action becomes
\begin{eqnarray}
S=\frac{1}{16\pi G}\int d^4x\epsilon_{mnkl}\tilde{\eta}^{\mu\nu\alpha\beta}e_{\mu}^me_{\nu}^nF_{\alpha\beta}^{kl}.\label{pal2}
\end{eqnarray}
$F$ is the curvature tensor of the self-dual connection $A$. Thus, it must be given by
\begin{eqnarray}
F_{\alpha\beta m}^{n}=\partial_{\alpha}A_{\beta m}^n-\partial_{\beta}A_{\alpha m}^n+G^4A_{\alpha m}^kA_{\beta k}^n-G^4A_{\beta m}^kA_{\alpha k}^n.
\end{eqnarray}
We can check that the classical equations of motion derived from the self-dual Palatini action (\ref{pal2}) are exactly equivalent to the classical equations of motion derived from the original Palatini action (\ref{pal1}). In particular, the variation of the action (\ref{pal2}) with respect to the connection $A_{\mu}^{mn}$ gives as equation of motion the result that $A_{\mu}^{mn}$ is the (self-dual part of the) spin connection $\omega_{\mu}^{mn}$ which is compatible with the vielbein field $e_{\mu}^m$, i.e. it is determined by the condition ${\cal D}_{\rho}e_{\mu}^m=0$. Hence, the connection $A_{\mu}^{mn}$ is completely determined by the vielbein field $e_{\mu}^m$. On the other hand, the variation of the action (\ref{pal2}) with respect to the vielbein field $e_{\mu}^m$ gives as equation of motion the result that the spacetime metric $g_{\mu\nu}=e_{\mu}^me_{\nu}^{n}\eta_{mn}$ solves Einstein's equations.
Thus the classical equations of motion derived from the self-dual Palatini action (\ref{pal2}) are exactly equivalent to the classical equations of motion derived from the standard Palatini action (\ref{pal1}). But this does not mean that the two actions are identical. Indeed, the difference between (\ref{pal2}) and (\ref{pal1}) is an imaginary term which is not a pure divergence but reproduces as a correction to the equation of motion the first Bianchi identity (the trace of the dual of the Riemann tensor vanishes) which thus holds automatically. This imaginary term leads however under the Legendre transform of the self-dual Palatini action to a different conjugate momentum which is linear instead of being quadratic in the vielbein field and that makes the self-dual Palatini action (\ref{pal2}) distinctly different and quite superior to the standard Palatini action (\ref{pal1}) which is nothing else but the Hilbert-Einstein action. See \cite{Ashtekar} and references therein.
General relativity as an SU(2) gauge theory of self-dual spin connections
We start with the Hilbert-Einstein action with variable given only by the metric tensor $g_{\mu\nu}$ (the affine Levi-Civita connection $\Gamma_{\mu\nu}^{\rho}$ is not an independent variable here). Then by performing a Legendre transform and an ADM analysis the canonically conjugate variables are from the one hand the three-dimensional metric $h_{\mu\nu}$ or equivalently the vielbein fields (or triads) $e_{\mu}^{m}$ and from the other hand we have the corresponding conjugate momentum $\Pi_{\mu\nu}$ defined in terms of the extrinsic curvature $K_{\mu\nu}$ by the relation $\Pi_{\mu\nu}=-\sqrt{h}(K_{\mu\nu}-Kh_{\mu\nu})$. We are here only dealing with geometrodynamics where only first class constraints are involved.
If we start on the other hand with the standard Palatini action (\ref{pal1}) with variables given by the real vielbein field $e_{\mu}^m$ and the real spin connection $\omega_{\mu}^{mn}$ then the canonically conjugate variables are the spin connection $\omega_{\mu}^{mn}$ and the conjugate momentum $\Pi^{\mu}_{mn}=\tilde{\eta}^{\mu\nu\alpha}\epsilon_{mnkl}e_{\nu}^ke_{\alpha}^l$. By solving the second class constraint (the momentum is decomposable as a product of two vielbein fields) we obtain new canonical variables in which the spin connection is lost as a dynamical variable and we end up again with geometrodynamics.
We also recall that the vielbein field $e_{\mu}^m$ is covariantly constant with respect to both the spin connection $\omega_{\mu}^{mn}$ and the Levi-Civita connection $ \Gamma_{\mu\nu}^{\alpha}$, viz
\begin{eqnarray}
{\cal D}_{\rho}e_{\mu}^m&=&{\partial}_{\rho}e_{\mu}^m-{\Gamma}_{\rho\mu}^{\nu}e_{\nu}^m+\omega_{\rho n}^{m}e_{\mu}^n\nonumber\\
&=& 0.
\end{eqnarray}
This compatibility condition gives the spin connection in terms of the Levi-Civita connection and the vielbein field.
