This is the fourth post of a series of four posts (plus a fifth post of exercises) concerned with the canonical quantization of general relativity.
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. Loop representation and spin network stats are then briefly discussed.
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.
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.
\begin{eqnarray}
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
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}.
Abstract
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. Loop representation and spin network stats are then briefly discussed.
Previous Posts
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
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} will constitue of holonomies of connections along paths in \Sigma_t.
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}
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).
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}
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 {\cal S} in the three-dimensional hypersurface \Sigma_t is given by
\begin{eqnarray} A_{\cal S}=\int dx^1dx^2\sqrt{{\rm det} h^{(2)}}. \end{eqnarray}
The metric h_{ab}^{(2)} is the induced metric on the surface {\cal S}. This 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_{\cal S} becomes an area operator \hat{A}_{\cal S} which admits the Wilson loops (\ref{WL}) as eigenvectors, viz \cite{Gambini:2011zz}
\begin{eqnarray} A_{\cal S}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 {\cal S}. The surface areas of {\cal S} are then quantized 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. Any Gauss gauge invariant function \Psi[A] in the connection representation is then expanded in terms of Wilson loops as
\begin{eqnarray} \Psi[A]=\sum_{\alpha}\Psi[\alpha]W_{\alpha}[A]. \end{eqnarray}
This is called the loop transform. The function \Psi[\alpha] defines the loop representation and it is given by the inverse loop transform
\begin{eqnarray} \Psi[\alpha]=\int [dA] \Psi[A] W_{\alpha}[A]. \end{eqnarray}
In the loop representation we can also solve the spatial diffeomorphism constraint by considering functions \Psi[\alpha] which are invariant under diffeomorphisms of the loop \alpha. Thes are knot invariants \cite{Rovelli:1987df}. The Hamiltonian constraint is solved on the other hand by non-intersecting Wilson loops.
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 also 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 \bigotimes_{e\in S(v)}V_e and \bigotimes_{e\in T(v)}V_e.
The intertwining operator I_v can also be understood as an invariant element of the representation \bigotimes_{e\in S(v)}V_e\bigotimes \bigotimes_{e\in T(v)}V_e^{\star}.
Thus, if \vec{j} 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 and 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 a 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}.
These are gauge invariant by construction. Any Gauss gauge invariant function \Psi[A] in the connection representation is then expanded in terms of Wilson loops as
\begin{eqnarray} \Psi[A]=\sum_{\alpha}\Psi[\alpha]W_{\alpha}[A]. \end{eqnarray}
This is called the loop transform. The function \Psi[\alpha] defines the loop representation and it is given by the inverse loop transform
\begin{eqnarray} \Psi[\alpha]=\int [dA] \Psi[A] W_{\alpha}[A]. \end{eqnarray}
In the loop representation we can also solve the spatial diffeomorphism constraint by considering functions \Psi[\alpha] which are invariant under diffeomorphisms of the loop \alpha. Thes are knot invariants \cite{Rovelli:1987df}. The Hamiltonian constraint is solved on the other hand by non-intersecting Wilson loops.
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 also 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 \bigotimes_{e\in S(v)}V_e and \bigotimes_{e\in T(v)}V_e.
The intertwining operator I_v can also be understood as an invariant element of the representation \bigotimes_{e\in S(v)}V_e\bigotimes \bigotimes_{e\in T(v)}V_e^{\star}.
Thus, if \vec{j} 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 and 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 a 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 {\cal S} in the three-dimensional hypersurface \Sigma_t is given by
\begin{eqnarray} A_{\cal S}=\int dx^1dx^2\sqrt{{\rm det} h^{(2)}}. \end{eqnarray}
The metric h_{ab}^{(2)} is the induced metric on the surface {\cal S}. This 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_{\cal S} becomes an area operator \hat{A}_{\cal S} which admits the Wilson loops (\ref{WL}) as eigenvectors, viz \cite{Gambini:2011zz}
\begin{eqnarray} A_{\cal S}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 {\cal S}. The surface areas of {\cal S} are then quantized 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.
References
%\cite{ADM}
\bibitem{ADM}
S.~ Deser,
``Arnowitt-Deser-Misner formalism,'' Scholarpedia, 3(10):7533 (2008).
%\cite{Ashtekar}
\bibitem{Ashtekar}
A.~ Ashtekar,
``Ashtekar variables,'' Scholarpedia, 10(6):32900 (2015).
%\cite{Thiemann:2006cf}
\bibitem{Thiemann:2006cf}
T.~Thiemann,
``Loop Quantum Gravity: An Inside View,''
Lect.\ Notes Phys.\ {\bf 721}, 185 (2007).
[hep-th/0608210].
%\cite{Ashtekar:1988sw}
\bibitem{Ashtekar:1988sw}
A.~Ashtekar, A.~P.~Balachandran and S.~Jo,
``The {CP} Problem in Quantum Gravity,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 4}, 1493 (1989).
doi:10.1142/S0217751X89000649
%%CITATION = doi:10.1142/S0217751X89000649;%%
%115 citations counted in INSPIRE as of 25 May 2020
%\cite{Thiemann:2007zz}
\bibitem{Thiemann:2007zz}
T.~Thiemann,
%``Modern canonical quantum general relativity,''
[arXiv:gr-qc/0110034 [gr-qc]].
