A Brief Review on Canonical Loop Quantum Gravity: The Kinematical Part

In this article, we briefly review the kinematical part of canonical loop quantum gravity. This article starts with tetradic formulation of gravity both in the covariant approach and canonical approach. The next step is to introduce Ashtekar new variables and to apply the Dirac canonical quantization procedure of gravity. By the holonomy representation, one obtains the loop representation of quantum gravity.


Introduction
The search for quantum gravity as a quantization of General Relativity has been carried for a long time and had not been completed recently. The research has evolved in various ways, ranging from the perturbative to the non-perturbative theory. The perturbative quantization meets several obstacles, one of them being the non-renormalizable property of the gravitational field. In the other hand, the non-perturbative theories provide possible ways to quantize gravity; they could be broadly categorized into two main branches: string and non-string approach. One of the candidates of non-string approach of quantum gravity is loop quantum gravity (LQG), which is based rigorously on Dirac quantization procedure.
In this article, we review the basic of canonical loop quantum gravity. Due to time and length constraint, we discuss only on the kinematical part. Section II consists of tetradic formulation of gravity as an attempt to treat gravity as a gauge theory. It contains three subsections, reviewing on the covariant approach, canonical approach, and Ashtekar new variables. In this section, we briefly review the Hamiltonian formulation of general relativity. The third section consists the Dirac canonical quantization procedure of gravity. It contains two subsections, which are the quantization via connection and holonomy representation. The previous are problematic, while the latter leads to loop quantum gravity. Section 4 consist the main subject of this article, namely a review of the kinematical part of loop quantum gravity. It consist four subsections which respectively review the cylindrical functions and the Hilbert space of quantum gravity, the Gauss constraint and the kinematical Hilbert space, the graph and spin-network states, and the geometric operators on quanta of space.
The review on this article is mainly based on reviews of the kinematical part of quantum gravity in [1,2,3,4], and can be viewed as a summary of these articles H is the Hamiltonian density; to obtain the Hamiltonian of the system, the corresponding constraint are smeared on the spatial hypersurface: G ( ω), C N , and C (N ) are, respectively, the Gauss, diffeomorphism, and Hamiltonian constraints.
For constrained system, it is important to check that the constraints are consistent to each other. This lead Dirac and Bergmann to propose an algorithm to check the consistency of the constraints [7]. Constraints on a Hamiltonian system are classified into the following conditions: (1) primary constraint, if a constraint is independent from the dynamical equation, otherwise it is called secondary; (2) first class, if a constraint commutes with all other existing constraints on the constraint hypersurface, otherwise it is called second class. First class constraints generates gauge transformation [7,8].
The constraints in GR (6)-(8) are primary, but second class. To do a quantization on each foliation of the constraint hypersurface, it is more convenient to have a set of first class constraint, which is done in the next subsection.

Ashtekar New Variables
The procedure proposed by Ashtekar, is an attempt to make the constraints (6)-(8) first class [9]. As another advantage to this, the constraints can be rewritten in nice polynomial forms, which make them easier to quantize.
2.3.1. Holst Term and Holst Action. A Lagrangian of a system is not unique, in the sense that it can be modified as long as the Euler-Lagrange equation is invariant. Using this fact, one could add an additional term labeled as Holst term in the Palatini action [10]: with δ IJKL = δ I[K δ L]J . The factor γ is a constant known as the Barbero-Immirzi parameter, and the action is called as Holst action [10].
Minimizing the variation of δS for any δe and δω gives the following Euler-Lagrange equations: The dynamical equation derived from the Holst action are still equivalent to EFE (2) as long as the tetrads are non-degenerate, e I = 0. In the original version, one starts with a complexification of sl(2, c) by setting γ = i, thus gauge group of general relativity is the complexified Lorentz group SO C (3, 1). With SO C (3, 1), one has two distinct SO(3, 1) elements (which consist the real SO(3, 1) group, its complex conjugate part, and their dual groups). But due to the reality issue, it is more convenient to use real variables, that is, restricting γ to be real.