However, the theory formulated in terms of the Ashtekar variables given by a self-dual spin connection $A_{\mu}^{mn}$ which is necessarily complex and a real vielbein field (or tetrads) $e_{\mu}^m$ with an action given by the self-dual Palatini action (\ref{pal2}) is equivalent to a complex $SU(2)$ gauge theory of the self-dual spin connections . Indeed, after Legendre transform the canonically conjugate variables are found to be the self-dual spin connection $A_{\mu}^{mn}$ with a corresponding conjugate momentum $\Pi^{\mu}_{mn}$ which is also self-dual and furthermore is proportional to a single vielbein field not two and hence second class constraints are avoided.
Here it is technically simpler to start with complex general relativity since the connection is necessarily complex. So we start with a complex vielbein field $e_{\mu}^m$ and a complex self-dual $SO(1,3)$ connection $A_{\mu}^{mn}$ with an action given by the Palatini action (\ref{pal2}). After Legendre transform we get as our canonically conjugate variables the connection $A_{\mu}^{mn}$ and the conjugate momentum $\Pi^{\mu}_{mn}$ which are both in the self-dual part of the complexified $so(1,3)$ Lie algebra.
The original spin connection $\omega_{\mu}^{mn}$ is a real $SO(1,3)$ connection and recall that $SL(2,\mathbb{C})$ is the universal cover of $SO(1,3)$. The self-dual connection $A_{\mu}^{mn}$ belongs however to the complexified group $SO(1,3)_{\mathbb{C}}$. We have the Lie algebra isomorphisms
\begin{eqnarray}
so(1,3)_{\mathbb{C}}=so(4)_{\mathbb{C}}=so(3)_{\mathbb{C}}\oplus so(3)_{\mathbb{C}}.
\end{eqnarray}
The first $so(3)_{\mathbb{C}}$ factor represents self-dual (chiral, right-handed) fields whereas the second factor represents anti-self-dual (anti-chiral, left-handed) fields. The connection $A_{\mu}^{mn}$ is a complex connection (thus belonging to $so(1,3)_{\mathbb{C}}$ not to $so(1,3)$ like the connection $\omega_{\mu}^{mn}$) which is also self-dual (thus it belongs to the first factor $so(3)_{\mathbb{C}}$).
We are therefore dealing with an $so(3)_{\mathbb{C}}-$valued one-form and since the universal cover of $SO(3)$ is $SU(2)$ the connection $A_{\mu}^{mn}$ is in fact an $su(2)_{\mathbb{C}}-$valued one-form.
Using the above isomorphism between the self-dual subalgebra of the complexified Lie algebra $so(1,3)_{\mathbb{C}}$ and the complexfied Lie algebra $so(3)_{\mathbb{C}}$ we can map the canonically conjugate variables $A_{\mu}^{mn}$ and $\Pi^{\mu}_{mn}$ to the $so(3)_{\mathbb{C}}-$valued fields $A_{\mu}^n$ and $\Pi_{\mu}^n$ given respectively by
\begin{eqnarray}
A_{\mu}^m=\frac{1}{2}A_{\mu kl}\epsilon^{klm}~,~\Pi_{\mu}^m=\frac{1}{2}\Pi_{\mu kl}\epsilon^{klm}.
\end{eqnarray}
The self-dual connection $A_{\mu}^m$ is also called the chiral spin connection. As it turns out, the canonical momentum $\Pi_{\mu}^m$ is precisely the densitized vielbein field (or triad) given by
\begin{eqnarray}
\Pi_{\mu}^m=\tilde{e}_{\mu}^m=\sqrt{h}e_{\mu}^m.
\end{eqnarray}
The Ashtekar variables are precisley the densitized triad $\tilde{e}_{\mu}^m$ and the self-dual connection $A_{\mu}^m$. The self-dual Palatini action in terms of these variables takes the form
\begin{eqnarray}
S=\int d^4x \bigg(-2i\tilde{e}_{\mu}^m{\cal L}_tA_m^{\mu}-2i(t^{\mu}A_{\mu}^m)G_m+2iN^{\mu}{\cal V}_{\mu}+\frac{N}{\sqrt{h}}{\cal S}\bigg). \label{pal3}
\end{eqnarray}
The quantities $G_m$, ${\cal V}_{\mu}$ and ${\cal S}$ are explicitly given by
\begin{eqnarray}
G_m={\cal D}_{\mu}\tilde{e}_m^{\mu}~,~{\cal V}_{\mu}=\tilde{e}_n^{\nu}F_{\mu\nu}^n~,~{\cal S}=\epsilon_{ijk}\tilde{e}_i^{\mu}\tilde{e}_j^{\nu}F_{\mu\nu}^k.
\end{eqnarray}
The curvature $F_{\mu\nu}^k$ of the gauge field $A_{\mu}^k$ is explicitly given by
\begin{eqnarray}
F_{\mu\nu}^l=\partial_{\mu}A_{\nu}^l-\partial_{\nu}A_{\mu}^l+G\epsilon_{lmn}A_{\mu}^mA_{\nu}^n.
\end{eqnarray}
This shows explicitly that we are indeed dealing with an $SU(2)$ gauge theory.