%642 citations counted in INSPIRE as of 05 Jun 2020
%\cite{Ashtekar:2014kba}
\bibitem{Ashtekar:2014kba}
A.~Ashtekar, M.~Reuter and C.~Rovelli,
``From General Relativity to Quantum Gravity,''
[arXiv:1408.4336 [gr-qc]].
%54 citations counted in INSPIRE as of 07 Jun 2020
%\cite{Penrose}
\bibitem{Penrose}
R.~Penrose,
``Angular momentum; an approach to combinatorial spacetime,''
In Quantum Theory and Beyond, ed. T. Bastin, Cambridge University Press, 1971.
%\cite{Rovelli:1995ac}
\bibitem{Rovelli:1995ac}
C.~Rovelli and L.~Smolin,
``Spin networks and quantum gravity,''
Phys. Rev. D \textbf{52}, 5743-5759 (1995)
doi:10.1103/PhysRevD.52.5743
[arXiv:gr-qc/9505006 [gr-qc]].
%442 citations counted in INSPIRE as of 07 Jun 2020
%\cite{Rovelli:1987df}
\bibitem{Rovelli:1987df}
C.~Rovelli and L.~Smolin,
``Knot Theory and Quantum Gravity,''
Phys. Rev. Lett. \textbf{61}, 1155 (1988)
doi:10.1103/PhysRevLett.61.1155
%376 citations counted in INSPIRE as of 08 Jun 2020
%\cite{Baez:1994hx}
\bibitem{Baez:1994hx}
J.~C.~Baez,
``Spin network states in gauge theory,''
Adv. Math. \textbf{117}, 253-272 (1996)
doi:10.1006/aima.1996.0012
[arXiv:gr-qc/9411007 [gr-qc]].
%185 citations counted in INSPIRE as of 07 Jun 2020
%\cite{Pullin:1993fw}
\bibitem{Pullin:1993fw}
J.~Pullin,
``Knot theory and quantum gravity in loop Space: A Primer,''
AIP Conf. Proc. \textbf{317}, 141-190 (1994)
[arXiv:hep-th/9301028 [hep-th]]. For a concise summary see also the wikipedia page
https://en.wikipedia.org/wiki/Self-dual_Palatini_action#cite_note-8
%\cite{Barbero:1994ap}
\bibitem{Barbero:1994ap}
J.~Barbero G.,
``Real Ashtekar variables for Lorentzian signature space times,''
Phys. Rev. D \textbf{51}, 5507-5510 (1995)
doi:10.1103/PhysRevD.51.5507
[arXiv:gr-qc/9410014 [gr-qc]].
%624 citations counted in INSPIRE as of 05 Jun 2020
%\cite{Immirzi:1996di}
\bibitem{Immirzi:1996di}
G.~Immirzi,
``Real and complex connections for canonical gravity,''
Class. Quant. Grav. \textbf{14}, L177-L181 (1997)
doi:10.1088/0264-9381/14/10/002
[arXiv:gr-qc/9612030 [gr-qc]].
%322 citations counted in INSPIRE as of 05 Jun 2020
%\cite{Loll:1993yz}
\bibitem{Loll:1993yz}
R.~Loll,
``Gauge theory and gravity in the loop formulation,''
Lect. Notes Phys. \textbf{434}, 254-288 (1994)
doi:10.1007/3-540-58339-4_21
%1 citations counted in INSPIRE as of 06 Jun 2020
%\cite{Gambini:2011zz}
\bibitem{Gambini:2011zz}
R.~Gambini and J.~Pullin,
``A first course in loop quantum gravity,''
%5 citations counted in INSPIRE as of 07 Jun 2020
%\cite{Huggett:1998sz}
\bibitem{Huggett:1998sz}
A.~Ashtekar,
``Geometric issues in quantum gravity,''
In S.~Huggett, L.~Mason, K.~Tod, S.~Tsou and N.~Woodhouse,
``The geometric universe: Science, geometry, and the work of Roger Penrose,"
Proceedings, Symposium, Geometric Issues in the Foundations of Science, Oxford, UK,
June 25-29, 1996.
%\cite{McCabe}
\bibitem{McCabe}
G.~McCabe,
``Loop quantum gravity and discrete space-time,`` philsci-archive.pitt.edu.
%\cite{Jora:2019sla}
\bibitem{Jora:2019sla}
R.~Jora,
``Note about the spin connection in general relativity,''
arXiv:1911.05283 [gr-qc].
%\cite{Yoon:2018jnq}
\bibitem{Yoon:2018jnq}
S.~Yoon,
``Lagrangian formulation of the Palatini action,''
arXiv:1805.01996 [gr-qc].
%\cite{tong}
\bibitem{tong}
F.~Tong,
''A Hamiltonian formulation of general relativity,'' www.math.toronto.edu.
%\cite{vielbein}
\bibitem{vielbein}
``The vielbein formalism for spinors in general relativity,''http://www.dkpi.at/.
%\cite{spinconnection}
\bibitem{spinconnection}
``The spin connection,'' empg.maths.ed.ac.uk.
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