Ashtekar Variables.
As already been discussed in the previous subsection, the triads 3 ω is an element of su(2) × 1 (Σ). Let us construct another variables which is also an element of su(2), with the following transformation: 3 ω is the Hodge-dual part (in the internal/fibre space) of 3 ω. A is the half of the Ashtekar 'new' variables [9]. Taking the derivative of connection, and using the Holst Lagrangian density (9), the corresponding momentum conjugate is exactly the densitized triads of the previous case: Using the time gauge and applying the Hodge star operator on the 3-connections and triads, one obtains: The symplectic structure is satisfied byẼ b j = E b j γ (or by E b j , but with a factor γ in the RHS of the following Poisson Bracket): with the following constraint density in terms of the new variables: For a special condition γ = i, the constraint greatly simplifies to a compact form as follows: Smearing the constraints as in the previous subsection: one could obtain the following constraints algebra: . The algebra, although they do not construct a true Lie algebra since RHS of (21) is not a structure constant, is closed. Therefore, by restricting on the constraint hypersurface, where G A ≈ 0, C N ≈ 0, and C (N ) ≈ 0, all the primary constraint commutes with themself on the constraint hypersurface. This means the sets of constraint arising from the Ashtekar formulation of GR are primary first class constraint. As a consequence to this, these constraints generate gauge transformation: Gauss constraint generates SU(2) gauge transformation, diffeomorphism constraint generates evolution in space, Hamiltonian constraint generates evolution in time.

Canonical Quantization
Since the set of constraints are primary and first class, one could proceed to the quantization procedure, based rigorously on [7]. The steps are:

Quantization via Connection Representation
One could immediately quantized the phase space, the Poisson bracket (11) becomes: The next step is to promote the fundamental variables to the following operators: δA is the functional derivatives with respect to A i a . The last step is to obtain the representation space. Normally, the representation space is the square-integrable space over the configuration space of the system. In our case, the configuration space is the space of connection A su(2)× 1 (Σ), therefore the representation space should be the functional space Moreover, one could equip the space with a well-defined inner product and completeness requirement; in this case, the functional space is a Hilbert space.
At this point, several problems arise: the properties of the representation space are not well-defined: has infinite (uncountable) dimensions, the measure and the inner product in this space is unknown. Let us neglect this problem for a while. Nevertheless, there exist a remaining problem when one tries to promote contraints (18)-(20) to an operator: the ordering of triadsẼ and curvature F (A) matters. Let us check what can be obtained from each ordering.
3.1.1. Triads on the right: Wilson Loops. In this ordering, the constraint becomes the following operator:Ĝ The Gauss constraint becomes an infinitesimal generator of gauge transformation for SU (2) and the diffeomorphism constraint becomes infinitesimal generator of 3D diffeomorphism. The wavefunctional that satisfies the Gauss constraint needs to be invariant under SU(2) transformation, thus the candidate for the solution is the Wilson loop of an SU(2) holonomy: In fact, the Wilson loop is also a solution to the Hamiltonian constraint. On the other hand, (1) is not a solution to Hamiltonian constraint if the loops contains kinks or intersections, (3) given a metric operator, the wavefunction represents a space with degenerate metric [4]. Unfortunately, research in this direction might be not really useful. Nevertheless, it is historically important since this gives motivational research to the loop direction.
For a reason which will be clear later, let us modify the Hamiltonian constraint by the existence of the cosmological constant: Chern-Simon Theory. At this point, let us review a different subject: the Yang-Mills field. The action of a Yang-Mills fields is given as follows: For a case without source, as an example, in the electromagnetism case, the electromagnetic field satisfy self-duality F = F , such that the action (28) becomes: It is clear that the action is independent from the use of metric, which is included in the definition of the Hodge-dual. This is favourable in the background independence perspective. In fact, one could generalize the action for arbitrary even 2n-dimension of M, n integers: The term F n is known as the n th Chern form, and (30) is the Chern-Simon action. One can prove that d D F n = 0. Since F n is closed, it is reasonable to ask if one could obtain a solution to d D F n = 0, namely a (2n − 1)-form potential φ such that d D φ = F n . This is equivalent with requiring the n th Chern form F n to be exact. An equivalence class of exact F n is known as the n th Chern class, and the corresponding potential φ as the Chern-Simon form. This solution will be important if one is interested to consider the bulk/boundary correspondence of the field. For n = 4 case, one could prove that φ = A ∧ dA + 2 3 A ∧ A ∧ A, a Chern-Simon 3-form. Therefore, using Stokes theorem, one could write (29) as: with ∂M is the boundary of M. Now, let us construct a Chern-Simon state as follows: It is widely known as Kodama state [11]. Remarkably, it is the solution to these three constraints, but with an existence of a positive cosmological constant. The Kodama state is interpreted as the ground state wavefunction for a deSitter space. Nevertheless, it is not a state for a gravitational field, one could attempt to find a solution based on the action of GR, and the research in this direction is still on progress.