In the first term of the self-dual Palatini action (\ref{pal3}) the operator ${\cal L}_t$ is the Lie derivative along the time direction and hence ${\cal L}_tA_m^{\mu}$ is the covariant time derivative of the field configuration $A_m^{\mu}$ along the vector field $t^{\mu}=Nn^{\mu}+N^{\mu}$ which defines the spacetime foliation with hypersurfaces $\Sigma_t$ whose normal vector field is given by $n^{\mu}$ ($N$ and $N^{\mu}$ are then the lapse function and the shift vector).
From the first term in the action (\ref{pal3}) which is then of the form $p\dot{q}$ we can immediately conclude that the densitized triad $\tilde{e}_{\mu}^m$ is precisely the conjugate momentum $p$ associated with the self-dual connection $A_{\mu}^m$ which acts exactly as the configuration variable $q$. Indeed, the fundamental Poisson brackets are of the form
\begin{eqnarray}
\{A_{\mu}^m(x), \tilde{e}^{\nu}_n(y)\}=\frac{i}{2}\delta_m^n\delta_{\mu}^{\nu}\delta^3(x-y).
\end{eqnarray}
In summary, we have gone from the ADM variables consisting of the three-dimensional metric $h_{\mu\nu}$ (or equivalently the densitized triads $\tilde{e}_{\mu}^m$) and the canonical momentum $\Pi_{\mu\nu}$ (or equivalently the extrinsic curvature $K_{\mu}^m$ defined by $K_{\mu}^m=K_{\mu\nu}e^{\nu m}$) to the complex Ashtekar variables consisting of $\tilde{e}_{\mu}^m$ and the connection $A_{\mu}^m$. The relation between the self-dual connection $A_{\mu}^m$ and the original variables $\tilde{e}_{\mu}^m$ and $K_{\mu}^m$ is given explicitly by
\begin{eqnarray}
GA_{\mu}^m=\Gamma_{\mu}^m-iK_{\mu}^m.\label{fundamental}
\end{eqnarray}
The spin connection $\Gamma_{\mu}^m$ which is compatible with the densitized triads $\tilde{e}_{\mu}^m$ is given obviously by the relation
\begin{eqnarray}
\Gamma_{\mu}^m=\frac{1}{2}\omega_{\mu kl}\epsilon^{klm}.
\end{eqnarray}
See \cite{Pullin:1993fw} and references therein.
At the end of all this we will naturally need to return to real (Lorentzian) general relativity and thus one must impose reality conditions. In terms of the geometrodynamic variables these reality conditions are simply the requirements that the three-dimensional metric $h_{\mu\nu}=e_{\mu}^me_{\nu}^n\eta_{mn}$ and the extrinsic curvature $K_{\mu\nu}$ must be real. Let us emphasize here that the self-dual spin connection $A_{\mu}^{mn}$ given by equation (\ref{sdc}) is necessarily complex (since the spacetime manifold is Lorentzian) and thus the reality conditions will not alter this fact. But, in Euclidean signature the self-dual connections are necessarily real and thus the reality conditions which are needed to be imposed on complex general relativity to recover the real phase space are the requirements that the triads must be real and the connections must also be real. In Lorentzian signature the connections will remain complex after imposing the reality conditions (which will cause other problems for the integration measure in the quantum theory).
The reality conditions can also be understood in a more illuminating way as follows.
We start with real general relativity, i.e. real vielbein field and real spin connection in the Palatini action. After Legendre transform we can take as our variables the densitized triads $\tilde{e}_{\mu}^m$ (instead of the three-dimensional metric $h_{\mu\nu}$) and the extrinsic curvature $e_{\mu}^m$ (instead of the momentum $\Pi_{\mu\nu}$).
On the real phase space $(q,p)\equiv (\tilde{e}_{\mu}^m,K_{\mu}^m)$ we perform then a complex canonical transformation which takes us to the complex Ashtekar variables $(q,z)$ where $z$ is the complex coordinate given by $z=f(q)-ip$. Explicitly, $f(q)$ is the spin connection $\Gamma_{\mu}^m$ which is determined by the densitized triads $\tilde{e}_{\mu}^m$ and $z$ is precisely the self-dual connection $GA_{\mu}^m=\Gamma_{\mu}^m-iK_{\mu}^m$. As we have seen $\tilde{e}_{\mu}^m$ and $A_{\mu}^m$ are canonically conjugate to each other where the densitized triads are what play the role of the conjugate momentum in the Ashtekar variables $(q,z)$ contrary to their role in the original real coordinates $(q,p)$, i.e. $(q^A,p^A)\equiv (A_{\mu}^m, \tilde{e}_{\mu}^m)$.
The reality conditions are now given by the requirement that the three-dimensional metric $h_{\mu\nu}=e_{\mu}^me_{\nu}^n\eta_{mn}$ is real and the requirement that $GA_{\mu}^m-\Gamma_{\mu}^m$ is pure imaginary.
In the Palatini action (\ref{pal3}) which is written in terms of Ashtekar variables the second, third and fourth terms lead to the constraints. Indeed, the lpase function $N$, the shift vector $N^{\mu}$ and the component of the connection $A_{\mu}$ along the time direction, i.e. $t^{\mu}A_{\mu}^m$ are all Legendre multipliers and the variation of the action with respect to them will lead to the constraints
\begin{eqnarray}
G_m={\cal D}_{\mu}\tilde{e}_m^{\mu}=0.\label{const1}
\end{eqnarray}
\begin{eqnarray}
{\cal V}_{\mu}=\tilde{e}_n^{\nu}F_{\mu\nu}^n=0.\label{const2}
\end{eqnarray}
\begin{eqnarray}
{\cal S}=\epsilon_{ijk}\tilde{e}_i^{\mu}\tilde{e}_j^{\nu}F_{\mu\nu}^k=0. \label{const3}
\end{eqnarray}
These seven first class constraints are simple polynomials in the basic variables (as opposed to what happens in geometrodynamics). And, they reduce the $9$ degrees of freedom of $A_{\mu}^i$ to the two degrees of freedom of the graviton.
The constraints (\ref{const2}) and (\ref{const3}) are the diffeomorphism and Hamiltonian constraints found in geometrodynamics which generate respectively spatial diffeomorphisms on each surface $\Sigma_t$ and time evolution between different surfaces $\Sigma_t$ and $\Sigma_{t+\delta t}$.
The first constraint (\ref{const1}) is the so-called Gauss constraint and it represents Gauss law in this gauge theory and generates local $SO(3)$ invariance of the triads. It arises from the fact that the time component of the connection $A_{\mu}$ is not a dynamical field. We are really dealing with an $SU(2)$ gauge theory on the the three-dimensional surfaces $\Sigma_t$ with gauge field $A_{\mu}^m$ and since $\tilde{e}^{\mu}_m$ is the conjugate momentum it will act as the electric field $E^{\mu}_m$ with a quantized flux leading to quantized geometry and discretized spacetime (in the form of discrete spectra of areas and volumes) and also leads to the absence of gravitational singularities.
Thus, the Gauss constraint and the diffeomorphism constraints generate the local invariance group which is the semi-direct product of the local $SO(3)$ rotation group of the triads and the spatial diffeomorphism group on $\Sigma_t$. On the other hand, the scalar constraint is of the form $G^{\alpha\beta}p_{\alpha}p_{\beta}=0$ where $G$ is the supermetric and thus this constraint generates null geodesics motion in the configuration space of the connection.
Similarly to the constraints, the Hamiltonian and the equations of motion are all low order polynomials in the basic variables of Ashtekar.
The real SU(2) gauge theory
The self-dual or chiral connection $A_{\mu}^m$ in Lorentzian signature is a complex $SU(2)$ gauge field which means in particular that the corresponding holonomies or Wilson loops (which define the obeservables of the quantum gauge theory) are non-compact, i.e. they belong to a non-compact subgroup $SL(2,\mathbb{C})_{\rm sd}$ of $SL(2,\mathbb{C})$ generated by the self-dual part of the Lie algebra $sl(2,\mathbb{C})$. As a consequence the path integrals defining the quantum theory are ill defined and require a regularization in the form of a Wick rotation in the internal space which sends the non-compact group $SL(2,\mathbb{C})_{\rm sd}$ to the compact group $SU(2)$, i.e. the chiral connection $A_{\mu}^m$ given by (\ref{fundamental}) is replaced with a real $SU(2)$ gauge field given by
GA_{\mu}^m=\Gamma_{\mu}^m+\beta K_{\mu}^m.\label{fundamental1}
\end{eqnarray}
In other words, we replace the "-i" in (\ref{fundamental}) with a real parameter $\beta$ in (\ref{fundamental1}) called the Barbero-Immirzi parameter \cite{Barbero:1994ap,Immirzi:1996di}. The fundamental Poisson brackets become
\begin{eqnarray}
\{A_{\mu}^m(x), \tilde{e}^{\nu}_n(y)\}=-\frac{\beta}{2}\delta_m^n\delta_{\mu}^{\nu}\delta^3(x-y).
\end{eqnarray}
The choice of the parameter does not alter the Gauss and the spatial diffeomorphism constraints. But the Hamiltonian (which is a linear combination of the constraints) acquires an additional term, viz
\begin{eqnarray}
H=\frac{\epsilon_{lmn}\tilde{e}^{\mu}_m\tilde{e}^{\nu}_nF_{\mu\nu}^l}{\sqrt{h}}+2\frac{\beta^2+1}{\beta^2}\frac{\tilde{e}^{\mu}_m\tilde{e}^{\nu}_n-\tilde{e}^{\mu}_n\tilde{e}^{\nu}_m}{\sqrt{h}}(A_{\mu}^m-\Gamma_{\mu}^m)(A_{\nu}^n-\Gamma_{\nu}^n).