Quantization via Holonomy Representation
It had been explained in the previous chapters that there exist a wavefunction of GR in the form of Kodama state. Nevertheless, the Hilbert space of this state, had not been obtained in a rigorous manner. One could argue that the Ashtekar new variables are not the best phase-space variables to quantize GR; one needs to search for another phase-space variables. Implementing the main idea of the construction of Lattice Gauge Theories, it turns out that the good variables compatible with the quantization procedure are the regularized version of the Ashtekar variables.
3.2.1. Regularization. The 3-connections A, which is an su(2)-valued 1-form, is smeared along a curve (line, or link) l, and is known as holonomy: This is also a solution to the following parallel transport equation along curve γ, with γ (τ ) = Meanwhile the 'electric' triadsẼ = E γ , which is a matrix-valued 2-form, is smeared on a portion of surface area S, known as the flux : The index l in the flux indicates that the (infinitesimal) surface S in 3D hypersurface Σ always posses a link l which is defined as its (lattice) dual, thus it is convenient to label the flux crossing the surface with the index of the link. The smeared phase-space variable associated on link l is now (U l , E l ) , where the U l is smeared along the link, andẼ l is smeared along the surface (lattice)-dual to the link. They satisfy the symplectic structure through the Poisson bracket: this algebra is known as the holonomy-flux algebra, labeled by J Γ , with Γ is a collection of links and node known as a graph.σ a is the Pauli matrices. There are some subleties concerning the orientation of the fluxẼ with respect to the surface S. In this article, we always take the flux to cross surface S positively. For a more subtle derivation consult [3]. Clearly, U l is an element of SU (2), since it is an 'exponential map' of 3 A ∈ su(2), whileẼ a l , which are components ofẼ l , from (35), is an element of su(2), since they satisfy the su(2) algebra. Therefore, these variables are conjugate to each other, forming an elements of 'regularized' phase space (U l , J a l ) ∈ T * SU (2) SU (2) × su(2) * (since su(2) ∼ su(2) * ). The smearing used for this regularization is different from the standard smearing use in QFT, namely, the smearing over a portion of a 3D volume. This happens because the smearing is defined with respect to a specific background metric. For general relativity, one use different type of smearing such that it does not assume a fix background metric, since the theory is expected to be background independent.

Quantization.
Repeating the quantization on the regularized variables, the holonomyflux algebra naturally satisfies the following algebra relation: is a constant equal to G c 3 , known as Planck length. Notice that we use E instead ofẼ, which give rise to the Immirzi factor γ on the RHS of (34)-(35). Promoting the phase-space elements as operators, one obtains: with J l is a left invariant vector field on SU(2) generated by Pauli matricesσ a if the orientation of the link points outward from the node and right invariant if it points inward: The flux is defined as an invariant vector field acting on SU (2) with a reason which will be clear later. The phase-space variables are automatically the basic operators of the canonical theory; any operatorÔ, including the constraints, can be written in the form of these basic operators. The last step of quantization is constructing the Hilbert space. Since the regularized phase space variables is now (U, J), the regularized configuration space is SU (2) U l . This made us possible to define the flux operator in (38) as an invariant vector field working on SU(2). Therefore, the representation space of quantum gravity could be build over SU (2) , instead of A. To be precise, the representation space is exactly the space of square-integrable function of SU (2): C ∞ [SU (2)] L 2 [SU (2)]. This allows us to define the Hilbert space, which will be explained in detail in the next section.