\end{eqnarray}
The second term vanishes for $\beta=\pm 1$. This Hamiltonian was simplified by Thiemann. See for example his book \cite{Thiemann:2007zz}.
Loop representation and spin networks
We have now a real $SU(2)$ gauge theory with a connection or gauge field $A_{\mu}^m$ living on a three-dimensional surface $\Sigma_t$. The classical configuration space (the space of all connections $A_{\mu}^m$ ) is denoted by ${\cal A}$ while the quantum configuration space is denoted by $\bar{\cal A}$ which is an extension of ${\cal A}$.
Each connection $A_{\mu}^m$ defines a holonomy $h_{\alpha}[A]$ along any oriented path on the surface $\Sigma_t$, i.e. $\alpha: [s_0,s_1]=[0,1]\longrightarrow \Sigma_t$ with an affine parameter $s$ by the relation \cite{Loll:1993yz}
\begin{eqnarray}
h_{\alpha}[A]=U(s_1,s_0)={\cal P}\exp\bigg(-\int_{0}^{1}ds\dot{\alpha}^{\mu}(s)A_{\mu}^m(\alpha(s))T_m\bigg).
\end{eqnarray}
h_{\alpha}[A]=U(s_1,s_0)={\cal P}\exp\bigg(-\int_{0}^{1}ds\dot{\alpha}^{\mu}(s)A_{\mu}^m(\alpha(s))T_m\bigg).
\end{eqnarray}
The ${\cal P}$ is the usual path ordering operation (operators with larger values of $s$ are placed to the left of the operators with to smaller values of $s$) and $T_m$ are the usual generators of $SU(2)$ which satisfy the Lie algebra
\begin{eqnarray}
[T_m,T_n]=i\epsilon_{mnl}T_l.
\end{eqnarray}
The set of all holonomies $h_{\alpha}[A]$ define the quantum configuration space $\bar{\cal A}$ in the same way that the set of all connections $A_{\mu}^m$ define the classical configuration space ${\cal A}$. A holonomy (called generalized connection in \cite{Huggett:1998sz}) is a map on the space of all paths in $\Sigma_t$ which assigns an element of the group $SU(2)$ to each path $\alpha(t)$ (in contrast to the connection which is a map on the hypersurface $\Sigma_t$ which assigns an element of the Lie algebra $su(2)$ to each point on the surface).
[T_m,T_n]=i\epsilon_{mnl}T_l.
\end{eqnarray}
The set of all holonomies $h_{\alpha}[A]$ define the quantum configuration space $\bar{\cal A}$ in the same way that the set of all connections $A_{\mu}^m$ define the classical configuration space ${\cal A}$. A holonomy (called generalized connection in \cite{Huggett:1998sz}) is a map on the space of all paths in $\Sigma_t$ which assigns an element of the group $SU(2)$ to each path $\alpha(t)$ (in contrast to the connection which is a map on the hypersurface $\Sigma_t$ which assigns an element of the Lie algebra $su(2)$ to each point on the surface).
The holonomy defines the parallel transport of a spinor in the background of the configuration $A_{\mu}^m$ along the curve $\alpha(t)$ between the start point $\alpha(0)$ and the end point $\alpha(1)$ and it measures the accumulated phase difference between the initial and final values of the spinor at the two points. More explicitly, under gauge transformations $g$ we must have
\begin{eqnarray}
h_{\alpha}\longrightarrow h_{\alpha}^{\prime}=g^{-1}(\alpha(1))h_{\alpha}g(\alpha(0)).
\end{eqnarray}
h_{\alpha}\longrightarrow h_{\alpha}^{\prime}=g^{-1}(\alpha(1))h_{\alpha}g(\alpha(0)).
\end{eqnarray}
This defines the so-called generalized gauge transformations \cite{Huggett:1998sz} which act on holonomies only at the end points of paths in contrast to ordinary gauge transformations which act on connections at every point of $\Sigma_t$. They are generated by the Gauss gauge constraint which can then be solved explicitly by using only Wilson loops which are holonomies traced out around closed paths (loops), viz
For example, the area of a two-dimensional surface $\Sigma$ with induced metric $h_{ab}^{(2)}$ in the three-dimensional hypersurface $\Sigma_t$ with induced metric $h_{ab}$ is given by
\begin{eqnarray}
A_{\Sigma}=\int dx^1dx^2\sqrt{{\rm det} h^{(2)}}.
\end{eqnarray}
The metric $h_{ab}^{(2)}$ is the induced metric on the surface $\Sigma$
which can be expressed in terms of the metric $h_{ab}$, then in terms of the densitized triads $\tilde{e}_m^{\mu}$ using the relation $\tilde{e}_m^{\mu}\tilde{e}^{\nu m}=h^{\mu\nu}{\rm det} h$. In the quantum theory the densitized triads become operators $\hat{\tilde{e}}_m^{\mu}$ which act (up to a numerical factor) as $\delta/\delta A_{\mu}^m$ and hence the area $A_{\Sigma}$ becomes an area operator $\hat{A}_{\Sigma}$ which admits the Wilson loops (\ref{WL}) as eigenvectors, viz \cite{Gambini:2011zz}
\begin{eqnarray}
A_{\Sigma}W_{\alpha}[A]=8\pi l_{\rm P}^2\beta \sum_I\sqrt{j_I(J_I+1)}W_{\alpha}[A].
\end{eqnarray}
The sum is over all edges of the Wilson loop $\alpha$ that intersect the surface $\Sigma$. The surface areas of $\Sigma$ is then given by $8\pi l_P^2\beta$ \sum_I\sqrt{j_I(J_I+1)}$ where $j_I$ is the spin associated with the edge $I$, $\beta$ is the Immirzi parameter and $l_P$ is the Planck length.
\begin{eqnarray}
W_{\alpha}[A]=Tr h_{\alpha}[A].\label{WL}
\end{eqnarray}
These are gauge invariant by construction.
If we denote the set of all generalized gauge transformations by $\bar{\cal G}$ then the gauge invariant quantum configuration space must be given by the quotient $\bar{\cal A}/\bar{\cal G}$. This space can be viewed as a projective limit of a family of compact, smooth and finite dimensional configuration spaces $\{\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}\}$.
Each configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ is labelled by a path or graph $\alpha$ characterized by $N$ edges and $V$ vertices (each edge $e$ starts at a vertex $v_{e_0}$ and ends at a vertex $v_{e_1}$). From the one hand, the space $\bar{\cal A}_{\alpha}$ is the space of generalized connections or holonomies over the graph $\alpha$ which consists of the mappings which assign to each edge of the graph an element of the group $SU(2)$, i.e. $\bar{\cal A}_{\alpha}$ is isomorphic to $SU(2)^N$. From the other hand, the space $\bar{\cal G}_{\alpha}$ is the space of generalized gauge transformations over the graph $\alpha$ which consists of all mappings which assign to each vertex an element of $SU(2)$. More precisely, the action of a given generalized gauge transformation $g$ on an edge $e$ of the graph $\alpha$ is given by $h_{\alpha}(e)\longrightarrow h_{\alpha}^{\prime}(e)=g^{-1}(v_{e_1})h_{\alpha}(e)g(v_{e_0})$. Hence the space $\bar{\cal G}_{\alpha}$ is isomorphic to $SU(2)^V$ and as a consequence the configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ is isomorphic to $SU(2)^{N-V}$.
The Hilbert space of state vectors on the configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ (which is compact and finite dimensional) is precisely the space of square-integrable functions ${\cal H}_{\alpha}=L^2(\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha})$ where the measure is obviously induced by the usual the Haar measure on $SU(2)$. Elementary quanta of geometry are elements of ${\cal H}_{\alpha}$ and they are of the form \cite{Huggett:1998sz}
\begin{eqnarray}
\Psi_{\alpha}(h_{\alpha})=\psi(h_{\alpha}(e_1),...,h_{\alpha}(e_N))~,~\psi\in SU(2)^N.
\end{eqnarray}
In summary, the members of the family $\{\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}\}$ are all compact and finite dimensional spaces with corresponding Hilbert spaces ${\cal H}_{\alpha}=L^2(\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha})$ and as a consequence the projective limit $\bar{\cal A}/\bar{\cal G}$ which is also compact admits a regular Borel measure allowing the construction of a corresponding Hilbert space of square-integrable functions ${\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$ where the measure is induced by the usual the Haar measure on $SU(2)$.
The Hilbert space ${\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$ admits a more interesting decomposition as a direct sum of finite dimensional orthogonal Hilbert spaces ${\cal H}_{\alpha,\vec{j}}$ characterized together with the graph $\alpha$ by a vector $\vec{j}$ of half-integers, i.e. $\vec{j}=(j_1,j_2,...,j_N)$ where the integer $j_i$ represents the irreducible representation of $SU(2)$ which labels the edge $i$ of the graph $\alpha$. We have then \cite{Ashtekar:2014kba}
\begin{eqnarray}
{\cal H}=L^2(\bar{\cal A}/\bar{\cal G})=\oplus_{\alpha,\vec{j}} {\cal H}_{\alpha,\vec{j}}.
\end{eqnarray}
${\cal H}_{\alpha,\vec{j}}$ are Hilbert spaces of spin network states on the configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$. A spin network state $\Psi_{\alpha,\vec{j},\vec{i}}(A)$ is then an element of ${\cal H}_{\alpha,\vec{j}}$ which is characterized by $1)$ the graph $\alpha$, $2)$ the $N$ irreducible representations $j_i$ associated with the edges of the graph and $3)$ the $V$ intertwining operators $i_i$ associated with the vertices of the graph.