Kinematical Part of Loop Quantum Gravity
4.1. Cylindrical Functions 4.1.1. Hilbert space of quantum gravity. As explained in the previous section, the representation space over the connection C ∞ [A] is problematic, i.e., one can not define an inner product, measure, and completeness, which are possesed by a Hilbert space. Therefore, with the connection representation, one cannot obtain the Hilbert space of the theory. Nevertheless, there is a possible way to obtain the Hilbert space using the holonomy representation.
Let a functional over connection A be written as:

Now let us consider a subset of C ∞ [A], say Cyl[A]
, such that each element is a function of the holonomy U = U [γ, A] , namely, the dependence on A enters through U . Such functional which are build from some subset of the field are generally known as cylindrical functions. We label this kind of function as:  10 Γ is a graph constructed from intersecting loops, where each link l contains the information of the holonomy U l . Notice that since A ∼ su(2) × 1 (Σ) , then the space of holonomies, new configuration space, is clearly SU (2) which is compact. Because of this, one has completeness, and could define the following inner product on Cyl[A] C ∞ [SU (2)] : with dU l is the Haar measure on SU (2). Moreover, Ashtekar and Lewandowski shows that Cyl[A] could be extended to define a Hilbert space over a connection, H AL L 2 [A, dµ AL ], using the Ashtekar-Lewandowski measure dµ AL [12,13]. Therefore, one obtains a candidate of Hilbert space for quantum gravity. Firstly, let us construct the Hilbert space for a single link of graph Γ. A link l = (n, n ) connecting node n to n of the graph is equipped by an element of phase space on the boundary, (U l , J l ) ∈ T * SU (2), together with the label of spin-j, the irreducible representation of SU (2) in (2j + 1)-dimension. The representation space for each link is build over SU (2), and from the theorem one has: Thus, the basis in H l L 2 [SU (2)] is: |j l , m l , n l = |j l , m l j l , n l | ∈ H l j l with each spin basis is related to one end of the link, see Figure. 1. For a graph containing n-links, the Hilbert space is the product of (39), namely: and the basis, which could be represented with n non-connected strains, is:

General states.
Let us write the vector state |ψ Γ,ϕ ∈ H AL of an arbitrary graph Γ having |N | nodes and |L| links, in the representation of holonomy as follows: |ϕ which is a function over SU (2). Using the completeness and orthogonality of (40), one could write ϕ (U l 1 [A] , .., U l L [A]) as a linear combination of irreducible representation of SU(2), as follows: Having the well-defined Ashtekar-Lewandowski Hilbert space and its complete basis, one could start to promote the constraints as operators and find the wavefunctional solving the constraints.
In this review, we only consider the Gauss constraint. The Gauss constraint (12) is a divergence theorem, it can be interpreted as the 'incompressibility' of the flux E a i . We can write the Gauss constraint without indices as follows: with d Σ is the exterior-covariant derivative on the spatial slice Σ. The quantity (43) is a 3-form, therefore, to define the regularization, one needs to smeared the quantity over a single finite region with volume R i , and by using Stokes theorem, one obtains: with ∂R is the closed surface enclosing the finite volume R i . Using the smeared variable of the infinitesimal momentum over a finite surface, one obtains: Defining a regulator or a cut-off which prevent a l → 0, the regularized Gauss constraint is obtained as follows: By promoting the Gauss constraint to an operator using (37), one has: The eigenvalue equation of (45) is translated as: 1..n |ϕ = 0, requiring the total spin and magnetic quantum number j 1...n = 0 and m 1..n = 0 (in fact, m 1..n is automatically zero if the first relation is satisfied, since −j ≤ m ≤ j). The state |ϕ ∈ H Γ satisfying (46) automatically satisfies the quantum Gauss constraint. The kinematical Hilbert space: is an invariant subspace of H which satisfy the quantum Gauss constraint on each node, and the basis which spans K is called as the spin network basis, which will be discussed in detail in the next subsections.

4.2.2.
Gauss constraint in holonomy basis: group averaging procedure. The construction of quantum Gauss operator in the previous subsection provides a clear geometrical interpretation. Nevertheless, a formal derivation on states satisfying the Gauss constraint use the procedure borrowed from Lattice Gauge Theory, namely, the group averaging procedure. From (15), it is known that the Gauss constraint is the (infinitesimal) generator of SU(2) gauge transformation. The finite gauge transformation acts on the holonomy in the following way: where source s is the origin of curve γ (or link l), and target t is the endpoint of curve γ (s and t are clearly nodes of the graph Γ). In the matrix representation, (47) acts as: .