More precisely, let $\rho_e$ be the irreducible representation associated with the edge $e$, i.e. $\rho_e$ is the homomorphism $\rho_e:SU(2)\longrightarrow {\rm End}(V_e)$ where ${\rm End}(V_e)$ is the group of endomorphisms of a vector space $V_e$. Then let $S(v)$ be the set of edges with the vertex $v$ as a source (start point) and let $T(v)$ be the set of edges with the vertex $v$ as a target (end point). An intertwining operator $I_v$ is a linear endomorphism between the two vector spaces $\Oplus_{e\in S(v)}V_e$ and $\Oplus_{e\in T(v)}V_e$. The intertwining operator $I_v$ can also be understood as an invariant element of the representation $\Oplus_{e\in S(v)}V_e\otimes \Oplus_{e\in T(v)}V_e^{\star}$. Thus, $\vec{i}$ is a labelling of the edges of the graph $\alpha$ by irreducible representations of $SU(2)$. Similarly, $\vec{i}$ is a labelling of the vertices of the graph $\alpha$ by intertwining operators from the tensor product of incoming representations to the tensor product of the outgoing representations \cite{Baez:1994hx}.
For example, in Penrose spin networks \cite{Penrose} we consider trivalent graphs labelled by spins $j$ satisfying the rule that if the spins of the edges at a given vertex are $j_1$, $j_2$ an d$j_3$ then these three spins must satisfy the rules of conservation of angular momentum, i.e. the Clebsch-Gordon condition $|j_1-j_2|\leq j_3\le j_1+j_2$ must hold. This Clebsch-Gordon condition is a necessary and sufficient condition for the existence of intertwining operators from $j_1\otimes j_2$ to $j_3$.
The set of all spin network states with all possible graphs $\alpha$, all assignements $\vec{j}$ of irreducible representations of $SU(2)$ to the edges, and all assignements $\vec{i}$ of intertwining operators to the vertices form the Hilbert space $\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$.
The spin network states $\Psi_{\alpha,\vec{j},\vec{i}}(A)$ are eigenvectors of area and volume operators in the hypersurfaces $\Sigma_t$ with discrete spectra \cite{Rovelli:1995ac}.
W_{\alpha}[A]=Tr h_{\alpha}[A].\label{WL}
\end{eqnarray}
These are gauge invariant by construction.
If we denote the set of all generalized gauge transformations by $\bar{\cal G}$ then the gauge invariant quantum configuration space must be given by the quotient $\bar{\cal A}/\bar{\cal G}$. This space can be viewed as a projective limit of a family of compact, smooth and finite dimensional configuration spaces $\{\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}\}$.
Each configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ is labelled by a path or graph $\alpha$ characterized by $N$ edges and $V$ vertices (each edge $e$ starts at a vertex $v_{e_0}$ and ends at a vertex $v_{e_1}$). From the one hand, the space $\bar{\cal A}_{\alpha}$ is the space of generalized connections or holonomies over the graph $\alpha$ which consists of the mappings which assign to each edge of the graph an element of the group $SU(2)$, i.e. $\bar{\cal A}_{\alpha}$ is isomorphic to $SU(2)^N$. From the other hand, the space $\bar{\cal G}_{\alpha}$ is the space of generalized gauge transformations over the graph $\alpha$ which consists of all mappings which assign to each vertex an element of $SU(2)$. More precisely, the action of a given generalized gauge transformation $g$ on an edge $e$ of the graph $\alpha$ is given by $h_{\alpha}(e)\longrightarrow h_{\alpha}^{\prime}(e)=g^{-1}(v_{e_1})h_{\alpha}(e)g(v_{e_0})$. Hence the space $\bar{\cal G}_{\alpha}$ is isomorphic to $SU(2)^V$ and as a consequence the configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ is isomorphic to $SU(2)^{N-V}$.
The Hilbert space of state vectors on the configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ (which is compact and finite dimensional) is precisely the space of square-integrable functions ${\cal H}_{\alpha}=L^2(\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha})$ where the measure is obviously induced by the usual the Haar measure on $SU(2)$. Elementary quanta of geometry are elements of ${\cal H}_{\alpha}$ and they are of the form \cite{Huggett:1998sz}
\begin{eqnarray}
\Psi_{\alpha}(h_{\alpha})=\psi(h_{\alpha}(e_1),...,h_{\alpha}(e_N))~,~\psi\in SU(2)^N.
\end{eqnarray}
In summary, the members of the family $\{\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}\}$ are all compact and finite dimensional spaces with corresponding Hilbert spaces ${\cal H}_{\alpha}=L^2(\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha})$ and as a consequence the projective limit $\bar{\cal A}/\bar{\cal G}$ which is also compact admits a regular Borel measure allowing the construction of a corresponding Hilbert space of square-integrable functions ${\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$ where the measure is induced by the usual the Haar measure on $SU(2)$.
The Hilbert space ${\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$ admits a more interesting decomposition as a direct sum of finite dimensional orthogonal Hilbert spaces ${\cal H}_{\alpha,\vec{j}}$ characterized together with the graph $\alpha$ by a vector $\vec{j}$ of half-integers, i.e. $\vec{j}=(j_1,j_2,...,j_N)$ where the integer $j_i$ represents the irreducible representation of $SU(2)$ which labels the edge $i$ of the graph $\alpha$. We have then \cite{Ashtekar:2014kba}
\begin{eqnarray}
{\cal H}=L^2(\bar{\cal A}/\bar{\cal G})=\oplus_{\alpha,\vec{j}} {\cal H}_{\alpha,\vec{j}}.
\end{eqnarray}
${\cal H}_{\alpha,\vec{j}}$ are Hilbert spaces of spin network states on the configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$. A spin network state $\Psi_{\alpha,\vec{j},\vec{i}}(A)$ is then an element of ${\cal H}_{\alpha,\vec{j}}$ which is characterized by $1)$ the graph $\alpha$, $2)$ the $N$ irreducible representations $j_i$ associated with the edges of the graph and $3)$ the $V$ intertwining operators $i_i$ associated with the vertices of the graph.
More precisely, let $\rho_e$ be the irreducible representation associated with the edge $e$, i.e. $\rho_e$ is the homomorphism $\rho_e:SU(2)\longrightarrow {\rm End}(V_e)$ where ${\rm End}(V_e)$ is the group of endomorphisms of a vector space $V_e$. Then let $S(v)$ be the set of edges with the vertex $v$ as a source (start point) and let $T(v)$ be the set of edges with the vertex $v$ as a target (end point). An intertwining operator $I_v$ is a linear endomorphism between the two vector spaces $\Oplus_{e\in S(v)}V_e$ and $\Oplus_{e\in T(v)}V_e$. The intertwining operator $I_v$ can also be understood as an invariant element of the representation $\Oplus_{e\in S(v)}V_e\otimes \Oplus_{e\in T(v)}V_e^{\star}$. Thus, $\vec{i}$ is a labelling of the edges of the graph $\alpha$ by irreducible representations of $SU(2)$. Similarly, $\vec{i}$ is a labelling of the vertices of the graph $\alpha$ by intertwining operators from the tensor product of incoming representations to the tensor product of the outgoing representations \cite{Baez:1994hx}.
For example, in Penrose spin networks \cite{Penrose} we consider trivalent graphs labelled by spins $j$ satisfying the rule that if the spins of the edges at a given vertex are $j_1$, $j_2$ an d$j_3$ then these three spins must satisfy the rules of conservation of angular momentum, i.e. the Clebsch-Gordon condition $|j_1-j_2|\leq j_3\le j_1+j_2$ must hold. This Clebsch-Gordon condition is a necessary and sufficient condition for the existence of intertwining operators from $j_1\otimes j_2$ to $j_3$.
The set of all spin network states with all possible graphs $\alpha$, all assignements $\vec{j}$ of irreducible representations of $SU(2)$ to the edges, and all assignements $\vec{i}$ of intertwining operators to the vertices form the Hilbert space $\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$.
The spin network states $\Psi_{\alpha,\vec{j},\vec{i}}(A)$ are eigenvectors of area and volume operators in the hypersurfaces $\Sigma_t$ with discrete spectra \cite{Rovelli:1995ac}.
For example, the area of a two-dimensional surface $\Sigma$ with induced metric $h_{ab}^{(2)}$ in the three-dimensional hypersurface $\Sigma_t$ with induced metric $h_{ab}$ is given by
\begin{eqnarray}
A_{\Sigma}=\int dx^1dx^2\sqrt{{\rm det} h^{(2)}}.
\end{eqnarray}
The metric $h_{ab}^{(2)}$ is the induced metric on the surface $\Sigma$
which can be expressed in terms of the metric $h_{ab}$, then in terms of the densitized triads $\tilde{e}_m^{\mu}$ using the relation $\tilde{e}_m^{\mu}\tilde{e}^{\nu m}=h^{\mu\nu}{\rm det} h$. In the quantum theory the densitized triads become operators $\hat{\tilde{e}}_m^{\mu}$ which act (up to a numerical factor) as $\delta/\delta A_{\mu}^m$ and hence the area $A_{\Sigma}$ becomes an area operator $\hat{A}_{\Sigma}$ which admits the Wilson loops (\ref{WL}) as eigenvectors, viz \cite{Gambini:2011zz}
\begin{eqnarray}
A_{\Sigma}W_{\alpha}[A]=8\pi l_{\rm P}^2\beta \sum_I\sqrt{j_I(J_I+1)}W_{\alpha}[A].
\end{eqnarray}
The sum is over all edges of the Wilson loop $\alpha$ that intersect the surface $\Sigma$. The surface areas of $\Sigma$ is then given by $8\pi l_P^2\beta$ \sum_I\sqrt{j_I(J_I+1)}$ where $j_I$ is the spin associated with the edge $I$, $\beta$ is the Immirzi parameter and $l_P$ is the Planck length.
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