Any wavefunction ϕ (U [A]
) ∈ H can be written as a linear combination of products of irreducible matrix representation of SU(2) as in (42). One would like to obtain a set of wavefunction ϕ inv (U [A]) ∈ K ⊂ H, such that the following condition holds:

Lattice-graph and Spin-network states
Constructing spin network states. The graphical interpretation of spin network is more clearer in the basis of matrix representation |j l , m l , n l instead of the holonomy basis |U l . The steps for constructing a spin network states can be written in the following, based on [14]: (i) Having an arbitrary connected graph Γ, with |N | nodes and |L| links, construct the Ashtekar-Lewandowski Hilbert space as the product of the Hilbert space of the link by relation (41). The basis functional on this space is ⊗ |j l , m l , n l , by Peter-Weyl theorem. We called this the ⊗-product basis. (ii) On each node, collect half of the |j l , m l , n l = |j l , m l j l , n l | basis, that is, either |j l , m l or j l , n l | depending on the graph, and transform them to the ⊕-sum basis, that is, sum all the half-spins on the node. The Hilbert space satisfies ⊗ j H j = ⊕ jmax j min H. The transformation matrix components are the Clebsh-Gordan coefficients, or more general, the recoupling coefficient i CG . This ⊕-sum basis equally spans the Hilbert space (41). (iii) Implement the Gauss constraint on each node, that is, by requiring the total spins on each node to satisfy (46). This procedure selects only the singlet spaces H j min = H 0 (if they exist). This is equal with giving a condition to the recoupling coefficient such that they are equal with the intertwiner, i.e., i CG = i. The singlet ⊕-sum basis is the spin network basis. (iv) Contract the spin network basis with the dual-basis in holonomy representation U l | to obtain the rotation matrices as in (42). One obtains the basis of spin network in terms of (50). Any general spin network states are a linear combination of the spin network basis, as in (51).
Example: 4-valent graph. Taking the invariant subspace of the spin network is implementing the quantum Gauss constraint on each node. In the ⊕-basis, implementing this constraint will set the quantum number j 1234 = 0, which is the Clebsch-Gordan condition. This will cause m 1234 = 0, and j 123 = j 4 for the 4-valent graph case: π n j 1234 , m 1234 , j 123 , j 12 ,

The geometric operators and quanta of space
The spin network state describes the state of the graph, which is embedded on the spatial hypersurface Σ. It contains the informations of the hypersurface. Since the graph has finite nodes and links, it describes a discretized space, where the nodes correspond to a polyhedra, and the links correspond to flat surfaces enclosing each polyhedron, see Figure 2. Therefore, one could construct the geometric operators such as area and volume operators. A (S) = ∆S ds 1 ds 2 E a i E bi n a n b .

A regularization gives:
A N (S) = which are discrete.

Volume operator.
The classical volume of region R on Σ is defined as: with q is the determinant of metric q ij of Σ. Using the relation between the 3-metric and the triads, one obtains: q = E. Therefore: Promoting the classical volume to operator gives: By a specific regularization defined by [13], the Ashtekar-Lewandowski volume operator is: The spectrum of the volume operator is quite complicated to obtain since they are in general, not analytical. For a complete derivation, consult [15]. Nevertheless one can conclude that the volume are discrete. The spin-network graph describe quantum polyhedron, which could be viewed as the quanta of space, having discrete surface areas and volumes and fuzzy shapes.

Conclusions
In this article, we have given a brief review on the kinematical part of canonical loop quantum gravity. The kinematical part is complete and well-understood. One of the consequences of LQG coming from the kinematical part is the discreteness of space in the Planck scale, namely the existence of quanta of space. The quanta, having discrete surface areas and volumes, as well as fuzzy shapes, are described by spin-networks: a lattice-graph labeled by spin representations of SU (2). The graph may contain loops, where the SU(2) holonomies describing the intrinsic curvature of the discrete space are located.
The second part of this article will briefly review the dynamical part of canonical loop quantum gravity.